,transcrib]s note3 ,_m ( ! puzzles 9 ?is book assume a famili>ity ) ! curr5cy ( ,great ,brita9 9 ! e>ly #aijjs4 ,z ?is is likely n common k = ^? outside ,brita9 7& possibly _m )917 ,i am includ+ a *>t ( relative values4 ,! mo/ common units us$ 73 ! ,p5ny1 abbreviat$3 d4 7f ! ,roman p5ny1 d5>ius7 ! ,%ill+1 abbreviat$3 s4 ! ,p\nd1 abbreviat$3 ,!re 0 #ab ,p5nies to a ,%ill+ & #bj ,%ill+s to a ,p\nd1 s !re 0 #bdj ,p5nies 9 a ,p\nd4 ,to fur!r complicate ?+s1 !re 7 _m co9s :i* 7 v>i\s fractional values ( ,p5nies1 ,%ill+s or ,p\nds4 ,f>?+ d4 ,half-p5ny d4 ,p5ny #a;d4 ,?ree-p5ny #c;d4 ,sixp5ce 7or tann]7 #f;d4 ,%ill+ 7or bob7 #as4 ,flor9 or two %ill+ piece #bs4 ,half-cr[n 7or half-doll>7 #bs4 #f;d4 ,d\ble-flor9 #ds4 ,cr[n 7or doll>7 #es4 ,half-,sov]eign #ajs4 ,sov]eign 7or ,p\nd7 #a or #bjs4 ,?is is by no means a compreh5sive li/1 b x %\ld be adequate to solve ! puzzles 9 ?is book4 ,expon5ts are repres5t$ 9 ?is text by 1 e4g4 '#c squ>$' is #cb4 ,numb]s ) fractional compon5ts 7o!r ?an 1 & 7 h a symbol sep>at+ ! :ole numb] compon5t f ! fraction4 ,x makes ! fraction look odd1 b yeilds correct solutions no matt] h[ x is int]pret$4 ,e4,g41 #d & elev5 tw5ty-?irds is #daa_/bc1 n #daa_/bc or #d-aa_/bc4 ,,amusem5ts ,,9 ,,ma!matics by ,,h5ry ,,erne/ ,,dud5ey ,9 ,ma!maticks he 0 great] ,?an ,ty*o ,brahe or ,erra ,pat]3 ,= he1 by geometrick scale1 ,c\ld take ! size ( pots ( ale2 ,resolve1 by s9es & tang5ts1 /raight1 ,if bread or butt] want$ weight2 ,& wisely tell :at h\r o' ?' day ,! clock does /rike by algebra4 ,,butl]',s ,hudibras4 #aiag ,,preface ,9 issu+ ?is volume ( my ,ma!matical ,puzzles1 ( :i* "s h appe>$ 9 p]iodicals & o!rs are giv5 "h = ! fir/ "t1 ,i mu/ ackn[l$ge ! enc\ragem5t t ,i h receiv$ f _m unkn[n correspond5ts1 at home & abroad1 :o h express$ a desire to h ! problems 9 a collect$ =m1 ) "s ( ! solutions giv5 at great] l5g? ?an is possible 9 magaz9es & newspap]s4 ,?\gh ,i h includ$ a few old puzzles t h int]e/$ ! _w = g5]ations1 :]e ,i felt t !re 0 some?+ new to be said ab\t !m1 ! problems are 9 ! ma9 orig9al4 ,x is true t "s ( ^! h become widely kn[n ?r\gh ! press1 & x is possible t ! read] may be glad to "k _! s\rce4 ,on ! "q ( ,ma!matical ,puzzles 9 g5]al !re is1 p]haps1 little m to be said ?an ,i h writt5 else:]e4 ,! hi/ory ( ! subject entails no?+ %ort ( ! actual /ory ( ! beg9n+s & developm5t ( exact ?9k+ 9 man4 ,! hi/orian mu/ />t f ! "t :5 man fir/ succe$$ 9 c\nt+ 8 t5 f+]s & 9 divid+ an apple into two approximately equal p>ts4 ,e puzzle t is wor?y ( consid]ation c be ref]r$ to ma!matics & logic4 ,e man1 woman1 & *ild :o tries to reason out ! answ] to ! simple/ puzzle is work+1 ?\gh n ( necessity consci\sly1 on ma!matical l9es4 ,ev5 ^? puzzles t we h no way ( attack+ except by haphaz>d attempts c be br\ght "u a me?od ( :at has be5 call$ glorifi$ trial--a sy/em ( %ort5+ our lab\rs by avoid+ or elim9at+ :at our reason tells u is useless4 ,x is1 9 fact1 n easy to say sometimes :]e ! empirical beg9s & :]e x ends4 ,:5 a man says1 ,i h nev] solv$ a puzzle 9 my life1 x is difficult to "k exactly :at he means1 = e intellig5t individual is do+ x e day4 ,! un=tunate inmates ( our lunatic asylums are s5t !re expressly because !y _c solve puzzles--because !y h lo/ _! p[]s ( reason4 ,if !re 7 no puzzles to solve1 !re w\ld be no que/ions to ask2 & if !re 7 no que/ions to be ask$1 :at a _w x w\ld be6 ,we %\ld all be equally omnisci5t1 & conv]sation w\ld be useless & idle4 ,x is possible t "s few exce$+ly sob]-m9d$ ma!maticians1 :o are impati5t ( any t]m9ology 9 _! fav\rite sci5ce b ! academic1 & :o object to ! elusive x & y appe>+ "u any o!r names1 w h wi%$ t v>i\s problems _h be5 pres5t$ 9 a less popul> dress & introduc$ ) a less flippant phraseology4 ,i c only ref] !m to ! fir/ ^w ( my title & rem9d !m t we are prim>ily out to be amus$--n1 x is true1 )\t "s hope ( pick+ up morsels ( k by ! way4 ,if ! mann] is light1 ,i c only say1 9 ! words ( ,t\*/one1 t x is an ill-fav\r$ ?+1 sir1 b my own2 a poor hum\r ( m9e1 sir4 ,z = ! "q ( difficulty1 "s ( ! puzzles1 especially 9 ! ,ari?metical & ,algebraical category1 are q easy4 ,yet "s ( ^? examples t look ! simple/ %\ld n be pass$ ov] )\t a little consid]ation1 = n[ & aga9 x w be f\nd t !re is "s m or less subtle pitfall or trap into :i* ! read] may be apt to fall4 ,x is good ex]cise to cultivate ! habit ( be+ v w>y ov] ! exact word+ ( a puzzle4 ,x tea*es exactitude & caution4 ,b "s ( ! problems are v h>d nuts inde$1 & n unwor?y ( ! att5tion ( ! advanc$ ma!matician4 ,read]s w d\btless select accord+ to _! individual ta/es4 ,9 _m cases only ! m]e answ]s are giv54 ,?is leaves ! beg9n] some?+ to d on 8 own behalf 9 work+ out ! me?od ( solution1 & saves space t w\ld be wa/$ f ! po9t ( view ( ! advanc$ /ud5t4 ,on ! o!r h&1 9 p>ticul> cases :]e x seem$ likely to int]e/1 ,i h giv5 r ext5sive solutions & treat$ problems 9 a g5]al mann]4 ,x w (t5 be f\nd t ! notes on "o problem w s]ve to elucidate a good _m o!rs 9 ! book2 s t ! read]'s difficulties w sometimes be f\nd cle>$ up z he advances4 ,:]e x is possible to say a ?+ 9 a mann] t may be und]/&$ ( ! p g5]ally1 ,i pref] to use ?is simple phraseology1 & s engage ! att5tion & int]e/ ( a l>g] public4 ,! ma!matician w 9 su* cases h no difficulty 9 express+ ! matt] "u consid]ation 9 t]ms ( 8 famili> symbols4 ,i h tak5 ! greate/ c>e 9 read+ ! pro(s1 & tru/ t any errors t may h crept 9 are v few4 ,if any su* %\ld occur1 ,i c only plead1 9 ! words ( ,horace1 t good ,hom] sometimes nods1 or1 z ! bi%op put x1 ,n ev5 ! y\nge/ curate 9 my diocese is infallible4 ,i h to express my ?anks 9 p>ticul> to ! proprietors ( ,! ,/r& ,magaz9e1 ,cassell's ,magaz9e1 ,! ,que51 ,tit-,bits1 & ,! ,weekly ,dispat* = _! c\rtesy 9 all[+ me to repr9t "s ( ! puzzles t h appe>$ 9 _! pages4 ,,! ,,au?ors' ,,club ,m>* #be1 #aiag ,,cont5ts ,,preface v ,,ari?metical ,,& ,,algebraical ,,problems #a ,money ,puzzles #a ,age & ,k9%ip ,puzzles #f ,clock ,puzzles #i ,locomotion & ,spe$ ,puzzles #aa ,digital ,puzzles #ac ,v>i\s ,ari?metical & ,algebraical ,problems #ag ,,geometrical ,,problems #bg ,dissection ,puzzles #bg ,greek ,cross ,puzzles #bh ,v>i\s ,dissection ,puzzles #ce ,pat*work ,puzzles #df ,v>i\s ,geometrical ,puzzles #di ,,po9ts ,,& ,,l9es ,,problems #ef ,,mov+ ,,c\nt] ,,problems #eh ,,unicursal ,,& ,,r\te ,,problems #fh ,,comb9ation ,,& ,,gr\p ,,problems #gf ,,*essbo>d ,,problems #he ,! ,*essbo>d #he ,/atical ,*ess ,puzzles #hh ,! ,gu>d$ ,*essbo>d #ie ,dynamical ,*ess ,puzzles #if ,v>i\s ,*ess ,puzzles #aje ,,measur+1 ,,weigh+1 ,,& ,,pack+ ,,puzzles #aji ,,cross+ ,,riv] ,,problems #aab ,,problems ,,conc]n+ ,,games #aad ,,puzzle ,,games #aag ,,magic ,,squ>e ,,problems #aai ,subtract+1 ,multiply+1 & ,divid+ ,magics #abd ,magic ,squ>es ( ,primes #abe ,,mazes ,,& ,,h[ ,,to ,,?read ,,!m #abg ,,! ,,p>adox ,,p>ty #acg ,,unclassifi$ ,,problems #adb ,,solutions #adh ,,index #bec ,,amusem5ts ,,9 ,,ma!matics4 ,,ari?metical ,,& ,,algebraical ,,problems4 ,& :at 0 he8 ,=soo?1 a great ari?metician4 ,o!llo1 ,i4 i4 ,! puzzles 9 ?is dep>tm5t are r\ghly ?r[n toge!r 9 classes = ! conv5i5ce ( ! read]4 ,"s are v easy1 o!rs q difficult4 ,b !y are n arrang$ 9 any ord] ( difficulty--& ?is is int5tional1 = x is well t ! solv] %\ld n be w>n$ t a puzzle is j :at x seems to be4 ,x may1 !re=e1 prove to be q z simple z x looks1 or x may conta9 "s pitfall into :i*1 ?r\gh want ( c>e or ov]-confid5ce1 we may /umble4 ,also1 ! ari?metical & algebraical puzzles are n sep>at$ 9 ! mann] adopt$ by "s au?ors1 :o arbitr>ily require c]ta9 problems to be solv$ by "o me?od or ! o!r4 ,! read] is left to make 8 own *oice & det]m9e :i* puzzles are capable ( be+ solv$ by him on purely ari?metical l9es4 ,,money ,,puzzles4 ,put n y\r tru/ 9 money1 b put y\r money 9 tru/4 ,,oliv] ,,w5dell ,,holmes4 #a4--;,a ,,po/-,,(fice ,,p]plexity4 ,9 e bus9ess ( life we are occasionally p]plex$ by "s *ance "q t = ! mom5t /agg]s u4 ,i q piti$ a "y lady 9 a bran* po/-(fice :5 a g5tleman ent]$ & deposit$ a cr[n on ! c\nt] ) ?is reque/3 ,please give me "s twop5ny /amps1 six times z _m p5ny /amps1 & make up ! re/ ( ! money 9 twop5ce-halfp5ny /amps4 ,= a mom5t %e seem$ bewild]$1 !n h] bra9 cle>$1 & ) a smile %e h&$ ov] /amps 9 exact fulfilm5t ( ! ord]4 ,h[ long w\ld x h tak5 y to ?9k x out8 #b4--,,y\?ful ,,precocity4 ,! precocity ( "s y\?s is surpris+4 ,"o is dispos$ to say on occasion1 ,t boy ( y\rs is a g5ius1 & he is c]ta9 to d great ?+s :5 he gr[s up2 b pa/ exp]i5ce has taught u t he inv>iably becomes q an ord9>y citiz54 ,x is s (t5 ! case1 on ! contr>y1 t ! dull boy becomes a great man4 ,y nev] c tell4 ,nature loves to pres5t to u ^! que] p>adoxes4 ,x is well kn[n t ^? wond]ful lightn+ calculators1 :o n[ & aga9 surprise ! _w by _! feats1 lose all _! my/]i\s p[]s directly !y are taught ! elem5t>y rules ( ari?metic4 ,a boy :o 0 demoli%+ a *oice banana 0 approa*$ by a "y fri5d1 :o1 reg>d+ him ) envi\s eyes1 ask$1 ,h[ mu* did y pay = t banana1 ,fr$8 ,! prompt answ] 0 q rem>kable 9 its way3 ,! man :at ,i b\ght x ( receives j half z _m sixp5ces = sixte5 doz5 doz5 bananas z he gives bananas = a fiv]4 ,n[1 h[ long w x take ! read] to say correctly j h[ mu* ,fr$ paid = 8 r>e & refre%+ fruit8 #c4--;,,at ,a ,,cattle ,,m>ket4 ,?ree c\ntrym5 met at a cattle m>ket4 ,look "h1 said ,hodge to ,jakes1 ,i'll give y six ( my pigs = "o ( y\r horses1 & !n y'll h twice z _m animals "h z ,i've got4 ,if t's y\r way ( do+ bus9ess1 said ,durrant to ,hodge1 ,i'll give y f\rte5 ( my %eep = a horse1 & !n y'll h ?ree times z _m animals z ,i4 ,well1 ,i'll g bett] ?an t1 said ,jakes to ,durrant2 ,i'll give y f\r c[s = a horse1 & !n y'll h six times z _m animals z ,i've got "h4 ,no d\bt ?is 0 a v primitive way ( b>t]+ animals1 b x is an int]e/+ little puzzle to discov] j h[ _m animals ,jakes1 ,hodge1 & ,durrant mu/ h tak5 to ! cattle m>ket4 #d4--,,! ,,beanfea/ ,,puzzle4 ,a numb] ( m5 w5t out toge!r on a bean-fea/4 ,!re 7 f\r p>ties invit$--namely1 #be cobbl]s1 #bj tailors1 #ah hatt]s1 & #ab glov]s4 ,!y sp5t altoge!r #f1 #acs4 ,x 0 f\nd t five cobbl]s sp5t z mu* z f\r tailors2 t twelve tailors sp5t z mu* z n9e hatt]s2 & t six hatt]s sp5t z mu* z eight glov]s4 ,! puzzle is to f9d out h[ mu* ea* ( ! f\r p>ties sp5t4 #e4--;,a ,,que] ,,co9cid5ce4 ,sev5 m51 ^: names 7 ,adams1 ,bak]1 ,c>t]1 ,dobson1 ,edw>ds1 ,francis1 & ,gudgeon1 7 rec5tly engag$ 9 play4 ,! "n ( ! p>ticul> game is ( no consequ5ce4 ,!y _h agre$ t :5ev] a play] won a game he %\ld d\ble ! money ( ea* ( ! o!r play]s--t is1 he 0 to give ! play]s j z mu* money z !y _h already 9 _! pockets4 ,!y play$ sev5 games1 &1 /range to say1 ea* won a game 9 turn1 9 ! ord] 9 :i* _! names are giv54 ,b a m curi\s co9cid5ce is ?is--t :5 !y _h f9i%$ play ea* ( ! sev5 m5 _h exactly ! same am\nt--two %ill+s & eightp5ce--9 8 pocket4 ,! puzzle is to f9d out h[ mu* money ea* man _h ) him be=e he sat d[n to play4 #f4--;,a ,,*>itable ,,beque/4 ,a man left in/ructions to 8 executors to di/ribute once a ye> exactly fifty-five %ill+s among ! poor ( 8 p>i%2 b !y 7 only to cont9ue ! gift s long z !y c\ld make x 9 diff]5t ways1 always giv+ eighte5p5ce ea* to a numb] ( wom5 & half a cr[n ea* to m54 ,dur+ h[ _m ye>s c\ld ! *>ity be adm9i/]$8 ,( c\rse1 by diff]5t ways is meant a diff]5t numb] ( m5 & wom5 e "t4 #g4--,,! ,,wid[',s ,,legacy4 ,a g5tleman :o rec5tly di$ left ! sum ( #h1jjj to be divid$ among 8 wid[1 five sons1 & f\r daught]s4 ,he direct$ t e son %\ld receive ?ree times z mu* z a daught]1 & t e daught] %\ld h twice z mu* z _! "m4 ,:at 0 ! wid['s %>e8 #h4--;,,indiscrim9ate ,,*>ity4 ,a *>itable g5tleman1 on 8 way home "o night1 0 appeal$ to by ?ree ne$y p]sons 9 succession = assi/ance4 ,to ! fir/ p]son he gave "o p5ny m ?an half ! money he _h 9 8 pocket2 to ! second p]son he gave twop5ce m ?an half ! money he !n _h 9 8 pocket2 & to ! ?ird p]son he h&$ ov] ?reep5ce m ?an half ( :at he _h left4 ,on ent]+ 8 h\se he _h only "o p5ny 9 8 pocket4 ,n[1 c y say exactly h[ mu* money t g5tleman _h on him :5 he />t$ = home8 #i4--,,! ,,two ,,a]oplanes4 ,a man rec5tly b\ght two a]oplanes1 b aft]w>ds f\nd t !y w\ld n answ] ! purpose = :i* he want$ !m4 ,s he sold !m = #fjj ea*1 mak+ a loss ( #bj p] c5t4 on "o ma*9e & a pr(it ( #bj p] c5t4 on ! o!r4 ,did he make a pr(it on ! :ole transaction1 or a loss8 ,& h[ mu*8 #aj4--;,,buy+ ,,pres5ts4 ,:om d y ?9k ,i met 9 t[n la/ week1 ,bro!r ,william8 said ,uncle ,b5jam94 ,t old sk9fl9t ,jork9s4 ,8 family _h be5 tak+ him ar\nd buy+ ,*ri/mas pres5ts4 ,he said to me1 ',:y _c ! gov]nm5t aboli% ,*ri/mas1 & make ! giv+ ( pres5ts puni%able by law8 ,i came out ?is morn+ ) a c]ta9 am\nt ( money 9 my pocket1 & ,i f9d ,i h sp5t j half ( x4 ,9 fact1 if y w believe me1 ,i take home j z _m %ill+s z ,i _h p\nds1 & half z _m p\nds z ,i _h %ill+s4 ,x is mon/r\s6' ,c y say exactly h[ mu* money ,jork9s _h sp5t on ^? pres5ts8 #aa4--,,! ,,cycli/s' ,,fea/4 ',twas la/ ,bank ,holiday1 s ,i've be5 told1 ,"s cycli/s rode abroad 9 glori\s wea!r4 ,re/+ at noon )9 a tav]n old1 ,!y all agre$ to h a fea/ toge!r4 ,put x all 9 "o bill1 m9e ho/1 !y said1 ,= e man an equal %>e w pay4 ,! bill 0 promptly on ! table laid1 ,& f\r p\nds 0 ! reckon+ t day4 ,b1 sad to /ate1 :5 !y prep>$ to squ>e1 ',twas f\nd t two _h sneak$ outside & fl$4 ,s1 = two %ill+s m ?an 8 due %>e ,ea* hone/ man :o _h rema9$ 0 bl$4 ,!y settl$ lat] ) ^? rogues1 no d\bt4 ,h[ _m 7 !y :5 !y fir/ set out8 #ab4--;,a ,,que] ,,?+ ,,9 ,,money4 ,x w be f\nd t #ff1 #fs4 #f;d4 equals #ae1iah p5ce4 ,n[1 ! f\r #f's add$ toge!r make #bd1 & ! figures 9 #ae1iah also add to #bd4 ,x is a curi\s fact t !re is only "o o!r sum ( money1 9 p\nds1 %ill+s1 & p5ce 7all simil>ly repetitions ( "o figure71 ( :i* ! digits %all add up ! same z ! digits ( ! am\nt 9 p5ce4 ,:at is ! o!r sum ( money8 #ac4--;,a ,,new ,,money ,,puzzle4 ,! l>ge/ sum ( money t c be writt5 9 p\nds1 %ill+s1 p5ce1 & f>?+s1 us+ ea* ( ! n9e digits once & only once1 is #ih1gfe1 #ds4 #c;d4 ,n[1 try to discov] ! smalle/ sum ( money t c be writt5 d[n "u precisely ! same conditions4 ,!re mu/ be "s value giv5 = ea* d5om9ation--p\nds1 %ill+s1 p5ce1 & f>?+s--& ! n\ght may n be us$4 ,x requires j a little judgm5t & ?\ght4 #ad4--,,squ>e ,,money4 ,?is is que]1 said ,mc,crank to 8 fri5d4 ,twop5ce add$ to twop5ce is f\rp5ce1 & twop5ce multipli$ by twop5ce is also f\rp5ce4 ,( c\rse1 he 0 wrong 9 ?9k+ y c multiply money by money4 ,! multipli] mu/ be reg>d$ z an ab/ract numb]4 ,x is true t two feet multipli$ by two feet w make f\r squ>e feet4 ,simil>ly1 two p5ce multipli$ by two p5ce w produce f\r squ>e p5ce6 ,& x w p]plex ! read] to say :at a squ>e p5ny is4 ,b we w assume = ! purposes ( our puzzle t twop5ce multipli$ by twop5ce is f\rp5ce4 ,n[1 :at two am\nts ( money w produce ! next smalle/ possible result1 ! same 9 bo? cases1 :5 add$ or multipli$ 9 ?is mann]8 ,! two am\nts ne$ n be alike1 b !y mu/ be ^? t c be paid 9 curr5t co9s ( ! realm4 #ae4--,,pocket ,,money4 ,:at is ! l>ge/ sum ( money--all 9 curr5t silv] co9s & no f\r-%ill+ piece--t ,i c\ld h 9 my pocket )\t be+ able to give *ange = a half-sov]eign8 #af4--,,! ,,millionaire',s ,,p]plexity4 ,mr4 ,morgan ,g4 ,bloomg>t51 ! millionaire1 kn[n 9 ! ,/ates z ! ,clam ,k+1 _h1 = 8 s9s1 m money ?an he knew :at to d )4 ,x bor$ him4 ,s he det]m9$ to p]secute "s ( 8 poor b happy fri5ds ) x4 ,!y _h nev] done him any h>m1 b he resolv$ to inoculate !m ) ! s\rce ( all evil4 ,he !re=e propos$ to di/ribute a million doll>s among !m & wat* !m g rapidly to ! bad4 ,b he 0 a man ( /range fancies & sup]/itions1 & x 0 an inviolable rule ) him nev] to make a gift t 0 n ei!r "o doll> or "s p[] ( sev5--su* z #g1 #di1 #cdc1 #b1dja1 :i* numb]s ( doll>s are produc$ by simply multiply+ sev5s toge!r4 ,ano!r rule ( 8 0 t he w\ld nev] give m ?an six p]sons exactly ! same sum4 ,n[1 h[ 0 he to di/ribute ! #a1jjj1jjj doll>s8 ,y may di/ribute ! money among z _m p z y l1 "u ! conditions giv54 #ag4--,,! ,,puzzl+ ,,money-,,boxes4 ,f\r bro!rs--nam$ ,john1 ,william1 ,*>les1 & ,?omas--_h ea* a money-box4 ,! boxes 7 all giv5 to !m on ! same day1 & !y at once put :at money !y _h into !m2 only1 z ! boxes 7 n v l>ge1 !y fir/ *ang$ ! money into z few co9s z possible4 ,aft] !y _h done ?is1 !y told "o ano!r h[ mu* money !y _h sav$1 & x 0 f\nd t if ,john _h _h #bs4 m 9 8 box ?an at pres5t1 if ,william _h _h #bs4 less1 if ,*>les _h _h twice z mu*1 & if ,?omas _h _h half z mu*1 !y w\ld all h _h exactly ! same am\nt4 ,n[1 :5 ,i add t all f\r boxes toge!r conta9$ #des41 & t !re 7 only six co9s 9 all 9 !m1 x becomes an ent]ta9+ puzzle to discov] j :at co9s 7 9 ea* box4 #ah4--,,! ,,m>ket ,,wom54 ,a numb] ( m>ket wom5 sold _! v>i\s products at a c]ta9 price p] p\nd 7diff]5t 9 e case71 & ea* receiv$ ! same am\nt--#bs4 #b;d4 ,:at is ! greate/ numb] ( wom5 !re c\ld h be58 ,! price p] p\nd 9 e case mu/ be su* z c\ld be paid 9 curr5t money4 #ai4--,,! ,,new ,,ye>',s ,,eve ,,supp]s4 ,! proprietor ( a small ,london caf has giv5 me "s int]e/+ figures4 ,he says t ! ladies :o come alone to 8 place = refre%m5t sp5d ea* on an av]age eighte5p5ce1 t ! unaccompani$ m5 sp5d half a cr[n ea*1 & t :5 a g5tleman br+s 9 a lady he sp5ds half a gu9ea4 ,on ,new ,ye>'s ,eve he suppli$ supp]s to tw5ty-five p]sons1 & took five p\nds 9 all4 ,n[1 assum+ 8 av]ages to h held good 9 e case1 h[ 0 8 company made up on t occasion8 ,( c\rse1 only s+le g5tlem51 s+le ladies1 & pairs 7a lady & g5tleman7 c be suppos$ to h be5 pres5t1 z we are n consid]+ l>g] p>ties4 #bj4--;,,beef ,,& ,,sausages4 ,a neighb\r ( m9e1 said ,aunt ,jane1 b\ght a c]ta9 quantity ( beef at two %ill+s a p\nd1 & ! same quantity ( sausages at eighte5p5ce a p\nd4 ,i po9t$ out to h] t if %e _h divid$ ! same money equally betwe5 beef & sausages %e w\ld h ga9$ two p\nds 9 ! total weight4 ,c y tell me exactly h[ mu* %e sp5t8 ,( c\rse1 x is no bus9ess ( m9e1 said ,mrs4 ,sunniborne2 b a lady :o c\ld pay su* prices mu/ be some:at inexp]i5c$ 9 dome/ic economy4 ,i q agree1 my de>1 ,aunt ,jane repli$1 b y see t is n ! precise po9t "u discussion1 any m ?an ! "n & morals ( ! tradesman4 #ba4--;,a ,,deal ,,9 ,,apples4 ,i paid a man a %ill+ = "s apples1 b !y 7 s small t ,i made him ?r[ 9 two extra apples4 ,i f9d t made !m co/ j a p5ny a doz5 less ?an ! fir/ price he ask$4 ,h[ _m apples did ,i get = my %ill+8 #bb4--;,a ,,deal ,,9 ,,eggs4 ,a man w5t rec5tly into a dairyman's %op to buy eggs4 ,he want$ !m ( v>i\s qualities4 ,! salesman _h new-laid eggs at ! high price ( fivep5ce ea*1 fre% eggs at "o p5ny ea*1 eggs at a halfp5ny ea*1 & eggs = electione]+ purposes at a greatly r$uc$ figure1 b z !re 0 no election on at ! "t ! buy] _h no use = ! la/4 ,h[ev]1 he b\ght "s ( ea* ( ! ?ree o!r k9ds & obta9$ exactly "o hundr$ eggs = eight & f\rp5ce4 ,n[1 z he br\ght away exactly ! same numb] ( eggs ( two ( ! ?ree qualities1 x is an int]e/+ puzzle to det]m9e j h[ _m he b\ght at ea* price4 #bc4--,,! ,,*ri/mas-,,boxes4 ,"s ye>s ago a man told me he _h sp5t "o hundr$ ,engli% silv] co9s 9 ,*ri/mas-boxes1 giv+ e p]son ! same am\nt1 & x co/ him exactly #a1 #ajs4 #a;d4 ,c y tell j h[ _m p]sons receiv$ ! pres5t1 & h[ he c\ld h manag$ ! di/ribution8 ,t odd p5ny looks que]1 b x is all "r4 #bd4--;,a ,,%opp+ ,,p]plexity4 ,two ladies w5t into a %op :]e1 ?r\gh "s curi\s ecc5tricity1 no *ange 0 giv51 & made pur*ases am\nt+ toge!r to less ?an five %ill+s4 ,d y "k1 said "o lady1 ,i f9d ,i %all require no few] ?an six curr5t co9s ( ! realm to pay = :at ,i h b\ght4 ,! o!r lady consid]$ a mom5t1 & !n exclaim$3 ,by a peculi> co9cid5ce1 ,i am exactly 9 ! same dilemma4 ,!n we w pay ! two bills toge!r4 ,b1 to _! a/oni%m5t1 !y /ill requir$ six co9s4 ,:at is ! smalle/ possible am\nt ( _! pur*ases--bo? diff]5t8 #be4--;,,*9ese ,,money4 ,! ,*9ese are a curi\s p1 & h /range inv]t$ ways ( do+ ?+s4 ,x is said t !y use a saw ) an upw>d pressure in/ead ( a d[nw>d "o1 t !y plane a deal bo>d by pull+ ! tool t[>d !m in/ead ( pu%+ x1 & t 9 build+ a h\se !y fir/ con/ruct ! ro( &1 hav+ rais$ t into position1 proce$ to "w d[nw>ds4 ,9 money ! curr5cy ( ! c\ntry consi/s ( taels ( fluctuat+ value4 ,! tael became ?9n] & ?9n] until #b1jjj ( !m pil$ toge!r made less ?an ?ree in*es 9 height4 ,! common ca% consi/s ( brass co9s ( v>y+ ?icknesses1 ) a r\nd1 squ>e1 or triangul> hole 9 ! c5tre1 z 9 our illu/ration4 ,illu/ration ,^! are /rung on wires l buttons4 ,suppos+ t elev5 co9s ) r\nd holes are wor? fifte5 *+-*angs1 t elev5 ) squ>e holes are wor? sixte5 *+-*angs1 & t elev5 ) triangul> holes are wor? sev5te5 *+-*angs1 h[ c a ,*9aman give me *ange = half a cr[n1 us+ no co9s o!r ?an ! ?ree m5tion$8 ,a *+-*ang is wor? exactly twop5ce & f\r-fifte5?s ( a *+-*ang4 #bf4--,,! ,,junior ,,cl]k',s ,,puzzle4 ,two y\?s1 be>+ ! pleasant names ( ,moggs & ,snoggs1 7 employ$ z junior cl]ks by a m]*ant 9 ,m9c+ ,lane4 ,!y 7 bo? engag$ at ! same sal>y--t is1 comm5c+ at ! rate ( #ej a ye>1 payable half-ye>ly4 ,moggs _h a ye>ly rise ( #aj1 & ,snoggs 0 (f]$ ! same1 only he ask$1 = reasons t d n conc]n our puzzle1 t he might take 8 rise at #b1 #ajs4 half-ye>ly1 to :i* 8 employ] 7n1 p]haps1 unnaturally67 _h no objection4 ,n[ we come to ! real po9t ( ! puzzle4 ,moggs put regul>ly into ! ,po/ ,(fice ,sav+s ,bank a c]ta9 proportion ( 8 sal>y1 :ile ,snoggs sav$ twice z great a proportion ( 81 & at ! end ( five ye>s !y _h toge!r sav$ #bfh1 #aes4 ,h[ mu* _h ea* sav$8 ,! "q ( int]e/ c be ignor$4 #bg4--;,,giv+ ,,*ange4 ,e "o is famili> ) ! difficulties t frequ5tly arise ov] ! giv+ ( *ange1 & h[ ! assi/ance ( a ?ird p]son ) a few co9s 9 8 pocket w sometimes help u to set ! matt] "r4 ,"h is an example4 ,an ,engli%man w5t into a %op 9 ,new ,york & b\ght goods at a co/ ( ?irty-f\r c5ts4 ,! only money he _h 0 a doll>1 a ?ree-c5t piece1 & a two-c5t piece4 ,! tradesman _h only a half-doll> & a qu>t]-doll>4 ,b ano!r cu/om] happ5$ to be pres5t1 & :5 ask$ to help produc$ two dimes1 a five-c5t piece1 a two-c5t piece1 & a "o-c5t piece4 ,h[ did ! tradesman manage to give *ange8 ,= ! b5efit ( ^? read]s :o are n famili> ) ! ,am]ican co9age1 x is only necess>y to say t a doll> is a hundr$ c5ts & a dime t5 c5ts4 ,a puzzle ( ?is k9d %\ld r>ely cause any difficulty if attack$ 9 a prop] mann]4 #bh4--;,,defective ,,obs]vation4 ,our obs]vation ( little ?+s is frequ5tly defective1 & our memories v liable to lapse4 ,a c]ta9 judge rec5tly rem>k$ 9 a case t he _h no recollection :atev] ( putt+ ! w$d+-r+ on 8 wife's f+]4 ,c y correctly answ] ^! que/ions )\t hav+ ! co9s 9 sight8 ,on :i* side ( a p5ny is ! date giv58 ,"s p are s unobs]vant t1 al?\gh !y are h&l+ ! co9 ne>ly e day ( _! lives1 !y are at a loss to answ] ?is simple "q4 ,if ,i lay a p5ny flat on ! table1 h[ _m o!r p5nies c ,i place ar\nd x1 e "o also ly+ flat on ! table1 s t !y all t\* ! fir/ "o8 ,! geometrician w1 ( c\rse1 give ! answ] at once1 & n ne$ to make any exp]im5t4 ,he w also "k t1 s9ce all circles are simil>1 ! same answ] w necess>ily apply to any co94 ,! next "q is a mo/ int]e/+ "o to ask a company1 ea* p]son writ+ d[n 8 answ] on a slip ( pap]1 s t no "o %all be help$ by ! answ]s ( o!rs4 ,:at is ! greate/ numb] ( ?ree-p5ny-pieces t may be laid flat on ! surface ( a half-cr[n1 s t no piece lies on ano!r or ov]laps ! surface ( ! half-cr[n8 ,x is amaz+ :at a v>iety ( diff]5t answ]s "o gets to ?is "q4 ,v few p w be f\nd to give ! correct numb]4 ,( c\rse ! answ] mu/ be giv5 )\t look+ at ! co9s4 #bi4--,,! ,,brok5 ,,co9s4 ,a man _h ?ree co9s--a sov]eign1 a %ill+1 & a p5ny--& he f\nd t exactly ! same fraction ( ea* co9 _h be5 brok5 away4 ,n[1 assum+ t ! orig9al intr9sic value ( ^! co9s 0 ! same z _! nom9al value--t is1 t ! sov]eign 0 wor? a p\nd1 ! %ill+ wor? a %ill+1 & ! p5ny wor? a p5ny--:at proportion ( ea* co9 has be5 lo/ if ! value ( ! ?ree rema9+ fragm5ts is exactly "o p\nd8 #cj4--,,two ,,que/ions ,,9 ,,probabilities4 ,!re is p]haps no class ( puzzle ov] :i* p s frequ5tly blund] z t :i* involves :at is call$ ! !ory ( probabilities4 ,i w give two simple examples ( ! sort ( puzzle ,i mean4 ,!y are really q easy1 & yet _m p]sons are tripp$ up by !m4 ,a fri5d rec5tly produc$ five p5nies & said to me3 ,9 ?r[+ ^! five p5nies at ! same "t1 :at are ! *ances t at lea/ f\r ( ! co9s w turn up ei!r all heads or all tails8 ,8 own solution 0 q wrong1 b ! correct answ] ought n to be h>d to discov]4 ,ano!r p]son got a wrong answ] to ! foll[+ little puzzle :i* ,i he>d him prop\nd3 ,a man plac$ ?ree sov]eigns & "o %ill+ 9 a bag4 ,h[ mu* %\ld be paid = p]mission to draw "o co9 f x8 ,x is1 ( c\rse1 und]/ood t y are z likely to draw any "o ( ! f\r co9s z ano!r4 #ca4--;,,dome/ic ,,economy4 ,"y ,mrs4 ,p]k9s1 ( ,putney1 writes to me z foll[s3 ,i %\ld be v glad if y c\ld give me ! answ] to a little sum t has be5 worry+ me a good deal lately4 ,"h x is3 ,we h only be5 m>ri$ a %ort "t1 & n[1 at ! end ( two ye>s f ! "t :5 we set up h\sekeep+1 my husb& tells me t he f9ds we h sp5t a ?ird ( 8 ye>ly income 9 r5t1 rates1 & taxes1 "o-half 9 dome/ic exp5ses1 & "o-n9? 9 o!r ways4 ,he has a balance ( #aij rema9+ 9 ! bank4 ,i "k ?is la/1 because he accid5tally left out 8 pass-book ! o!r day1 & ,i peep$ into x4 ,don't y ?9k t a husb& ought to give 8 wife 8 entire confid5ce 9 8 money matt]s8 ,well1 ,i d2 &--w y believe x8--he has nev] told me :at 8 income really is1 & ,i want1 v naturally1 to f9d out4 ,c y tell me :at x is f ! figures ,i h giv5 y8 ,yes2 ! answ] c c]ta9ly be giv5 f ! figures conta9$ 9 ,mrs4 ,p]k9s's lett]4 ,& my read]s1 if n w>n$1 w be practically unanim\s 9 decl>+ ! income to be--some?+ absurdly 9 excess ( ! correct answ]6 #cb4--,,! ,,excursion ,,ticket ,,puzzle4 ,:5 ! big flam+ plac>ds 7 exhibit$ at ! little prov9cial railway /ation1 ann\nc+ t ! ,great ---- ,company w\ld run *eap excursion tra9s to ,london = ! ,*ri/mas holidays1 ! inhabitants ( ,mudley-cum-,turmits 7 9 q a flutt] ( excitem5t4 ,half an h\r be=e ! tra9 came 9 ! little book+ (fice 0 cr[d$ ) c\ntry pass5g]s1 all b5t on visit+ _! fri5ds 9 ! great ,metropolis4 ,! book+ cl]k 0 unaccu/om$ to deal+ ) cr[ds ( su* a dim5sion1 & he told me aft]w>ds1 :ile wip+ 8 manly br[1 t :at caus$ him s mu* tr\ble 0 ! fact t ^! ru/ics paid _! f>es 9 su* a lot ( small money4 ,he said t he _h en\gh f>?+s to supply a ,we/ ,end drap] ) *ange = a week1 & a suffici5t numb] ( ?reep5ny pieces = ! congregations ( ?ree p>i% *ur*es4 ,t excursion f>e1 said he1 is n9ete5 %ill+s & n9ep5ce1 & ,i %\ld l to "k 9 j h[ _m diff]5t ways x is possible = su* an am\nt to be paid 9 ! curr5t co9 ( ?is realm4 ,"h1 !n1 is a puzzle3 ,9 h[ _m diff]5t ways may n9ete5 %ill+s & n9ep5ce be paid 9 our curr5t co98 ,rememb] t ! f\rp5ny-piece is n n[ curr5t4 #cc4--,,puzzle ,,9 ,,rev]sals4 ,mo/ p "k t if y take any sum ( money 9 p\nds1 %ill+s1 & p5ce1 9 :i* ! numb] ( p\nds 7less ?an #ab7 exce$s t ( ! p5ce1 rev]se x 7call+ ! p\nds p5ce & ! p5ce p\nds71 f9d ! diff]5ce1 !n rev]se & add ?is diff]5ce1 ! result is always #ab1 #ahs4 #aa;d4 ,b if we omit ! condition1 less ?an #ab1 & all[ n\ght to repres5t %ill+s or p5ce--7#a7 ,:at is ! l[e/ am\nt to :i* ! rule w n apply8 7#b7 ,:at is ! highe/ am\nt to :i* x w apply8 ,( c\rse1 :5 rev]s+ su* a sum z #ad1 #aes4 #c;d4 x may be writt5 #c1 #afs4 #b;d41 :i* is ! same z #c1 #aes4 #ad;d4 #cd4--,,! ,,groc] ,,& ,,drap]4 ,a c\ntry groc] & drap] _h two rival assi/ants1 :o prid$ !mselves on _! rapidity 9 s]v+ cu/om]s4 ,! "y man on ! groc]y side c\ld weigh up two "o-p\nd p>cels ( sug> p] m9ute1 :ile ! drap]y assi/ant c\ld cut ?ree "o-y>d l5g?s ( clo? 9 ! same "t4 ,_! employ]1 "o slack day1 set !m a race1 giv+ ! groc] a b>rel ( sug> & tell+ him to weigh up =ty-eight "o-p\nd p>cels ( sug> ,:ile ! drap] divid$ a roll ( =ty-eight y>ds ( clo? into y>d pieces4 ,! two m5 7 int]rupt$ toge!r by cu/om]s = n9e m9utes1 b ! drap] 0 di/urb$ sev5te5 times z long z ! groc]4 ,:at 0 ! result ( ! race8 #ce4--;,,judk9s',s ,,cattle4 ,hiram ,b4 ,judk9s1 a cattle-deal] ( ,texas1 _h five droves ( animals1 consi/+ ( ox51 pigs1 & %eep1 ) ! same numb] ( animals 9 ea* drove4 ,"o morn+ he sold all t he _h to eight deal]s4 ,ea* deal] b\ght ! same numb] ( animals1 pay+ sev5te5 doll>s = ea* ox1 f\r doll>s = ea* pig1 & two doll>s = ea* %eep2 & ,hiram receiv$ 9 all ?ree hundr$ & "o doll>s4 ,:at is ! greate/ numb] ( animals he c\ld h _h8 ,& h[ _m w\ld !re be ( ea* k9d8 #cf4--;,,buy+ ,,apples4 ,z ! pur*ase ( apples 9 small quantities has always pres5t$ consid]able difficulties1 ,i ?9k x well to (f] a few rem>ks on ?is subject4 ,we all "k ! /ory ( ! sm>t boy :o1 on be+ told by ! old woman t %e 0 sell+ h] apples at f\r = ?reep5ce1 said3 ,let me see6 ,f\r = ?reep5ce2 t's ?ree = twop5ce1 two = a p5ny1 "o = no?+--,i'll take "o6 ,!re are simil> cases ( p]plexity4 ,= example1 a boy once pick$ up a p5ny apple f a /all1 b :5 he le>nt t ! woman's pe>s 7 ! same price he ex*ang$ x1 & 0 ab\t to walk (f4 ,/op6 said ! woman4 ,y hav5't paid me = ! pe>6 ,no1 said ! boy1 ( c\rse n4 ,i gave y ! apple = x4 ,b y didn't pay = ! apple6 ,bless ! woman6 ,y don't expect me to pay = ! apple & ! pe> too6 ,& be=e ! poor creature c\ld get out ( ! tangle ! boy _h disappe>$4 ,!n1 aga91 we h ! case ( ! man :o gave a boy sixp5ce & promis$ to repeat ! gift z soon z ! y\ng/] _h made x into n9ep5ce4 ,five m9utes lat] ! boy return$4 ,i h made x into n9ep5ce1 he said1 at ! same "t h&+ 8 b5efactor ?reep5ce4 ,h[ d y make t out8 he 0 ask$4 ,i b\ght ?reep5nywor? ( apples4 ,b t does n make x into n9ep5ce6 ,i %\ld r ?9k x did1 0 ! boy's reply4 ,! apple woman has ?reep5ce1 hasn't %e8 ,v well1 ,i h ?reep5nywor? ( apples1 & ,i h j giv5 y ! o!r ?reep5ce4 ,:at's t b n9ep5ce8 ,i cite ^! cases j to %[ t ! small boy really /&s 9 ne$ ( a little in/ruction 9 ! art ( buy+ apples4 ,s ,i w give a simple pos] deal+ ) ?is bran* ( comm]ce4 ,an old woman _h apples ( ?ree sizes = sale--"o a p5ny1 two a p5ny1 & ?ree a p5ny4 ,( c\rse two ( ! second size & ?ree ( ! ?ird size 7 respectively equal to "o apple ( ! l>ge/ size4 ,n[1 a g5tleman :o _h an equal numb] ( boys & girls gave 8 *ildr5 sev5p5ce to be sp5t among/ !m all on ^! apples4 ,! puzzle is to give ea* *ild an equal di/ribution ( apples4 ,h[ 0 ! sev5p5ce sp5t1 & h[ _m *ildr5 7 !re8 #cg4--;,,buy+ ,,*e/nuts4 ,?\gh ! foll[+ little puzzle deals ) ! pur*ase ( *e/nuts1 x is n itself ( ! *e/nut type4 ,x is q new4 ,at fir/ sight x has c]ta9ly ! appe>ance ( be+ ( ! nons5se puzzle "*1 b x is all "r :5 prop]ly consid]$4 ,a man w5t to a %op to buy *e/nuts4 ,he said he want$ a p5nywor?1 & 0 giv5 five *e/nuts4 ,x is n en\gh2 ,i ought to h a six?1 he rem>k$6 ,b if ,i give y "o *e/nut m4 ! %opman repli$1 y w h five too _m4 ,n[1 /range to say1 !y 7 bo? "r4 ,h[ _m *e/nuts %\ld ! buy] receive = half a cr[n8 #ch4--,,! ,,bicycle ,,?ief4 ,"h is a little tangle t is p]petually cropp+ up 9 v>i\s guises4 ,a cycli/ b\ght a bicycle = #ae & gave 9 paym5t a *eque = #be4 ,! sell] w5t to a neighb\r+ %opkeep] & got him to *ange ! *eque = him1 & ! cycli/1 hav+ receiv$ 8 #aj *ange1 m\nt$ ! ma*9e & disappe>$4 ,! *eque prov$ to be valueless1 & ! salesman 0 reque/$ by 8 neighb\r to refund ! am\nt he _h receiv$4 ,to d ?is1 he 0 compell$ to borr[ ! #be f a fri5d1 z ! cycli/ =got to leave 8 address1 & c\ld n be f\nd4 ,n[1 z ! bicycle co/ ! salesman #aa1 h[ mu* money did he lose altoge!r8 #ci4--,,! ,,co/]mong]',s ,,puzzle4 ,h[ mu* did y] pay = !m oranges1 ,bill8 ,i a9't a-go9' to tell y]1 ,jim4 ,b ,i beat ! old cove d[n f\rp5ce a hundr$4 ,:at good did t d y]8 ,well1 x meant five m oranges on e t5 %ill9's-wor?4 ,n[1 :at price did ,bill actually pay = ! oranges8 ,!re is only "o rate t w fit 9 ) 8 /atem5ts4 ,,age ,,& ,,k9%ip ,,puzzles4 ,! days ( our ye>s are ?reescore ye>s & t54 --,psalm xc4 #aj4 ,= c5turies x has be5 a fav\rite me?od ( prop\nd+ ari?metical puzzles to pose !m 9 ! =m ( que/ions z to ! age ( an individual4 ,!y g5]ally l5d !mselves to v easy solution by ! use ( algebra1 ?\gh (t5 ! difficulty lies 9 /at+ !m correctly4 ,!y may be made v complex & may dem& consid]able +5uity1 b no g5]al laws c well be laid d[n = _! solution4 ,! solv] mu/ use 8 own sagacity4 ,z = puzzles 9 relation%ip or k9%ip1 x is q curi\s h[ bewild]+ _m p f9d ^! ?+s4 ,ev5 9 ord9>y conv]sation1 "s /atem5t z to relation%ip1 :i* is q cle> 9 ! m9d ( ! speak]1 w imm$iately tie ! bra9s ( o!r p into knots4 ,su* expressions z ,he is my uncle's son-9-law's si/] convey absolutely no?+ to "s p )\t a detail$ & lab\r$ explanation4 ,9 su* cases ! be/ c\rse is to sket* a brief g5ealogical table1 :5 ! eye comes imm$iately to ! assi/ance ( ! bra94 ,9 ^! days1 :5 we h a gr[+ lack ( respect = p$igrees1 mo/ p h got out ( ! habit ( rapidly draw+ su* tables1 :i* is to be regrett$1 z !y w\ld save a lot ( "t & bra9 rack+ on occasions4 #dj4--,,mamma',s ,,age4 ,tommy3 ,h[ old are y1 mamma8 ,mamma3 ,let me ?9k1 ,tommy4 ,well1 our ?ree ages add up to exactly sev5ty ye>s4 ,tommy3 ,t's a lot1 isn't x8 ,& h[ old are y1 papa8 ,papa3 ,j six times z old z y1 my son4 ,tommy3 ,%all ,i ev] be half z old z y1 papa8 ,papa3 ,yes1 ,tommy2 & :5 t happ5s our ?ree ages w add up to exactly twice z mu* z to-day4 ,tommy3 ,& suppos+ ,i 0 born be=e y1 papa2 & suppos+ mamma _h =got all ab\t x1 & hadn't be5 at home :5 ,i came2 & suppos+-- ,mamma3 ,suppos+1 ,tommy1 we talk ab\t b$4 ,come along1 d>l+4 ,y'll h a heada*e4 ,n[1 if ,tommy _h be5 "s ye>s old] he might h calculat$ ! exact ages ( 8 p>5ts f ! in=mation !y _h giv5 him4 ,c y f9d out ! exact age ( mamma8 #da4--,,_! ,,ages4 ,my husb&'s age1 rem>k$ a lady ! o!r day1 is repres5t$ by ! figures ( my own age rev]s$4 ,he is my s5ior1 & ! diff]5ce betwe5 our ages is "o-elev5? ( _! sum4 #db4--,,! ,,family ,,ages4 ,:5 ! ,smileys rec5tly receiv$ a visit f ! fav\rite uncle1 ! fond p>5ts _h all ! five *ildr5 br\ght into 8 pres5ce4 ,fir/ came ,billie & little ,g]trude1 & ! uncle 0 in=m$ t ! boy 0 exactly twice z old z ! girl4 ,!n ,h5rietta arriv$1 & x 0 po9t$ out t ! comb9$ ages ( h]self & ,g]trude equall$ twice ! age ( ,billie4 ,!n ,*>lie came runn+ 91 & somebody rem>k$ t n[ ! comb9$ ages ( ! two boys 7 exactly twice ! comb9$ ages ( ! two girls4 ,! uncle 0 express+ 8 a/oni%m5t at ^! co9cid5ces :5 ,janet came 94 ,ah6 uncle1 %e exclaim$1 y h actually arriv$ on my tw5ty-fir/ bir?day6 ,to ?is ,mr4 ,smiley add$ ! f9al /agg]]3 ,yes1 & n[ ! comb9$ ages ( ! ?ree girls are exactly equal to twice ! comb9$ ages ( ! two boys4 ,c y give ! age ( ea* *ild8 #dc4--,,mrs4 ,,timpk9s',s ,,age4 ,edw93 ,d y "k1 :5 ! ,timpk9ses m>ri$ eighte5 ye>s ago ,timpk9s 0 ?ree times z old z 8 wife1 & to-day he is j twice z old z %e8 ,angel9a3 ,!n h[ old 0 ,mrs4 ,timpk9s on ! w$d+ day8 ,c y answ] ,angel9a's "q8 #dd--;,a ,,c5sus ,,puzzle4 ,mr4 & ,mrs4 ,jork9s h fifte5 *ildr51 all born at int]vals ( "o ye> & a half4 ,miss ,ada ,jork9s1 ! elde/1 _h an objection to /ate h] age to ! c5sus man1 b %e admitt$ t %e 0 j sev5 times old] ?an little ,johnnie1 ! y\nge/ ( all4 ,:at 0 ,ada's age8 ,d n too ha/ily assume t y h solv$ ?is little pos]4 ,y may f9d t y h made a bad blund]6 #de4--,,"m ,,& ,,daught]4 ,"m1 ,i wi% y w\ld give me a bicycle1 said a girl ( twelve ! o!r day4 ,i d n ?9k y are old en\gh yet1 my de>1 0 ! reply4 ,:5 ,i am only ?ree times z old z y are y %all h "o4 ,n[1 ! "m's age is =ty-five ye>s4 ,:5 may ! "y lady expect to receive h] pres5t8 #df4--,,m>y ,,& ,,m>maduke4 ,m>maduke3 ,d y "k1 de>1 t 9 sev5 ye>s' "t our comb9$ ages w be sixty-?ree ye>s8 ,m>y3 ,is t really s8 ,& yet x is a fact t :5 y 7 my pres5t age y 7 twice z old z ,i 0 !n4 ,i work$ x out la/ night4 ,n[1 :at are ! ages ( ,m>y & ,m>maduke8 #dg--,,rov]',s ,,age4 ,n[1 !n1 ,tommy1 h[ old is ,rov]8 ,mildr$'s "y man ask$ h] bro!r4 ,well1 five ye>s ago1 0 ! y\ng/]'s reply1 si/] 0 f\r times old] ?an ! dog1 b n[ %e is only ?ree times z old4 ,c y tell ,rov]'s age8 #dh4--;,,conc]n+ ,,tommy',s ,,age4 ,tommy ,sm>t 0 rec5tly s5t to a new s*ool4 ,on ! fir/ day ( 8 arrival ! tea*] ask$ him 8 age1 & ?is 0 8 curi\s reply3 ,well1 y see1 x is l ?is4 ,at ! "t ,i 0 born--,i =get ! ye>--my only si/]1 ,ann1 happ5$ to be j "o-qu>t] ! age ( "m1 & %e is n[ "o-?ird ! age ( fa!r4 ,t's all v well1 said ! tea*]1 b :at ,i want is n ! age ( y\r si/] ,ann1 b y\r own age4 ,i 0 j com+ to t1 ,tommy answ]$2 ,i am j a qu>t] ( "m's pres5t age1 & 9 f\r ye>s' "t ,i %all be a qu>t] ! age ( fa!r4 ,isn't t funny8 ,?is 0 all ! in=mation t ! tea*] c\ld get out ( ,tommy ,sm>t4 ,c\ld y h told1 f ^! facts1 :at 0 8 precise age8 ,x is c]ta9ly a little puzzl+4 #di4--,,next-,,door ,,neighb\rs4 ,!re 7 two families liv+ next door to "o ano!r at ,toot+ ,bec--! ,jupps & ! ,simk9s4 ,! unit$ ages ( ! f\r ,jupps am\nt$ to "o hundr$ ye>s1 & ! unit$ ages ( ! ,simk9s also am\nt$ to ! same4 ,x 0 f\nd 9 ! case ( ea* family t ! sum obta9$ by add+ ! squ>es ( ea* ( ! *ildr5's ages to ! squ>e ( ! "m's age equall$ ! squ>e ( ! fa!r's age4 ,9 ! case ( ! ,jupps1 h[ev]1 ,julia 0 "o ye> old] ?an h] bro!r ,joe1 :]eas ,sophy ,simk9 0 two ye>s old] ?an h] bro!r ,sammy4 ,:at 0 ! age ( ea* ( ! eight individuals8 #ej4--,,! ,,bag ,,( ,,nuts4 ,?ree boys 7 giv5 a bag ( nuts z a ,*ri/mas pres5t1 & x 0 agre$ t !y %\ld be divid$ 9 proportion to _! ages1 :i* toge!r am\nt$ to #ag ye>s4 ,n[ ! bag conta9$ #ggj nuts1 & z (t5 z ,h]b]t took f\r ,rob]t took ?ree1 & z (t5 z ,h]b]t took six ,*ri/oph] took sev54 ,! puzzle is to f9d out h[ _m nuts ea* _h1 & :at 7 ! boys' respective ages4 #ea4--;,,h[ ,,old ,,0 ,,m>y8 ,"h is a funny little age problem1 by ! late ,sam ,loyd1 :i* has be5 v popul> 9 ! ,unit$ ,/ates4 ,c y unravel ! my/]y8 ,! comb9$ ages ( ,m>y & ,ann are =ty-f\r ye>s1 & ,m>y is twice z old z ,ann 0 :5 ,m>y 0 half z old z ,ann w be :5 ,ann is ?ree times z old z ,m>y 0 :5 ,m>y 0 ?ree times z old z ,ann4 ,h[ old is ,m>y8 ,t is all1 b c y "w x out8 ,if n1 ask y\r fri5ds to help y1 & wat* ! %ad[ ( bewild]m5t creep ov] _! faces z !y attempt to grip ! intricacies ( ! "q4 #eb4--,,que] ,,relation%ips4 ,speak+ ( relation%ips1 said ! ,p>son at a c]ta9 d9n]-p>ty1 our legislators are gett+ ! m>riage law into a frightful tangle1 ,"h1 = example1 is a puzzl+ case t has come "u my notice4 ,two bro!rs m>ri$ two si/]s4 ,"o man di$ & ! o!r man's wife also di$4 ,!n ! survivors m>ri$4 ,! man m>ri$ 8 deceas$ wife's si/] "u ! rec5t ,act8 put 9 ! ,lawy]4 ,exactly4 ,& !re=e1 "u ! civil law1 he is legally m>ri$ & 8 *ild is legitimate4 ,b1 y see1 ! man is ! woman's deceas$ husb&'s bro!r1 & !re=e1 also "u ! civil law1 %e is n m>ri$ to him & h] *ild is illegitimate4 ,he is m>ri$ to h] & %e is n m>ri$ to him6 said ! ,doctor4 ,q s4 ,& ! *ild is ! legitimate son ( 8 fa!r1 b ! illegitimate son ( 8 "m4 ,und\bt$ly '! law is a hass1' ! ,arti/ exclaim$1 if ,i may be p]mitt$ to say s1 he add$1 ) a b[ to ! ,lawy]4 ,c]ta9ly1 0 ! reply4 ,we lawy]s try our be/ to break 9 ! bea/ to ! s]vice ( man4 ,our legislators are responsible = ! bre$4 ,& ?is rem9ds me1 w5t on ! ,p>son1 ( a man 9 my p>i% :o m>ri$ ! si/] ( 8 wid[4 ,?is man-- ,/op a mom5t1 sir1 said ! ,pr(essor4 ,m>ri$ ! si/] ( 8 wid[8 ,d y m>ry dead m5 9 y\r p>i%8 ,no2 b ,i w expla9 t lat]4 ,well1 ?is man has a si/] ( 8 own4 ,_! names are ,/eph5 ,br[n & ,jane ,br[n4 ,la/ week a "y fell[ turn$ up :om ,/eph5 introduc$ to me z 8 nephew4 ,naturally1 ,i spoke ( ,jane z 8 aunt1 b1 to my a/oni%m5t1 ! y\? correct$ me1 assur+ me t1 ?\gh he 0 ! nephew ( ,/eph51 he 0 n ! nephew ( ,jane1 ! si/] ( ,/eph54 ,?is p]plex$ me a good deal1 b x is q correct4 ,! ,lawy] 0 ! fir/ to get at ! he>t ( ! my/]y4 ,:at 0 8 solution8 #ec4--;,,he>d ,,on ,,! ,,tube ,,railway4 ,fir/ ,lady3 ,& 0 he relat$ to y1 de>8 ,second ,lady3 ,oh1 yes4 ,y see1 t g5tleman's "m 0 my "m's "m-9-law1 b he is n on speak+ t]ms ) my papa4 ,fir/ ,lady3 ,oh1 inde$6 7,b y c\ld see t %e 0 n mu* wis]47 ,h[ 0 ! g5tleman relat$ to ! ,second ,lady8 #ed4--;,a ,,family ,,p>ty4 ,a c]ta9 family p>ty consi/$ ( #a gr&fa!r1 #a gr&mo!r1 #b fa!rs1 #b mo!rs1 #d *ildr51 #c gr&*ildr51 #a bro!r1 #b si/]s1 #b sons1 #b daught]s1 #a fa!r-9-law1 #a "m-9-law1 & #a daught]-9-law4 ,tw5ty-?ree p1 y w say4 ,no2 !re 7 only sev5 p]sons pres5t4 ,c y %[ h[ ?is might be8 #ee4--;,a ,,mix$ ,,p$igree4 ,joseph ,bloggs3 ,i c't foll[ x1 my de> boy4 ,x makes me dizzy6 ,john ,snoggs3 ,x's v simple4 ,li/5 aga96 ,y happ5 to be my fa!r's bro!r-9-law1 my bro!r's fa!r-9-law1 & also my fa!r-9-law's bro!r4 ,y see1 my fa!r 0-- ,b ,mr4 ,bloggs refus$ to he> any m4 ,c ! read] %[ h[ ?is extraord9>y triple relation%ip might h come ab\t8 #ef4--,,wilson',s ,,pos]4 ,speak+ ( p]plexities-- said ,mr4 ,wilson1 ?r[+ d[n a magaz9e on ! table 9 ! comm]cial room ( ! ,railway ,hotel4 ,:o 0 speak+ ( p]plexities8 inquir$ ,mr4 ,/ubbs4 ,well1 !n1 read+ ab\t !m1 if y want to be exact--x j occurr$ to me t p]haps y ?ree m5 may be int]e/$ 9 a little matt] connect$ ) myself4 ,x 0 ,*ri/mas ,eve1 & ! f\r comm]cial travell]s 7 sp5d+ ! holiday at ,grassm9/]4 ,probably ea* suspect$ t ! o!rs _h no homes1 & p]haps ea* 0 consci\s ( ! fact t he 0 9 t pr$icam5t himself4 ,9 any case !y seem$ to be p]fectly com=table1 & z !y drew r\nd ! *e]ful fire ! conv]sation became g5]al4 ,:at is ! difficulty8 ask$ ,mr4 ,packhur/4 ,!re's no difficulty 9 ! matt]1 :5 y rightly und]/& x4 ,x is l ?is4 ,a man nam$ ,p>k] _h a fly+-ma*9e t w\ld c>ry two4 ,he 0 a v5turesome sort ( *ap--reckless1 ,i %\ld call him--& he _h "s bo!r 9 f9d+ a man will+ to risk 8 life 9 mak+ an asc5t ) him4 ,h[ev]1 an uncle ( m9e ?\ght he w\ld *ance x1 & "o f9e morn+ he took 8 seat 9 ! ma*9e & %e />t$ (f well4 ,:5 !y 7 up ab\t a ?\s& feet1 my nephew sudd5ly-- ,"h1 /op1 ,wilson6 ,:at 0 y\r nephew do+ !re8 ,y said y\r uncle1 int]rupt$ ,mr4 ,/ubbs4 ,did ,i8 ,well1 x does n matt]4 ,my nephew sudd5ly turn$ to ,p>k] & said t ! eng9e wasn't runn+ well1 s ,p>k] call$ out to my uncle-- ,look "h1 broke 9 ,mr4 ,wat]son1 we are gett+ mix$4 ,0 x y\r uncle or y\r nephew8 ,let's h x "o way or ! o!r4 ,:at ,i said is q "r4 ,p>k] call$ out to my uncle to d some?+ or o!r1 :5 my nephew-- ,!re y are aga91 ,wilson1 cri$ ,mr4 ,/ubbs2 once = all1 are we to und]/& t bo? y\r uncle & y\r nephew 7 on ! ma*9e8 ,c]ta9ly4 ,i ?\ght ,i made t cle>4 ,:]e 0 ,i8 ,well1 my nephew %\t$ back to ,p>k]-- ,phew6 ,i'm sorry to int]rupt y aga91 ,wilson1 b we c't get on l ?is4 ,is x true t ! ma*9e w\ld only c>ry two8 ,( c\rse4 ,i said at ! />t t x only c>ri$ two4 ,!n :at 9 ! "n ( a]o/ation d y mean by say+ t !re 7 ?ree p]sons on bo>d8 %\t$ ,mr4 ,/ubbs4 ,:o said !re 7 ?ree8 ,y h told u t ,p>k]1 y\r uncle1 & y\r nephew w5t up on ?is bless$ fly+-ma*9e4 ,t's "r4 ,& ! ?+ w\ld only c>ry two6 ,"r aga94 ,wilson1 ,i h kn[n y = "s "t z a tru?ful man & a temp]ate man1 said ,mr4 ,/ubbs1 solemnly4 ,b ,i am afraid s9ce y took up t new l9e ( goods y h ov]work$ y\rself4 ,half a m9ute1 ,/ubbs1 int]pos$ ,mr4 ,wat]son4 ,i see cle>ly :]e we all slipp$ a cog4 ,( c\rse1 ,wilson1 y meant u to und]/& t ,p>k] is ei!r y\r uncle or y\r nephew4 ,n[ we %all be all "r if y w j tell u :e!r ,p>k] is y\r uncle or nephew4 ,he is no relation to me :atev]4 ,! ?ree m5 sigh$ & look$ anxi\sly at "o ano!r4 ,mr4 ,/ubbs got up f 8 *air to rea* ! mat*es1 ,mr4 ,packhur/ proce$$ to w9d up 8 wat*1 & ,mr4 ,wat]son took up ! pok] to att5d to ! fire4 ,x 0 an awkw>d mom5t1 = at ! season ( goodwill nobody wi%$ to tell ,mr4 ,wilson exactly :at 0 9 8 m9d4 ,x's curi\s1 said ,mr4 ,wilson1 v delib]ately1 & x's r sad1 h[ ?ick-head$ "s p are4 ,y don't seem to grip ! facts4 ,x nev] seems to h occurr$ to ei!r ( y t my uncle & my nephew are "o & ! same man4 ,:at6 exclaim$ all ?ree toge!r4 ,yes2 ,david ,george ,l9klat] is my uncle1 & he is also my nephew4 ,consequ5tly1 ,i am bo? 8 uncle & nephew4 ,que]1 isn't x8 ,i'll expla9 h[ x comes ab\t4 ,mr4 ,wilson put ! case s v simply t ! ?ree m5 saw h[ x might happ5 )\t any m>riage )9 ! prohibit$ degrees4 ,p]haps ! read] c "w x out = himself4 ,,clock ,,puzzles4 ,look at ! clock6 ,+oldsby ,leg5ds4 ,9 consid]+ a few puzzles conc]n+ clocks & wat*es1 & ! times record$ by _! h&s "u giv5 conditions1 x is well t a p>ticul> conv5tion %\ld always be kept 9 m9d4 ,x is frequ5tly ! case t a solution requires ! assumption t ! h&s c actually record a "t involv+ a m9ute fraction ( a second4 ,su* a "t1 ( c\rse1 _c be really indicat$4 ,is ! puzzle1 !re=e1 impossible ( solution8 ,! conclusion d$uc$ f a logical syllogism dep5ds = its tru? on ! two premises assum$1 & x is ! same 9 ma!matics4 ,c]ta9 ?+s are antec$5tly assum$1 & ! answ] dep5ds entirely on ! tru? ( ^? assumptions4 ,if two horses1 says ,lagrange1 c pull a load ( a c]ta9 weight1 x is natural to suppose t f\r horses c\ld pull a load ( d\ble t weight1 six horses a load ( ?ree times t weight4 ,yet1 /rictly speak+1 su* is n ! case4 ,= ! inf]5ce is bas$ on ! assumption t ! f\r horses pull alike 9 am\nt & direction1 :i* 9 practice c sc>cely ev] be ! case4 ,x s happ5s t we are frequ5tly l$ 9 our reckon+s to results :i* div]ge widely f reality4 ,b ! fault is n ! fault ( ma!matics2 = ma!matics always gives back to u exactly :at we h put into x4 ,! ratio 0 con/ant accord+ to t supposition4 ,! result is f\nd$ ^u t supposition4 ,if ! supposition is false ! result is necess>ily false4 ,if "o man c reap a field 9 six days1 we say two m5 w reap x 9 ?ree days1 & ?ree m5 w d ! "w 9 two days4 ,we "h assume1 z 9 ! case ( ,lagrange's horses1 t all ! m5 are exactly equally capable ( "w4 ,b we assume ev5 m ?an ?is4 ,= :5 ?ree m5 get toge!r !y may wa/e "t 9 gossip or play2 or1 on ! o!r h&1 a _s ( rivalry may spur !m on to great] dilig5ce4 ,we may assume any conditions we l 9 a problem1 provid$ !y be cle>ly express$ & und]/ood1 & ! answ] w be 9 accordance ) ^? conditions4 #eg4--,,:at ,,0 ,,! ,,"t8 ,i say1 ,rackbrane1 :at is ! "t8 an acqua9tance ask$ our fri5d ! pr(essor ! o!r day4 ,! answ] 0 c]ta9ly curi\s4 ,if y add "o qu>t] ( ! "t f noon till n[ to half ! "t f n[ till noon to-morr[1 y w get ! "t exactly4 ,:at 0 ! "t ( day :5 ! pr(essor spoke8 #eh4--;,a ,,"t ,,puzzle4 ,h[ _m m9utes is x until six o'clock if fifty m9utes ago x 0 f\r times z _m m9utes pa/ ?ree o'clock8 #ei4--;,a ,,puzzl+ ,,wat*4 ,a fri5d pull$ out 8 wat* & said1 ,?is wat* ( m9e does n keep p]fect "t2 ,i mu/ h x se5 to4 ,i h notic$ t ! m9ute h& & ! h\r h& are exactly toge!r e sixty-five m9utes4 ,does t wat* ga9 or lose1 & h[ mu* p] h\r8 #fj4--,,! ,,wap%aw',s ,,:>f ,,my/]y4 ,!re 0 a great commotion 9 ,l[] ,?ames ,/reet on ! morn+ ( ,janu>y #ab1 #ahhg4 ,:5 ! e>ly memb]s ( ! /aff arriv$ at ,wap%aw's ,:>f !y f\nd t ! safe _h be5 brok5 op51 a consid]able sum ( money remov$1 & ! (fices left 9 great disord]4 ,! night wat*man 0 n[h]e to be f\nd1 b nobody :o _h be5 acqua9t$ ) him = "o mom5t suspect$ him to be guilty ( ! robb]y4 ,9 ?is belief ! proprietors 7 confirm$ :51 lat] 9 ! day1 !y 7 in=m$ t ! poor fell['s body _h be5 pick$ up by ! ,riv] ,police4 ,c]ta9 m>ks ( viol5ce po9t$ to ! fact t he _h be5 brutally attack$ & ?r[n into ! riv]4 ,a wat* f\nd 9 8 pocket _h /opp$1 z is inv>iably ! case 9 su* circum/ances1 & ?is 0 a valuable clue to ! "t ( ! outrage4 ,b a v /upid (fic] 7& we inv>iably f9d "o or two /upid individuals 9 ! mo/ intellig5t bodies ( m57 _h actually amus$ himself by turn+ ! h&s r\nd & r\nd1 try+ to set ! wat* go+ aga94 ,aft] he _h be5 sev]ely reprim&$ = ?is s]i\s indiscretion1 he 0 ask$ :e!r he c\ld rememb] ! "t t 0 indicat$ by ! wat* :5 f\nd4 ,he repli$ t he c\ld n1 b he recollect$ t ! h\r h& & m9ute h& 7 exactly toge!r1 "o above ! o!r1 & ! second h& _h j pass$ ! =ty-n9? second4 ,m ?an ?is he c\ld n rememb]4 ,:at 0 ! exact "t at :i* ! wat*man's wat* /opp$8 ,! wat* is1 ( c\rse1 assum$ to h be5 an accurate "o4 #fa4--;,,*ang+ ,,places4 ,illu/ration ,! above clock face indicates a little be=e #db m9utes pa/ #d4 ,! h&s w aga9 po9t at exactly ! same spots a little aft] #bc m9utes pa/ #h4 ,9 fact1 ! h&s w h *ang$ places4 ,h[ _m times d ! h&s ( a clock *ange places betwe5 ?ree o'clock p4m4 & midnight8 ,& out ( all ! pairs ( times indicat$ by ^! *anges1 :at is ! exact "t :5 ! m9ute h& w be ne>e/ to ! po9t ,,ix8 #fb4--,,! ,,club ,,clock4 ,"o ( ! big clocks 9 ! ,cogitators' ,club 0 f\nd ! o!r night to h /opp$ j :51 z w be se5 9 ! illu/ration1 ! second h& 0 exactly midway betwe5 ! o!r two h&s4 ,"o ( ! memb]s propos$ to "s ( 8 fri5ds t !y %\ld tell him ! exact "t :5 7if ! clock _h n /opp$7 ! second h& w\ld next aga9 h be5 midway betwe5 ! m9ute h& & ! h\r h&4 ,c y f9d ! correct "t t x w\ld happ58 ,illu/ration #fc4--,,! ,,/op-,,wat*4 ,illu/ration ,we h "h a /op-wat* ) ?ree h&s4 ,! second h&1 :i* travels once r\nd ! face 9 a m9ute1 is ! "o ) ! little r+ at its end ne> ! c5tre4 ,our dial indicates ! exact "t :5 its own] /opp$ ! wat*4 ,y w notice t ! ?ree h&s are ne>ly equidi/ant4 ,! h\r & m9ute h&s po9t to spots t are exactly a ?ird ( ! circumf]5ce ap>t1 b ! second h& is a little too advanc$4 ,an exact equidi/ance = ! ?ree h&s is n possible4 ,n[1 we want to "k :at ! "t w be :5 ! ?ree h&s are next at exactly ! same di/ances z %[n f "o ano!r4 ,c y /ate ! "t8 #fd4--,,! ,,?ree ,,clocks4 ,on ,friday1 ,april #a1 #ahih1 ?ree new clocks 7 all set go+ precisely at ! same "t--twelve noon4 ,at noon on ! foll[+ day x 0 f\nd t clock ,a _h kept p]fect "t1 t clock ,b _h ga9$ exactly "o m9ute1 & t clock ,c _h lo/ exactly "o m9ute4 ,n[1 suppos+ t ! clocks ,b & ,c _h n be5 regulat$1 b all ?ree all[$ to g on z !y _h begun1 & t !y ma9ta9$ ! same rates ( progress )\t /opp+1 on :at date & at :at "t ( day w\ld all ?ree pairs ( h&s aga9 po9t at ! same mom5t at twelve o'clock8 #fe4--,,! ,,railway ,,/ation ,,clock4 ,a clock hangs on ! wall ( a railway /ation1 #ga ft4 #i 94 long & #aj ft4 #d 94 high4 ,^? are ! dim5sions ( ! wall1 n ( ! clock6 ,:ile wait+ = a tra9 we notic$ t ! h&s ( ! clock 7 po9t+ 9 opposite directions1 & 7 p>allel to "o ( ! diagonals ( ! wall4 ,:at 0 ! exact "t8 #ff4--,,! ,,village ,,simpleton4 ,a faceti\s individual :o 0 tak+ a long walk 9 ! c\ntry came ^u a yokel sitt+ on a /ile4 ,z ! g5tleman 0 n q sure ( 8 road1 he ?\ght he w\ld make inquiries ( ! local inhabitant2 b at ! fir/ glance he jump$ too ha/ily to ! conclusion t he _h dropp$ on ! village idiot4 ,he !re=e decid$ to te/ ! fell['s intellig5ce by fir/ putt+ to him ! simple/ "q he c\ld ?9k (1 :i* 01 ,:at day ( ! week is ?is1 my good man8 ,! foll[+ is ! sm>t answ] t he receiv$3-- ,:5 ! day aft] to-morr[ is ye/]day1 to-day w be z f> f ,sunday z to-day 0 f ,sunday :5 ! day be=e ye/]day 0 to-morr[4 ,c ! read] say :at day ( ! week x 08 ,x is pretty evid5t t ! c\ntryman 0 n su* a fool z he look$4 ,! g5tleman w5t on 8 road a puzzl$ b a wis] man4 ,,locomotion ,,& ,,spe$ ,,puzzles4 ,! race is n to ! swift4--,ecclesia/es ix4 ,,ii4 #fg4--;,,av]age ,,spe$4 ,9 a rec5t motor ride x 0 f\nd t we _h gone at ! rate ( t5 miles an h\r1 b we did ! return j\rney ov] ! same r\te1 ow+ to ! roads be+ m cle> ( traffic1 at fifte5 miles an h\r4 ,:at 0 our av]age spe$8 ,d n be too ha/y 9 y\r answ] to ?is simple little "q1 or x is pretty c]ta9 t y w be wrong4 #fh4--,,! ,,two ,,tra9s4 ,i put ?is little "q to a /ationma/]1 & 8 correct answ] 0 s prompt t ,i am conv9c$ !re is no necessity to seek tal5t$ railway (ficials 9 ,am]ica or else:]e4 ,two tra9s />t at ! same "t1 "o f ,london to ,liv]pool1 ! o!r f ,liv]pool to ,london4 ,if !y arrive at _! de/9ations "o h\r & f\r h\rs respectively aft] pass+ "o ano!r1 h[ mu* fa/] is "o tra9 runn+ ?an ! o!r8 #fi4--,,! ,,?ree ,,villages4 ,i set out ! o!r day to ride 9 a motor-c> f ,acrefield to ,butt]=d1 b by mi/ake ,i took ! road go+ via ,*eesebury1 :i* is ne>] ,acrefield ?an ,butt]=d1 & is twelve miles to ! left ( ! direct road ,i %\ld h travell$4 ,aft] arriv+ at ,butt]=d ,i f\nd t ,i _h gone ?irty-five miles4 ,:at are ! ?ree di/ances betwe5 ^! villages1 ea* be+ a :ole numb] ( miles8 ,i may m5tion t ! ?ree roads are q /raight4 #gj4--;,,draw+ ,,h] ,,p5sion4 ,speak+ ( odd figures1 said a g5tleman :o occupies "s po/ 9 a ,gov]nm5t (fice1 "o ( ! que]e/ *>act]s ,i "k is an old lame wid[ :o climbs up a hill e week to draw h] p5sion at ! village po/ (fice4 ,%e crawls up at ! rate ( a mile & a half an h\r & comes d[n at ! rate ( f\r & a half miles an h\r1 s t x takes h] j six h\rs to make ! d\ble j\rney4 ,c any ( y tell me h[ f> x is f ! bottom ( ! hill to ! top8 ,illu/ration #ga4--,,sir ,,edwyn ,,de ,,tudor4 ,9 ! illu/ration we h a sket* ( ,sir ,edwyn de ,tudor go+ to rescue 8 lady-love1 ! fair ,isabella1 :o 0 held a captive by a neighb\r+ wick$ b>on4 ,sir ,edwyn calculat$ t if he rode fifte5 miles an h\r he w\ld arrive at ! ca/le an h\r too soon1 :ile if he rode t5 miles an h\r he w\ld get !re j an h\r too late4 ,n[1 x 0 ( ! fir/ importance t he %\ld arrive at ! exact "t appo9t$1 9 ord] t ! rescue t he _h plann$ %\ld be a success1 & ! "t ( ! try/ 0 five o'clock1 :5 ! captive lady w\ld be tak+ h] aft]noon tea4 ,! puzzle is to discov] exactly h[ f> ,sir ,edwyn de ,tudor _h to ride4 #gb4--,,! ,,hydroplane ,,"q4 ,! inhabitants ( ,slocomb-on-,sea 7 greatly excit$ ov] ! visit ( a c]ta9 fly+ man4 ,all ! t[n turn$ out to see ! flight ( ! wond]ful hydroplane1 &1 ( c\rse1 ,dobson & 8 family 7 !re4 ,ma/] ,tommy 0 9 good =m1 & in=m$ 8 fa!r t ,engli%m5 made bett] airm5 ?an ,scotsm5 & ,iri%m5 because !y are n s heavy4 ,h[ d y make t out8 ask$ ,mr4 ,dobson4 ,well1 y see1 ,tommy repli$1 x is true t 9 ,irel& !re are m5 ( ,cork & 9 ,scotl& m5 ( ,ayr1 :i* is bett] /ill1 b 9 ,engl& !re are light]m54 ,un=tunately x _h to be expla9$ to ,mrs4 ,dobson1 & ?is took ! edge (f ! ?+4 ,! hydroplane flight 0 f ,slocomb to ! neighb\r+ wat]+-place ,poodleville--five miles di/ant4 ,b !re 0 a /rong w9d1 :i* s help$ ! airman t he made ! outw>d j\rney 9 ! %ort "t ( t5 m9utes1 ?\gh x took him an h\r to get back to ! />t+ po9t at ,slocomb1 ) ! w9d dead aga9/ him4 ,n[1 h[ long w\ld ! t5 miles h tak5 him if !re _h be5 a p]fect calm8 ,( c\rse1 ! hydroplane's eng9e work$ uni=mly ?r\gh\t4 #gc4--;,,donkey ,,rid+4 ,dur+ a visit to ! seaside ,tommy & ,evangel9e insi/$ on hav+ a donkey race ov] ! mile c\rse on ! s&s4 ,mr4 ,dobson & "s ( 8 fri5ds :om he _h met on ! bea* act$ z judges1 b1 z ! donkeys 7 famili> acqua9tances & decl9$ to "p company ! :ole way1 a dead heat 0 unavoidable4 ,h[ev]1 ! judges1 be+ /ation$ at diff]5t po9ts on ! c\rse1 :i* 0 m>k$ (f 9 qu>t]-miles1 not$ ! foll[+ results3--,! fir/ ?ree-qu>t]s 7 run 9 six & ?ree-qu>t] m9utes1 ! fir/ half-mile took ! same "t z ! second half1 & ! ?ird qu>t] 0 run 9 exactly ! same "t z ! la/ qu>t]4 ,f ^! results ,mr4 ,dobson amus$ himself 9 discov]+ j h[ long x took ^? two donkeys to run ! :ole mile4 ,c y give ! answ]8 #gd4--,,! ,,basket ,,( ,,potatoes4 ,a man _h a basket conta9+ fifty potatoes4 ,he propos$ to 8 son1 z a little recreation1 t he %\ld place ^! potatoes on ! gr\nd 9 a /raight l9e4 ,! di/ance betwe5 ! fir/ & second potatoes 0 to be "o y>d1 betwe5 ! second & ?ird ?ree y>ds1 betwe5 ! ?ird & f\r? five y>ds1 betwe5 ! f\r? & fif? sev5 y>ds1 & s on--an increase ( two y>ds = e successive potato laid d[n4 ,!n ! boy 0 to pick !m up & put !m 9 ! basket "o at a "t1 ! basket be+ plac$ beside ! fir/ potato4 ,h[ f> w\ld ! boy h to travel to accompli% ! feat ( pick+ !m all up8 ,we w n consid] ! j\rney involv$ 9 plac+ ! potatoes1 s t he />ts f ! basket ) !m all laid out4 #ge4--,,! ,,pass5g]',s ,,f>e4 ,at fir/ sight y w\ld h>dly ?9k !re 0 matt] = dispute 9 ! "q involv$ 9 ! foll[+ little incid5t1 yet x took ! two p]sons conc]n$ "s little "t to come to an agreem5t4 ,mr4 ,smi!rs hir$ a motor-c> to take him f ,addle=d to ,cl9k]ville & back aga9 = #c4 ,at ,bak5ham1 j midway1 he pick$ up an acqua9tance1 ,mr4 ,tompk9s1 & agre$ to take him on to ,cl9k]ville & br+ him back to ,bak5ham on ! return j\rney4 ,h[ mu* %\ld he h *>g$ ! pass5g]8 ,t is ! "q4 ,:at 0 a reasonable f>e = ,mr4 ,tompk9s8 ,,digital ,,puzzles4 ,n9e wor?ies 7 !y call$4 ,,dryd53 ,! ,fl[] & ! ,leaf4 ,i give ^! puzzles1 deal+ ) ! n9e digits1 a class to !mselves1 because ,i h always ?\ght t !y des]ve m consid]ation ?an !y usually receive4 ,beyond ! m]e trick ( ca/+ out n9es1 v little seems to be g5]ally kn[n ( ! laws involv$ 9 ^! problems1 & yet an acqua9tance ) ! prop]ties ( ! digits (t5 supplies1 among o!r uses1 a c]ta9 numb] ( ari?metical *ecks t are ( real value 9 ! sav+ ( lab\r4 ,let me give j "o example--! fir/ t occurs to me4 ,if ! read] 7 requir$ to det]m9e :e!r or n #ae1gfc1ecj1afc1bhi is a squ>e numb]1 h[ w\ld he proce$8 ,if ! numb] _h end$ ) a #b1 #c1 #g1 or #h 9 ! digits place1 ( c\rse he w\ld "k t x c\ld n be a squ>e1 b !re is no?+ 9 its app>5t =m to prev5t its be+ "o4 ,i suspect t 9 su* a case he w\ld set to "w1 ) a sigh or a groan1 at ! labori\s task ( extract+ ! squ>e root4 ,yet if he _h giv5 a little att5tion to ! /udy ( ! digital prop]ties ( numb]s1 he w\ld settle ! "q 9 ?is simple way4 ,! sum ( ! digits is #ei1 ! sum ( :i* is #ad1 ! sum ( :i* is #e 7:i* ,i call ! digital root71 & !re=e ,i "k t ! numb] _c be a squ>e1 & = ?is reason4 ,! digital root ( successive squ>e numb]s f #a upw>ds is always #a1 #d1 #g1 or #i1 & c nev] be any?+ else4 ,9 fact1 ! s]ies1 #a1 #d1 #i1 #g1 #g1 #i1 #d1 #a1 #i1 is repeat$ into inf9ity4 ,! analog\s s]ies = triangul> numb]s is #a1 #c1 #f1 #a1 #f1 #c1 #a1 #i1 #i4 ,s "h we h a simil> negative *eck1 = a numb] _c be triangul> 7t is1 7n n7_/#b7 if its digital root be #b1 #d1 #e1 #g1 or #h4 #gf4--,,! ,,b>rel ,,( ,,be]4 ,a man b\ght an odd lot ( w9e 9 b>rels & "o b>rel conta9+ be]4 ,^! are %[n 9 ! illu/ration1 m>k$ ) ! numb] ( gallons t ea* b>rel conta9$4 ,he sold a quantity ( ! w9e to "o man & twice ! quantity to ano!r1 b kept ! be] to himself4 ,! puzzle is to po9t out :i* b>rel conta9s be]4 ,c y say :i* "o x is8 ,( c\rse1 ! man sold ! b>rels j z he b\ght !m1 )\t manipulat+ 9 any way ! cont5ts4 ,illu/ration3 7 #ae ,gals 7 7#ca ,gals7 7#ai ,gals7 7#bj ,gals7 7#af ,gals7 7#ah ,gals7 #gg4--;,,digits ,,& ,,squ>es4 ,illu/ration3 --------- #a #i #b --------- #c #h #d --------- #e #g #f --------- ,x w be se5 9 ! diagram t we h s arrang$ ! n9e digits 9 a squ>e t ! numb] 9 ! second r[ is twice t 9 ! fir/ r[1 & ! numb] 9 ! bottom r[ ?ree times t 9 ! top r[4 ,!re are ?ree o!r ways ( arrang+ ! digits s z to produce ! same result4 ,c y f9d !m8 #gh4--,,odd ,,& ,,ev5 ,,digits4 ,! odd digits1 #a1 #c1 #e1 #g1 & #i1 add up #be1 :ile ! ev5 figures1 #b1 #d1 #f1 & #h1 only add up #bj4 ,arrange ^! figures s t ! odd ones & ! ev5 ones add up alike4 ,complex & improp] fractions & recurr+ decimals are n all[$4 #gi4--,,! ,,lock]s ,,puzzle4 ,illu/ration3 ,a ,b ,c -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- ------------------ ------------------ ------------------ ,a man _h 9 8 (fice ?ree cupbo>ds1 ea* conta9+ n9e lock]s1 z %[n 9 ! diagram4 ,he told 8 cl]k to place a diff]5t "o-figure numb] on ea* lock] ( cupbo>d ,a1 & to d ! same 9 ! case ( ,b1 & ( ,c4 ,z we are "h all[$ to call n\ght a digit1 & he 0 n prohibit$ f us+ n\ght z a numb]1 he cle>ly _h ! option ( omitt+ any "o ( t5 digits f ea* cupbo>d4 ,n[1 ! employ] did n say ! lock]s 7 to be numb]$ 9 any num]ical ord]1 & he 0 surpris$ to f9d1 :5 ! "w 0 done1 t ! figures _h app>5tly be5 mix$ up indiscrim9ately4 ,call+ ^u 8 cl]k = an explanation1 ! ecc5tric lad /at$ t ! notion _h occurr$ to him s to arrange ! figures t 9 ea* case !y =m$ a simple addition sum1 ! two upp] r[s ( figures produc+ ! sum 9 ! l[e/ r[4 ,b ! mo/ surpris+ po9t 0 ?is3 t he _h s arrang$ !m t ! addition 9 ,a gave ! smalle/ possible sum1 t ! addition 9 ,c gave ! l>ge/ possible sum1 & t all ! n9e digits 9 ! ?ree totals 7 diff]5t4 ,! puzzle is to %[ h[ ?is c\ld be done4 ,no decimals are all[$ & ! n\ght may n appe> 9 ! hundr$s place4 #hj4--,,! ,,?ree ,,gr\ps4 ,!re appe>$ 9 ,n\velles ,annales de ,ma?matiques ! foll[+ puzzle z a modification ( "o ( my ,cant]bury ,puzzles4 ,arrange ! n9e digits 9 ?ree gr\ps ( two1 ?ree1 & f\r digits1 s t ! fir/ two numb]s :5 multipli$ toge!r make ! ?ird4 ,?us1 #ab #dhc #e1gif4 ,i n[ also propose to include ! cases :]e !re are "o1 f\r1 & f\r digits1 su* z #d #a1gch #f1ieb4 ,c y f9d all ! possible solutions 9 bo? cases8 #ha4--,,! ,,n9e ,,c\nt]s4 ,illu/ration3 7#a77e77h7 7#g77i7 7#b77c7 7#d77f7 ,i h n9e c\nt]s1 ea* be>+ "o ( ! n9e digits1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h & #i4 ,i arrang$ !m on ! table 9 two gr\ps1 z %[n 9 ! illu/ration1 s z to =m two multiplication sums1 & f\nd t bo? sums gave ! same product4 ,y w f9d t #aeh multipli$ by #bc is #c1fcd1 & t #gi multipli$ by #df is also #c1fcd4 ,n[1 ! puzzle ,i propose is to re>range ! c\nt]s s z to get z l>ge a product z possible4 ,:at is ! be/ way ( plac+ !m8 ,rememb] bo? gr\ps mu/ multiply to ! same am\nt1 & !re mu/ be ?ree c\nt]s multipli$ by two 9 "o case1 & two multipli$ by two c\nt]s 9 ! o!r1 j z at pres5t4 #hb4--,,! ,,t5 ,,c\nt]s4 ,9 ?is case we use ! n\ght 9 addition to ! #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i4 ,! puzzle is1 z 9 ! la/ case1 s to arrange ! t5 c\nt]s t ! products ( ! two multiplications %all be ! same1 & y may "h h "o or m figures 9 ! multipli]1 z y *oose4 ,! above is a v easy feat2 b x is also requir$ to f9d ! two arrangem5ts giv+ pairs ( ! highe/ & l[e/ products possible4 ,( c\rse e c\nt] mu/ be us$1 & ! ciph] may n be plac$ to ! left ( a r[ ( figures :]e x w\ld h no effect4 ,vulg> fractions or decimals are n all[$4 #hc4--;,,digital ,,multiplication4 ,"h is ano!r ent]ta9+ problem ) ! n9e digits1 ! n\ght be+ exclud$4 ,us+ ea* figure once1 & only once1 we c =m two multiplication sums t h ! same product1 & ?is may be done 9 _m ways4 ,= example1 #g #feh & #ad #cbi conta9 all ! digits once1 & ! product 9 ea* case is ! same--#d1fjf4 ,n[1 x w be se5 t ! sum ( ! digits 9 ! product is #af1 :i* is nei!r ! highe/ nor ! l[e/ sum s obta9able4 ,c y f9d ! solution ( ! problem t gives ! l[e/ possible sum ( digits 9 ! common product8 ,also t :i* gives ! highe/ possible sum8 #hd4--,,! ,,pi]rot',s ,,puzzle4 ,illu/ration ,! ,pi]rot 9 ! illu/ration is /&+ 9 a po/ure t repres5ts ! sign ( multiplication4 ,he is indicat+ ! peculi> fact t #ae multipli$ by #ic produces exactly ! same figures 7#a1cie71 diff]5tly arrang$4 ,! puzzle is to take any f\r digits y l 7all diff]5t7 & simil>ly arrange !m s t ! numb] =m$ on "o side ( ! ,pi]rot :5 multipli$ by ! numb] on ! o!r side %all produce ! same figures4 ,!re are v few ways ( do+ x1 & ,i %all give all ! cases possible4 ,c y f9d !m all8 ,y are all[$ to put two figures on ea* side ( ! ,pi]rot z 9 ! example %[n1 or to place a s+le figure on "o side & ?ree figures on ! o!r4 ,if we only us$ ?ree digits in/ead ( f\r1 ! only possible ways are ^!3 #c multipli$ by #ea equals #aec1 & #f multipli$ by #ba equals #abf4 #he4--,,! ,,cab ,,numb]s4 ,a ,london policeman "o night saw two cabs drive (f 9 opposite directions "u suspici\s circum/ances4 ,?is (fic] 0 a p>ticul>ly c>eful & wide-awake man1 & he took out 8 pocket-book to make an entry ( ! numb]s ( ! cabs1 b discov]$ t he _h lo/ 8 p5cil4 ,luckily1 h[ev]1 he f\nd a small piece ( *alk1 ) :i* he m>k$ ! two numb]s on ! gateway ( a :>f close by4 ,:5 he return$ to ! same spot on 8 beat he /ood & look$ aga9 at ! numb]s1 & notic$ ?is peculi>ity1 t all ! n9e digits 7no n\ght7 7 us$ & t no figure 0 repeat$1 b t if he multipli$ ! two numb]s toge!r !y aga9 produc$ ! n9e digits1 all once1 & once only4 ,:5 "o ( ! cl]ks arriv$ at ! :>f 9 ! e>ly morn+1 he obs]v$ ! *alk m>ks & c>efully rubb$ !m out4 ,z ! policeman c\ld n rememb] !m1 c]ta9 ma!maticians 7 !n consult$ z to :e!r !re 0 any kn[n me?od = discov]+ all ! pairs ( numb]s t h ! peculi>ity t ! (fic] _h notic$2 b !y knew ( none4 ,! inve/igation1 h[ev]1 0 int]e/+1 & ! foll[+ "q out ( _m 0 propos$3 ,:at two numb]s1 conta9+ toge!r all ! n9e digits1 w1 :5 multipli$ toge!r1 produce ano!r numb] 7! highe/ possible7 conta9+ also all ! n9e digits8 ,! n\ght is n all[$ any:]e4 #hf4--,,que] ,,multiplication4 ,if ,i multiply #ea1bdi1hgf by #c 7?us us+ all ! n9e digits once1 & once only71 ,i get #aec1gdi1fbh 7:i* aga9 conta9s all ! n9e digits once74 ,simil>ly1 if ,i multiply #af1ehc1gdb by #i ! result is #adi1bec1fgh1 :]e 9 ea* case all ! n9e digits are us$4 ,n[1 take #f z y\r multipli] & try to arrange ! rema9+ eight digits s z to produce by multiplication a numb] conta9+ all n9e once1 & once only4 ,y w f9d x f> f easy1 b x c be done4 #hg4--,,! ,,numb]-,,*ecks ,,puzzle4 ,illu/ration ,:]e a l>ge numb] ( workm5 are employ$ on a build+ x is cu/om>y to provide e man ) a little disc be>+ 8 numb]4 ,^! are hung on a bo>d by ! m5 z !y arrive1 & s]ve z a *eck on punctuality4 ,n[1 ,i once notic$ a =eman remove a numb] ( ^! *ecks f 8 bo>d & place !m on a split-r+ :i* he c>ri$ 9 8 pocket4 ,?is at once gave me ! idea = a good puzzle4 ,9 fact1 ,i w confide to my read]s t ?is is j h[ ideas = puzzles arise4 ,y _c really create an idea3 x happ5s--& y h to be on ! al]t to seize x :5 x does s happ54 ,x w be se5 f ! illu/ration t !re are t5 ( ^! *ecks on a r+1 numb]$ #a to #i & #j4 ,! puzzle is to divide !m into ?ree gr\ps )\t tak+ any (f ! r+1 s t ! fir/ gr\p multipli$ by ! second makes ! ?ird gr\p4 ,= example1 we c divide !m into ! ?ree gr\ps1 #b--h #i #j #g--a #e #d #f #c1 by br++ ! #f & ! #c r\nd to ! #d1 b un=tunately ! fir/ two :5 multipli$ toge!r d n make ! ?ird4 ,c y sep>ate !m correctly8 ,( c\rse y may h z _m ( ! *ecks z y l 9 any gr\p4 ,! puzzle calls = "s +5uity1 unless y h ! luck to hit on ! answ] by *ance4 #hh4--;,,digital ,,division4 ,x is ano!r good puzzle s to arrange ! n9e digits 7! n\ght exclud$7 into two gr\ps s t "o gr\p :5 divid$ by ! o!r produces a giv5 numb] )\t rema9d]4 ,= example1 #a #c #d #e #h divid$ by #f #g #b #i gives #b4 ,c ! read] f9d simil> arrangem5ts produc+ #c1 #d1 #e1 #f1 #g1 #h1 & #i respectively8 ,also1 c he f9d ! pairs ( smalle/ possible numb]s 9 ea* case8 ,?us1 #a #d #f #e #h divid$ by #g #c #b #i is j z correct = #b z ! o!r example we h giv51 b ! numb]s are high]4 #hi4--;,,add+ ,,! ,,digits4 ,if ,i write ! sum ( money1 #ihg1 #es4 #d;d41 & add up ! digits1 !y sum to #cf4 ,no digit has ?us be5 us$ a second "t 9 ! am\nt or addition4 ,?is is ! l>ge/ am\nt possible "u ! conditions4 ,n[ f9d ! smalle/ possible am\nt1 p\nds1 %ill+s1 p5ce1 & f>?+s be+ all repres5t$4 ,y ne$ n use m ( ! n9e digits ?an y *oose1 b no digit may be repeat$ ?r\gh\t4 ,! n\ght is n all[$4 #ij4--,,! ,,c5tury ,,puzzle4 ,c y write #ajj 9 ! =m ( a mix$ numb]1 us+ all ! n9e digits once1 & only once8 ,! late di/+ui%$ ,fr5* ma!matician1 ,ed\>d ,lucas1 f\nd sev5 diff]5t ways ( do+ x1 & express$ 8 d\bts z to !re be+ any o!r ways4 ,z a matt] ( fact !re are j elev5 ways & no m4 ,"h is "o ( !m1 #iaegdb_/fch4 ,n9e ( ! o!r ways h simil>ly two figures 9 ! integral "p ( ! numb]1 b ! elev5? expression has only "o figure !re4 ,c ! read] f9d ?is la/ =m8 #ia4--,,m ,,mix$ ,,fractions4 ,:5 ,i fir/ publi%$ my solution to ! la/ puzzle1 ,i 0 l$ to attempt ! expression ( all numb]s 9 turn up to #ajj by a mix$ fraction conta9+ all ! n9e digits4 ,"h are twelve numb]s = ! read] to try 8 h& at3 #ac1 #ad1 #ae1 #af1 #ah1 #bj1 #bg1 #cf1 #dj1 #fi1 #gb1 #id4 ,use e "o ( ! n9e digits once1 & only once1 9 e case4 #ib4--;,,digital ,,squ>e ,,numb]s4 ,"h are ! n9e digits s arrang$ t !y =m f\r squ>e numb]s3 #i1 #ha1 #cbd1 #egf4 ,n[1 c y put !m all toge!r s z to =m a s+le squ>e numb]--7,i7 ! smalle/ possible1 & 7,,ii7 ! l>ge/ possible8 #ic4--,,! ,,my/ic ,,elev54 ,c y f9d ! l>ge/ possible numb] conta9+ any n9e ( ! t5 digits 7call+ n\ght a digit7 t c be divid$ by #aa )\t a rema9d]8 ,c y also f9d ! smalle/ possible numb] produc$ 9 ! same way t is divisible by #aa8 ,"h is an example1 :]e ! digit #e has be5 omitt$3 #hifgdcjab4 ,?is numb] conta9s n9e ( ! digits & is divisible by #aa1 b x is nei!r ! l>ge/ nor ! smalle/ numb] t w "w4 #id4--,,! ,,digital ,,c5tury4 #a #b #c #d #e #f #g #h #i #ajj4 ,x is requir$ to place ari?metical signs betwe5 ! n9e figures s t !y %all equal #ajj4 ,( c\rse1 y mu/ n alt] ! pres5t num]ical arrangem5t ( ! figures4 ,c y give a correct solution t employs 7#a7 ! fewe/ possible signs1 & 7#b7 ! fewe/ possible sep>ate /rokes or dots ( ! p58 ,t is1 x is necess>y to use z few signs z possible1 & ^? signs %\ld be ( ! simple/ =m4 ,! signs ( addition & multiplication 7 & 7 w ?us c\nt z two /rokes1 ! sign ( subtraction 7-7 z "o /roke1 ! sign ( division 77 z ?ree1 & s on4 #ie4--,,! ,,f\r ,,sev5s4 ,illu/ration ,9 ! illu/ration ,pr(essor ,rackbrane is se5 demon/rat+ "o ( ! little pos]s ) :i* he is accu/om$ to ent]ta9 8 class4 ,he believes t by tak+ 8 pupils (f ! beat5 tracks he is ! bett] able to secure _! att5tion1 & to induce orig9al & +5i\s me?ods ( ?\ght4 ,he has1 x w be se51 j %[n h[ f\r #e's may be writt5 ) simple ari?metical signs s z to repres5t #ajj4 ,e juv5ile read] w see at a glance t 8 example is q correct4 ,n[1 :at he wants y to d is ?is3 ,arrange f\r #g's 7nei!r m nor less7 ) ari?metical signs s t !y %all repres5t #ajj4 ,if he _h said we 7 to use f\r #i's we might at once h writt5 #iii_/i1 b ! f\r #g's call = r m +5uity4 ,c y discov] ! little trick8 #if4--,,! ,,dice ,,numb]s4 ,illu/ration ,i h a set ( f\r dice1 n m>k$ ) spots 9 ! ord9>y way1 b ) ,arabic figures1 z %[n 9 ! illu/ration4 ,ea* die1 ( c\rse1 be>s ! numb]s #a to #f4 ,:5 put toge!r !y w =m a good _m1 diff]5t numb]s4 ,z repres5t$ !y make ! numb] #abdf4 ,n[1 if ,i make all ! diff]5t f\r-figure numb]s t are possible ) ^! dice 7nev] putt+ ! same figure m ?an once 9 any numb]71 :at w !y all add up to8 ,y are all[$ to turn ! #f upside d[n1 s z to repres5t a #i4 ,i d n ask1 or expect1 ! read] to g to all ! lab\r ( writ+ out ! full li/ ( numb]s & !n add+ !m up4 ,life is n long en\gh = su* wa/$ en]gy4 ,c y get at ! answ] 9 any o!r way8 ,,v>i\s ,,ari?metical ,,& ,,algebraical ,,problems4 ,v>iety's ! v spice ( life1 ,t gives x all its flav\r4 ,,c[p]3 ,! ,task4 #ig4--,,! ,,spot ,,on ,,! ,,table4 ,a boy1 rec5tly home f s*ool1 wi%$ to give 8 fa!r an exhibition ( 8 precocity4 ,he pu%$ a l>ge circul> table into ! corn] ( ! room1 z %[n 9 ! illu/ration1 s t x t\*$ bo? walls1 & he !n po9t$ to a spot ( ink on ! extreme edge4 ,illu/ration ,"h is a little puzzle = y1 pat]1 said ! y\?4 ,t spot is exactly eight in*es f "o wall & n9e in*es f ! o!r4 ,c y tell me ! diamet] ( ! table )\t measur+ x8 ,! boy 0 ov]he>d to tell a fri5d1 ,x fairly beat ! guv'nor2 b 8 fa!r is kn[n to h rem>k$ to a ,city acqua9tance t he solv$ ! ?+ 9 8 head 9 a m9ute4 ,i (t5 wond] :i* spoke ! tru?4 #ih4--;,,academic ,,c\rtesies4 ,9 a c]ta9 mix$ s*ool1 :]e a special feature 0 made ( ! inculcation ( good mann]s1 !y _h a curi\s rule on assembl+ e morn+4 ,!re 7 twice z _m girls z boys4 ,e girl made a b[ to e o!r girl1 to e boy1 & to ! tea*]4 ,e boy made a b[ to e o!r boy1 to e girl1 & to ! tea*]4 ,9 all !re 7 n9e hundr$ b[s made 9 t model academy e morn+4 ,n[1 c y say exactly h[ _m boys !re 7 9 ! s*ool8 ,if y are n v c>eful1 y are likely to get a good deal out 9 y\r calculation4 #ii4--,,! ,,?irty-,,?ree ,,pe>ls4 ,illu/ration ,a man ,i "k1 said ,t$dy ,ni*olson at a c]ta9 family p>ty1 possesses a /r+ ( ?irty-?ree pe>ls4 ,! middle pe>l is ! l>ge/ & be/ ( all1 & ! o!rs are s select$ & arrang$ t1 />t+ f "o end1 ea* successive pe>l is wor? #ajj m ?an ! prec$+ "o1 "r up to ! big pe>l4 ,f ! o!r end ! pe>ls increase 9 value by #aej up to ! l>ge pe>l4 ,! :ole /r+ is wor? #fe1jjj4 ,:at is ! value ( t l>ge pe>l8 ,pe>ls & o!r articles ( clo?+1 said ,uncle ,walt]1 :5 ! price ( ! preci\s gem _h be5 discov]$1 rem9d me ( ,adam & ,eve4 ,au?orities1 y may n "k1 diff] z to ! numb] ( apples t 7 eat5 by ,adam & ,eve4 ,x is ! op9ion ( "s t ,eve #h 7ate7 & ,adam #b 7too71 a total ( #aj only4 ,b c]ta9 ma!maticians h figur$ x out diff]5tly1 & hold t ,eve #h & ,adam a total ( #af4 ,yet ! mo/ rec5t inve/igators ?9k ! above figures entirely wrong1 = if ,eve #h & ,adam #h #b1 ! total mu/ be #ij4 ,well1 said ,h>ry1 x seems to me t if !re 7 giants 9 ^? days1 probably ,eve #h #a & ,adam #h #b1 :i* w\ld give a total ( #afc4 ,i am n at all satisfi$1 said ,maud4 ,x seems to me t if ,eve #h #a & ,adam #h #a #b1 !y toge!r consum$ #hic4 ,i am sure y are all wrong1 insi/$ ,mr4 ,wilson1 = ,i consid] t ,eve #h #a #d ,adam1 & ,adam #h #a #b #d ,eve1 s we get a total ( #h1ich4 ,b1 look "h1 broke 9 ,h]b]t4 ,if ,eve #h #a #d ,adam & ,adam #h #a #b #d #b oblige ,eve1 surely ! total mu/ h be5 #hb1jef6 ,at ?is po9t ,uncle ,walt] sugge/$ t !y might let ! matt] re/4 ,he decl>$ x to be cle>ly :at ma!maticians call an indet]m9ate problem4 #ajj4--,,! ,,lab\r]',s ,,puzzle4 ,pr(essor ,rackbrane1 dur+ "o ( 8 rambles1 *anc$ to come ^u a man digg+ a deep hole4 ,good morn+1 he said4 ,h[ deep is t hole8 ,guess1 repli$ ! lab\r]4 ,my height is exactly five feet t5 in*es4 ,h[ mu* deep] are y go+8 said ! pr(essor4 ,i am go+ twice z deep1 0 ! answ]1 & !n my head w be twice z f> bel[ gr\nd z x is n[ above gr\nd4 ,rackbrane n[ asks if y c\ld tell h[ deep t hole w\ld be :5 f9i%$4 #aja4--,,! ,,trusses ,,( ,,hay4 ,f>m] ,tompk9s _h five trusses ( hay1 :i* he told 8 man ,hodge to weigh be=e deliv]+ !m to a cu/om]4 ,! /upid fell[ weigh$ !m two at a "t 9 all possible ways1 & in=m$ 8 ma/] t ! weights 9 p\nds 7 #aaj1 #aab1 #aac1 #aad1 #aae1 #aaf1 #aag1 #aah1 #abj1 & #aba4 ,n[1 h[ 0 ,f>m] ,tompk9s to f9d out f ^! figures h[ mu* e "o ( ! five trusses weigh$ s+ly8 ,! read] may at fir/ ?9k t he ought to be told :i* pair is :i* pair1 or some?+ ( t sort1 b x is q unnecess>y4 ,c y give ! five correct weights8 #ajb4--,,mr4 ,,gubb9s ,,9 ,a ,,fog4 ,mr4 ,gubb9s1 a dilig5t man ( bus9ess1 0 mu* inconv5i5c$ by a ,london fog4 ,! electric light happ5$ to be out ( ord] & he _h to manage z be/ he c\ld ) two c&les4 ,8 cl]k assur$ him t ?\gh bo? 7 ( ! same l5g? "o c&le w\ld burn = f\r h\rs & ! o!r = five h\rs4 ,aft] he _h be5 work+ "s "t he put ! c&les out z ! fog _h lift$1 & he !n notic$ t :at rema9$ ( "o c&le 0 exactly f\r times ! l5g? ( :at 0 left ( ! o!r4 ,:5 he got home t night ,mr4 ,gubb9s1 :o lik$ a good puzzle1 said to himself1 ,( c\rse x is possible to "w out j h[ long ^? two c&les 7 burn+ to-day4 ,i'll h a %ot at x4 ,b he soon f\nd himself 9 a worse fog ?an ! atmosph]ic "o4 ,c\ld y h assi/$ him 9 8 dilemma8 ,h[ long 7 ! c&les burn+8 #ajc4--,,pa9t+ ,,! ,,lamp-,,po/s4 ,tim ,murphy & ,pat ,donovan 7 engag$ by ! local au?orities to pa9t ! lamp-po/s 9 a c]ta9 /reet4 ,tim1 :o 0 an e>ly ris]1 arriv$ fir/ on ! job1 & _h pa9t$ ?ree on ! s\? side :5 ,pat turn$ up & po9t$ out t ,tim's contract 0 = ! nor? side4 ,s ,tim />t$ afre% on ! nor? side & ,pat cont9u$ on ! s\?4 ,:5 ,pat _h f9i%$ 8 side he w5t across ! /reet & pa9t$ six po/s = ,tim1 & !n ! job 0 f9i%$4 ,z !re 0 an equal numb] ( lamp-po/s on ea* side ( ! /reet1 ! simple "q is3 ,:i* man pa9t$ ! m lamp-po/s1 & j h[ _m m8 #ajd4--;,,cat*+ ,,! ,,?ief4 ,n[1 con/able1 said ! def5dant's c\nsel 9 cross-exam9ation1 y say t ! prison] 0 exactly tw5ty-sev5 /eps ahead ( y :5 y />t$ to run aft] him8 ,yes1 sir4 ,& y swe> t he takes eight /eps to y\r five8 ,t is s4 ,!n ,i ask y1 con/able1 z an intellig5t man1 to expla9 h[ y ev] caught him1 if t is ! case8 ,well1 y see1 ,i h got a long] /ride4 ,9 fact1 two ( my /eps are equal 9 l5g? to five ( ! prison]'s4 ,if y "w x out1 y w f9d t ! numb] ( /eps ,i requir$ w\ld br+ me exactly to ! spot :]e ,i captur$ him4 ,"h ! =eman ( ! jury ask$ = a few m9utes to figure out ! numb] ( /eps ! con/able mu/ h tak54 ,c y also say h[ _m /eps ! (fic] ne$$ to cat* ! ?ief8 #aje4--,,! ,,p>i% ,,c\ncil ,,election4 ,"h is an easy problem = ! novice4 ,at ! la/ election ( ! p>i% c\ncil ( ,tittlebury-9-!-,m>% !re 7 tw5ty-?ree c&idates = n9e seats4 ,ea* vot] 0 qualifi$ to vote = n9e ( ^! c&idates or = any less numb]4 ,"o ( ! electors wants to "k 9 j h[ _m diff]5t ways x 0 possible = him to vote4 #ajf4--,,! ,,muddlet[n ,,election4 ,at ! la/ ,p>liam5t>y election at ,muddlet[n #e1dgc votes 7 poll$4 ,! ,lib]al 0 elect$ by a majority ( #ah ov] ! ,cons]vative1 by #adf ov] ! ,indep5d5t1 & by #ege ov] ! ,sociali/4 ,c y give a simple rule = figur+ out h[ _m votes 7 poll$ = ea* c&idate8 #ajg4--,,! ,,suffragi/s' ,,meet+4 ,at a rec5t secret meet+ ( ,suffragi/s a s]i\s diff]5ce ( op9ion arose4 ,?is l$ to a split1 & a c]ta9 numb] left ! meet+4 ,i _h half a m9d to g myself1 said ! *air-woman1 & if ,i _h done s1 two-?irds ( u w\ld h retir$4 ,true1 said ano!r memb]2 b if ,i _h p]suad$ my fri5ds ,mrs4 ,wild & ,*ri/9e ,arm/rong to rema9 we %\ld only h lo/ half our numb]4 ,c y tell h[ _m 7 pres5t at ! meet+ at ! />t8 #ajh4--,,! ,,leap-,,ye> ,,ladies4 ,la/ leap-ye> ladies lo/ no "t 9 ex]cis+ ! privilege ( mak+ proposals ( m>riage4 ,if ! figures t rea*$ me f an occult s\rce are correct1 ! foll[+ repres5ts ! /ate ( affairs 9 ?is c\ntry4 ,a numb] ( wom5 propos$ once ea*1 ( :om "o-eigh? 7 wid[s4 ,9 consequ5ce1 a numb] ( m5 7 to be m>ri$ ( :om "o-elev5? 7 wid[]s4 ,( ! proposals made to wid[]s1 "o-fif? 7 decl9$4 ,all ! wid[s 7 accept$4 ,?irty-five =ty-f\r?s ( ! wid[s m>ri$ ba*elors4 ,"o ?\s& two hundr$ & tw5ty-"o sp9/]s 7 decl9$ by ba*elors4 ,! numb] ( sp9/]s accept$ by ba*elors 0 sev5 times ! numb] ( wid[s accept$ by ba*elors4 ,^? are all ! p>ticul>s t ,i 0 able to obta94 ,n[1 h[ _m wom5 propos$8 #aji4--,,! ,,great ,,scramble4 ,aft] d9n]1 ! five boys ( a h\sehold happ5$ to f9d a p>cel ( sug>-plums4 ,x 0 q unexpect$ loot1 & an excit+ scramble ensu$1 ! full details ( :i* ,i w rec\nt ) accuracy1 z x =ms an int]e/+ puzzle4 ,y see1 ,&rew manag$ to get possession ( j two-?irds ( ! p>cel ( sug>-plums4 ,bob at once grabb$ ?ree-eigh?s ( ^!1 & ,*>lie manag$ to seize ?ree-t5?s also4 ,!n "y ,david da%$ ^u ! sc5e1 & captur$ all t ,&rew _h left1 except "o-sev5?1 :i* ,edg> artfully secur$ = himself by a cunn+ trick4 ,n[ ! fun began 9 real e>ne/1 = ,&rew & ,*>lie jo9tly set ^u ,bob1 :o /umbl$ aga9/ ! f5d] & dropp$ half ( all t he _h1 :i* 7 equally pick$ up by ,david & ,edg>1 :o _h crawl$ "u a table & 7 wait+4 ,next1 ,bob sprang on ,*>lie f a *air1 & upset all ! latt]'s collection on to ! floor4 ,( ?is prize ,&rew got j a qu>t]1 ,bob ga!r$ up "o-?ird1 ,david got two-sev5?s1 :ile ,*>lie & ,edg> divid$ equally :at 0 left ( t /ock4 ,illu/ration ,!y 7 j ?9k+ ! fray 0 ov] :5 ,david sudd5ly /ruck out 9 two directions at once1 upsett+ ?ree-qu>t]s ( :at ,bob & ,&rew _h la/ acquir$4 ,! two latt]1 ) ! greate/ difficulty1 recov]$ five-eigh?s ( x 9 equal %>es1 b ! ?ree o!rs ea* c>ri$ (f "o-fif? ( ! same4 ,e sug>-plum 0 n[ acc\nt$ =1 & !y call$ a truce1 & divid$ equally among/ !m ! rema9d] ( ! p>cel4 ,:at is ! smalle/ numb] ( sug>-plums !re c\ld h be5 at ! />t1 & :at proportion did ea* boy obta98 #aaj4--,,! ,,abbot',s ,,puzzle4 ,! fir/ ,engli% puzzli/ ^: "n has come d[n to u 0 a ,york%ireman--no o!r ?an ,alcu91 ,abbot ( ,cant]bury 7,a4,d4 #gce-hjd74 ,"h is a little puzzle f 8 works1 :i* is at lea/ int]e/+ on acc\nt ( its antiquity4 ,if #ajj bu%els ( corn 7 di/ribut$ among #ajj p 9 su* a mann] t ea* man receiv$ ?ree bu%els1 ea* woman two1 & ea* *ild half a bu%el1 h[ _m m51 wom51 & *ildr5 7 !re8 ,n[1 !re are six diff]5t correct answ]s1 if we exclude a case :]e !re w\ld be no wom54 ,b let u say t !re 7 j five times z _m wom5 z m51 !n :at is ! correct solution8 #aaa4--,,reap+ ,,! ,,corn4 ,a f>m] _h a squ>e cornfield4 ,! corn 0 all ripe = reap+1 &1 z he 0 %ort ( m51 x 0 arrang$ t he & 8 son %\ld %>e ! "w betwe5 !m4 ,! f>m] fir/ cut "o rod wide all r\nd ! squ>e1 ?us leav+ a small] squ>e ( /&+ corn 9 ! middle ( ! field4 ,n[1 he said to 8 son1 ,i h cut my half ( ! field1 & y c d y\r %>e4 ,! son 0 n q satisfi$ z to ! propos$ division ( lab\r1 & z ! village s*oolma/] happ5$ to be pass+1 he appeal$ to t p]son to decide ! matt]4 ,he f\nd ! f>m] 0 q correct1 provid$ !re 0 no dispute z to ! size ( ! field1 & on ?is po9t !y 7 agre$4 ,c y tell ! area ( ! field1 z t +5i\s s*oolma/] succe$$ 9 do+8 #aab4--;,a ,,puzzl+ ,,legacy4 ,a man left a hundr$ acres ( l& to be divid$ among 8 ?ree sons--,alfr$1 ,b5jam91 & ,*>les--9 ! proportion ( "o-?ird1 "o-f\r?1 & "o-fif? respectively4 ,b ,*>les di$4 ,h[ 0 ! l& to be divid$ fairly betwe5 ,alfr$ & ,b5jam98 #aac4--,,! ,,torn ,,numb]4 ,illu/ration ,i _h ! o!r day 9 my possession a label be>+ ! numb] #c #j #b #e 9 l>ge figures4 ,?is got accid5tally torn 9 half1 s t #c #j 0 on "o piece & #b #e on ! o!r1 z %[n on ! illu/ration4 ,on look+ at ^! pieces ,i began to make a calculation1 sc>cely consci\s ( :at ,i 0 do+1 :5 ,i discov]$ ?is little peculi>ity4 ,if we add ! #c #j & ! #b #e toge!r & squ>e ! sum we get z ! result ! complete orig9al numb] on ! label6 ,?us1 #cj add$ to #be is #ee1 & #ee multipli$ by #ee is #cjbe4 ,curi\s1 is x n8 ,n[1 ! puzzle is to f9d ano!r numb]1 compos$ ( f\r figures1 all diff]5t1 :i* may be divid$ 9 ! middle & produce ! same result4 #aad4--;,,curi\s ,,numb]s4 ,! numb] #dh has ?is peculi>ity1 t if y add #a to x ! result is a squ>e numb] 7#di1 ! squ>e ( #g71 & if y add #a to its half1 y also get a squ>e numb] 7#be1 ! squ>e ( #e74 ,n[1 !re is no limit to ! numb]s t h ?is peculi>ity1 & x is an int]e/+ puzzle to f9d ?ree m ( !m--! smalle/ possible numb]s4 ,:at are !y8 #aae4--;,a ,,pr9t]',s ,,error4 ,9 a c]ta9 article a pr9t] _h to set up ! figures #edbc1 :i*1 ( c\rse1 means t ! f\r? p[] ( #e 7#fbe7 is to be multipli$ by ! cube ( #b 7#h71 ! product ( :i* is #e1jjj4 ,b he pr9t$ #edbc z #e #d #b #c1 :i* is n correct4 ,c y place f\r digits 9 ! mann] %[n1 s t x w be equally correct if ! pr9t] sets x up aright or makes ! same blund]8 #aaf4--,,! ,,conv]t$ ,,mis]4 ,mr4 ,jasp] ,bullyon 0 "o ( ! v few mis]s :o h ev] be5 conv]t$ to a s5se ( _! duty t[>ds _! less =tunate fell[-m54 ,"o ev5tful night he c\nt$ out 8 accumulat$ weal?1 & resolv$ to di/ribute x among/ ! des]v+ poor4 ,he f\nd t if he gave away ! same numb] ( p\nds e day 9 ! ye>1 he c\ld exactly spread x ov] a twelvemon? )\t !re be+ any?+ left ov]2 b if he re/$ on ! ,sundays1 & only gave away a fix$ numb] ( p\nds e weekday1 !re w\ld be "o sov]eign left ov] on ,new ,ye>'s ,eve4 ,n[1 putt+ x at ! l[e/ possible1 :at 0 ! exact numb] ( p\nds t he _h to di/ribute8 ,c\ld any "q be simpl]8 ,a sum ( p\nds divid$ by "o numb] ( days leaves no rema9d]1 b divid$ by ano!r numb] ( days leaves a sov]eign ov]4 ,t is all2 & yet1 :5 y come to tackle ?is little "q1 y w be surpris$ t x c become s puzzl+4 #aag4--;,a ,,f5ce ,,problem4 ,illu/ration ,! practical usefulness ( puzzles is a po9t t we are liable to ov]look4 ,yet1 z a matt] ( fact1 ,i h f "t to "t receiv$ q a l>ge numb] ( lett]s f individuals :o h f\nd t ! ma/]+ ( "s little pr9ciple ^u :i* a puzzle 0 built has prov$ ( consid]able value to !m 9 a mo/ unexpect$ way4 ,inde$1 x may be accept$ z a good maxim t a puzzle is ( little real value unless1 z well z be+ amus+ & p]plex+1 x conceals "s in/ructive & possibly useful feature4 ,x is1 h[ev]1 v curi\s h[ ^! little bits ( acquir$ k dovetail into ! occasional requirem5ts ( ev]yday life1 & equally curi\s to :at /range & my/]i\s uses "s ( our read]s seem to apply !m4 ,:at1 = example1 c be ! object ( ,mr4 ,wm4 ,oxley1 :o writes to me all ! way f ,i[a1 9 wi%+ to asc]ta9 ! dim5sions ( a field t he proposes to enclose1 conta9+ j z _m acres z !re %all be rails 9 ! f5ce8 ,! man wi%es to f5ce 9 a p]fectly squ>e field :i* is to conta9 j z _m acres z !re are rails 9 ! requir$ f5ce4 ,ea* hurdle1 or portion ( f5ce1 is sev5 rails high1 & two l5g?s w\ld ext5d "o pole 7#af ft473 t is to say1 !re are f\rte5 rails to ! pole1 l9eal measure4 ,n[1 :at mu/ be ! size ( ! field8 #aah4--;,,circl+ ,,! ,,squ>es4 ,illu/ration ,! puzzle is to place a diff]5t numb] 9 ea* ( ! t5 squ>es s t ! sum ( ! squ>es ( any two adjac5t numb]s %all be equal to ! sum ( ! squ>es ( ! two numb]s diametrically opposite to !m4 ,! f\r numb]s plac$1 z examples1 mu/ /& z !y are4 ,! squ>e ( #af is #bef1 & ! squ>e ( #b is #d4 ,add ^! toge!r1 & ! result is #bfj4 ,also--! squ>e ( #ad is #aif1 & ! squ>e ( #h is #fd4 ,^! toge!r also make #bfj4 ,n[1 9 precisely ! same way1 ,b & ,c %\ld be equal to ,g & ,h 7! sum w n necess>ily be #bfj71 ,a & ,k to ,f & ,e1 ,h & ,i to ,c & ,d1 & s on1 ) any two adjo9+ squ>es 9 ! circle4 ,all y h to d is to fill 9 ! rema9+ six numb]s4 ,fractions are n all[$1 & ,i %all %[ t no numb] ne$ conta9 m ?an two figures4 #aai4--,,rackbrane',s ,,little ,,loss4 ,pr(essor ,rackbrane 0 sp5d+ an ev5+ ) 8 old fri5ds1 ,mr4 & ,mrs4 ,potts1 & !y engag$ 9 "s game 7he does n say :at game7 ( c>ds4 ,! pr(essor lo/ ! fir/ game1 :i* result$ 9 d\bl+ ! money t bo? ,mr4 & ,mrs4 ,potts _h laid on ! table4 ,! second game 0 lo/ by ,mrs4 ,potts1 :i* d\bl$ ! money !n held by h] husb& & ! pr(essor4 ,curi\sly en\gh1 ! ?ird game 0 lo/ by ,mr4 ,potts1 & _h ! effect ( d\bl+ ! money !n held by 8 wife & ! pr(essor4 ,x 0 !n f\nd t ea* p]son _h exactly ! same money1 b ! pr(essor _h lo/ five %ill+s 9 ! c\rse ( play4 ,n[1 ! pr(essor asks1 :at 0 ! sum ( money ) :i* he sat d[n at ! table8 ,c y tell him8 #abj4--,,! ,,f>m] ,,& ,,8 ,,%eep4 ,illu/ration ,f>m] ,longmore _h a curi\s aptitude = ari?metic1 & 0 kn[n 9 8 di/rict z ! ma!matical f>m]4 ,! new vic> 0 n aw>e ( ?is fact :51 meet+ 8 wor?y p>i%ion] "o day 9 ! lane1 he ask$ him 9 ! c\rse ( a %ort conv]sation1 ,n[1 h[ _m %eep h y altoge!r8 ,he 0 !re=e r surpris$ at ,longmore's answ]1 :i* 0 z foll[s3 ,y c divide my %eep into two diff]5t p>ts1 s t ! diff]5ce betwe5 ! two numb]s is ! same z ! diff]5ce betwe5 _! squ>es4 ,maybe1 ,mr4 ,p>son1 y w l to "w out ! little sum = y\rself4 ,c ! read] say j h[ _m %eep ! f>m] _h8 ,suppos+ he _h possess$ only tw5ty %eep1 & he divid$ !m into ! two p>ts #ab & #h4 ,n[1 ! diff]5ce betwe5 _! squ>es1 #add & #fd1 is #hj4 ,s t w n d1 = #d & #hj are c]ta9ly n ! same4 ,if y c f9d numb]s t "w out correctly1 y w "k exactly h[ _m %eep ,f>m] ,longmore own$4 #aba4--;,,heads ,,or ,,tails4 ,crooks1 an invet]ate gambl]1 at ,goodwood rec5tly said to a fri5d1 ,i'll bet y half ! money 9 my pocket on ! toss ( a co9--heads ,i w91 tails ,i lose4 ,! co9 0 toss$ & ! money h&$ ov]4 ,he repeat$ ! (f] aga9 & aga91 ea* "t bett+ half ! money !n 9 8 possession4 ,we are n told h[ long ! game w5t on1 or h[ _m times ! co9 0 toss$1 b ?is we "k1 t ! numb] ( times t ,crooks lo/ 0 exactly equal to ! numb] ( times t he won4 ,n[1 did he ga9 or lose by ?is little v5ture8 #abb4--,,! ,,see-,,saw ,,puzzle4 ,necessity is1 inde$1 ! "m ( inv5tion4 ,i 0 amus$ ! o!r day 9 wat*+ a boy :o want$ to play see-saw &1 9 8 failure to f9d ano!r *ild to %>e ! sport ) him1 _h be5 driv5 back ^u ! +5i\s resort ( ty+ a numb] ( bricks to "o end ( ! plank to balance 8 weight at ! o!r4 ,z a matt] ( fact1 he j balanc$ aga9/ sixte5 bricks1 :5 ^! 7 fix$ to ! %ort end ( plank1 b if he fix$ !m to ! long end ( plank he only ne$$ elev5 z balance4 ,n[1 :at 0 t boy's weight1 if a brick weighs equal to a ?ree-qu>t] brick & ?ree-qu>t]s ( a p\nd8 #abc4--;,a ,,legal ,,difficulty4 ,a cli5t ( m9e1 said a lawy]1 0 on ! po9t ( dea? :5 8 wife 0 ab\t to pres5t him ) a *ild4 ,i drew up 8 w1 9 :i* he settl$ two-?irds ( 8 e/ate ^u 8 son 7if x %\ld happ5 to be a boy7 & "o-?ird on ! "m4 ,b if ! *ild %\ld be a girl1 !n two-?irds ( ! e/ate %\ld g to ! "m & "o-?ird to ! daught]4 ,z a matt] ( fact1 aft] 8 dea? tw9s 7 born--a boy & a girl4 ,a v nice po9t !n arose4 ,h[ 0 ! e/ate to be equitably divid$ among ! ?ree 9 ! close/ possible accordance ) ! _s ( ! dead man's w8 #abd4--;,a ,,"q ,,( ,,def9ition4 ,my prop]ty is exactly a mile squ>e1 said "o l&[n] to ano!r4 ,curi\sly en\gh1 m9e is a squ>e mile1 0 ! reply4 ,!n !re is no diff]5ce8 ,is ?is la/ /atem5t correct8 #abe4--,,! ,,m9]s' ,,holiday4 ,sev5 coal-m9]s took a holiday at ! seaside dur+ a big /rike4 ,six ( ! p>ty sp5t exactly half a sov]eign ea*1 b ,bill ,h>ris 0 m extravagant4 ,bill sp5t ?ree %ill+s m ?an ! av]age ( ! p>ty4 ,:at 0 ! actual am\nt ( ,bill's exp5diture8 #abf4--,,simple ,,multiplication4 ,if we numb] six c>ds #a1 #b1 #d1 #e1 #g1 & #h1 & arrange !m on ! table 9 ?is ord]3-- #a #d #b #h #e #g ,we c demon/rate t 9 ord] to multiply by #c all t is necess>y is to remove ! #a to ! o!r end ( ! r[1 & ! ?+ is done4 ,! answ] is #dbhega4 ,c y f9d a numb] t1 :5 multipli$ by #c & divid$ by #b1 ! answ] w be ! same z if we remov$ ! fir/ c>d 7:i* 9 ?is case is to be a #c7 ,f ! beg9n+ ( ! r[ to ! end8 #abg4--,,simple ,,division4 ,sometimes a v simple "q 9 elem5t>y ari?metic w cause a good deal ( p]plexity4 ,= example1 ,i want to divide ! f\r numb]s1 #gja1 #a1jei1 #a1dag1 & #b1cab1 by ! l>ge/ numb] possible t w leave ! same rema9d] 9 e case4 ,h[ am ,i to set to "w ,( c\rse1 by a labori\s sy/em ( trial "o c 9 "t discov] ! answ]1 b !re is q a simple me?od ( do+ x if y c only f9d x4 #abh4--;,a ,,problem ,,9 ,,squ>es4 ,we possess ?ree squ>e bo>ds4 ,! surface ( ! fir/ conta9s five squ>e feet m ?an ! second1 & ! second conta9s five squ>e feet m ?an ! ?ird4 ,c y give exact measurem5ts = ! sides ( ! bo>ds8 ,if y c solve ?is little puzzle1 !n try to f9d ?ree squ>es 9 ari?metical progression1 ) a common diff]5ce ( #g & also ( #ac4 #abi4--,,! ,,battle ,,( ,,ha/+s4 ,all hi/orians "k t !re is a great deal ( my/]y & unc]ta9ty conc]n+ ! details ( ! ev]-memorable battle on t fatal day1 ,octob] #ad1 #ajff4 ,my puzzle deals ) a curi\s passage 9 an anci5t monki% *ronicle t may nev] receive ! att5tion t x des]ves1 & if ,i am unable to v\* = ! au!nticity ( ! docum5t x w none ! less s]ve to furni% u ) a problem t c h>dly fail to int]e/ ^? ( my read]s :o h ari?metical pr$ilections4 ,"h is ! passage 9 "q4 ,! m5 ( ,h>old /ood well toge!r1 z _! wont 01 & =m$ sixty & "o squ>es1 ) a l numb] ( m5 9 e squ>e !re(1 & woe to ! h>dy ,norman :o v5tur$ to ent] _! r$\bts2 = a s+le bl[ ( a ,saxon w>-hat*et w\ld break 8 lance & cut ?r\gh 8 coat ( mail4444 ,:5 ,h>old ?rew himself into ! fray ! ,saxons 7 "o mighty squ>e ( m51 %\t+ ! battle-cries1 ',ut6' ',olicrosse6' ',godemit6' ,n[1 ,i f9d t all ! contempor>y au?orities agree t ! ,saxons did actually fight 9 ?is solid ord]4 ,= example1 9 ! ,c>m5 de ,bello ,ha/+5si1 a poem attribut$ to ,guy1 ,bi%op ( ,ami5s1 liv+ at ! "t ( ! battle1 we are told t ! ,saxons /ood fix$ 9 a d5se mass1 & ,h5ry ( ,hunt+don records t !y 7 l unto a ca/le1 imp5etrable to ! ,normans2 :ile ,rob]t ,wace1 a c5tury aft]1 tells u ! same ?+4 ,s 9 ?is respect my newly-discov]$ *ronicle may n be greatly 9 error4 ,b ,i h reason to believe t !re is some?+ wrong ) ! actual figures4 ,let ! read] see :at he c make ( !m4 ,! numb] ( m5 w\ld be sixty-"o times a squ>e numb]2 b :5 ,h>old himself jo9$ 9 ! fray !y 7 !n able to =m "o l>ge squ>e4 ,:at is ! smalle/ possible numb] ( m5 !re c\ld h be58 ,9 ord] to make cle> to ! read] ! simplicity ( ! "q1 ,i w give ! l[e/ solutions 9 ! case ( #fj & #fb1 ! numb]s imm$iately prec$+ & foll[+ #fa4 ,!y are #fj #d #a #ca1 & #fb #h #a #fc4 ,t is1 #fj squ>es ( #af m5 ea* w\ld be #ifj m51 & :5 ,h>old jo9$ !m !y w\ld be #ifa 9 numb]1 & s =m a squ>e ) #ca m5 on e side4 ,simil>ly 9 ! case ( ! figures ,i h giv5 = #fb4 ,n[1 f9d ! l[e/ answ] = #fa4 #acj4--,,! ,,sculptor',s ,,problem4 ,an anci5t sculptor 0 commission$ to supply two /atues1 ea* on a cubical p$e/al4 ,x is ) ^! p$e/als t we are conc]n$4 ,!y 7 ( unequal sizes1 z w be se5 9 ! illu/ration1 & :5 ! "t arriv$ = paym5t a dispute arose z to :e!r ! agreem5t 0 bas$ on l9eal or cubical measurem5t4 ,b z soon z !y came to measure ! two p$e/als ! matt] 0 at once settl$1 because1 curi\sly en\gh1 ! numb] ( l9eal feet 0 exactly ! same z ! numb] ( cubical feet4 ,! puzzle is to f9d ! dim5sions = two p$e/als hav+ ?is peculi>ity1 9 ! smalle/ possible figures4 ,y see1 if ! two p$e/als1 = example1 measure respectively #c ft4 & #a ft4 on e side1 !n ! l9eal measurem5t w\ld be #d ft4 & ! cubical cont5ts #bh ft41 :i* are n ! same1 s ^! measurem5ts w n d4 ,illu/ration #aca4--,,! ,,spani% ,,mis]4 ,!re once liv$ 9 a small t[n 9 ,new ,ca/ile a not$ mis] nam$ ,don ,manuel ,rodriguez4 ,8 love ( money 0 only equall$ by a /rong passion = ari?metical problems4 ,^! puzzles usually dealt 9 "s way or o!r ) 8 accumulat$ treasure1 & 7 prop\nd$ by him solely 9 ord] t he might h ! pleasure ( solv+ !m himself4 ,un=tunately v few ( !m h surviv$1 & :5 travell+ ?r\gh ,spa91 collect+ mat]ial = a propos$ "w on ,! ,spani% ,onion z a ,cause ( ,national ,decad5ce1 ,i only discov]$ a v few4 ,"o ( ^! conc]ns ! ?ree boxes t appe> 9 ! accompany+ au!ntic portrait4 ,illu/ration ,ea* box conta9$ a diff]5t numb] ( gold5 d\bloons4 ,! diff]5ce betwe5 ! numb] ( d\bloons 9 ! upp] box & ! numb] 9 ! middle box 0 ! same z ! diff]5ce betwe5 ! numb] 9 ! middle box & ! numb] 9 ! bottom box4 ,& if ! cont5ts ( any two ( ! boxes 7 unit$ !y w\ld =m a squ>e numb]4 ,:at is ! smalle/ numb] ( d\bloons t !re c\ld h be5 9 any "o ( ! boxes8 #acb4--,,! ,,n9e ,,treasure ,,boxes4 ,! foll[+ puzzle w illu/rate ! importance on occasions ( be+ able to fix ! m9imum & maximum limits ( a requir$ numb]4 ,?is c v frequ5tly be done4 ,= example1 x has n yet be5 asc]ta9$ 9 h[ _m diff]5t ways ! knight's t\r c be p]=m$ on ! *ess bo>d2 b we "k t x is few] ?an ! numb] ( comb9ations ( #afh ?+s tak5 #fc at a "t & is great] ?an #ca1jed1add--;= ! latt] is ! numb] ( r\tes ( a p>ticul> type4 ,or1 to take a m famili> case1 if y ask a man h[ _m co9s he has 9 8 pocket1 he may tell y t he has n ! slighte/ idea4 ,b on fur!r que/ion+ y w get out ( him "s su* /atem5t z ! foll[+3 ,yes1 ,i am positive t ,i h m ?an ?ree co9s1 & equally c]ta9 t !re are n s _m z tw5ty-five4 ,n[1 ! k t a c]ta9 numb] lies betwe5 #b & #ab 9 my puzzle w enable ! solv] to f9d ! exact answ]2 )\t t in=mation !re w\ld be an inf9ite numb] ( answ]s1 f :i* x w\ld be impossible to select ! correct "o4 ,?is is ano!r puzzle receiv$ f my fri5d ,don ,manuel ,rodriguez1 ! cranky mis] ( ,new ,ca/ile4 ,on ,new ,ye>'s ,eve 9 #ahgi he %[$ me n9e treasure boxes1 & aft] in=m+ me t e box conta9$ a squ>e numb] ( gold5 d\bloons1 & t ! diff]5ce betwe5 ! cont5ts ( ,a & ,b 0 ! same z betwe5 ,b & ,c1 ,d & ,e1 ,e & ,f1 ,g & ,h1 or ,h & ,i1 he reque/$ me to tell him ! numb] ( co9s 9 e "o ( ! boxes4 ,at fir/ ,i ?\ght ?is 0 impossible1 z !re w\ld be an inf9ite numb] ( diff]5t answ]s1 b on consid]ation ,i f\nd t ?is 0 n ! case4 ,i discov]$ t :ile e box conta9$ co9s1 ! cont5ts ( ,a1 ,b1 ,c increas$ 9 weight 9 alphabetical ord]2 s did ,d1 ,e1 ,f2 & s did ,g1 ,h1 ,i2 b ,d or ,e ne$ n be heavi] ?an ,c1 nor ,g or ,h heavi] ?an ,f4 ,x 0 also p]fectly c]ta9 t box ,a c\ld n conta9 m ?an a doz5 co9s at ! outside2 !re might n be half t numb]1 b ,i 0 positive t !re 7 n m ?an twelve4 ,) ?is k ,i 0 able to arrive at ! correct answ]4 ,9 %ort1 we h to discov] n9e squ>e numb]s su* t ,a1 ,b1 ,c2 & ,d1 ,e1 ,f2 & ,g1 ,h1 ,i are ?ree gr\ps 9 ari?metical progression1 ! common diff]5ce be+ ! same 9 ea* gr\p1 & ,a be+ less ?an #ab4 ,h[ _m d\bloons 7 !re 9 e "o ( ! n9e boxes8 #acc4--,,! ,,five ,,brig&s4 ,! five ,spani% brig&s1 ,alfonso1 ,b5ito1 ,c>los1 ,diego1 & ,e/eban1 7 c\nt+ _! spoils aft] a raid1 :5 x 0 f\nd t !y _h captur$ altoge!r exactly #bjj d\bloons4 ,"o ( ! b& po9t$ out t if ,alfonso _h twelve times z mu*1 ,b5ito ?ree times z mu*1 ,c>los ! same am\nt1 ,diego half z mu*1 & ,e/eban "o-?ird z mu*1 !y w\ld /ill h altoge!r j #bjj d\bloons4 ,h[ _m d\bloons _h ea*8 ,!re are a good _m equally correct answ]s to ?is "q4 ,"h is "o ( !m3 ,a #f #ab #gb ,b #ab #c #cf ,c #ag #a #ag ,d #abj #fj ,e #de #a_/c #ae #bjj #bjj ,! puzzle is to discov] exactly h[ _m diff]5t answ]s !re are1 x be+ und]/ood t e man _h some?+ & t !re is to be no fractional money--only d\bloons 9 e case4 ,?is problem1 word$ some:at diff]5tly1 0 prop\nd$ by ,t>taglia 7di$ #aeei71 & he flatt]$ himself t he _h f\nd "o solution2 b a ,fr5* ma!matician ( note 7,m4,a4 ,labosne71 9 a rec5t "w1 says t 8 read]s w be a/oni%$ :5 he assures !m t !re are #f1fci diff]5t correct answ]s to ! "q4 ,is ?is s8 ,h[ _m answ]s are !re8 #acd4--,,! ,,bank]',s ,,puzzle4 ,a bank] _h a sport+ cu/om] :o 0 always anxi\s to wag] on any?+4 ,hop+ to cure him ( 8 bad habit1 he propos$ z a wag] t ! cu/om] w\ld n be able to divide up ! cont5ts ( a box conta9+ only sixp5ces into an exact numb] ( equal piles ( sixp5ces4 ,! bank] 0 fir/ to put 9 "o or m sixp5ces 7z _m z he lik$72 !n ! cu/om] 0 to put 9 "o or m 7b 9 8 case n m ?an a p\nd 9 value71 nei!r kn[+ :at ! o!r put 94 ,la/ly1 ! cu/om] 0 to transf] f ! bank]'s c\nt] to ! box z _m sixp5ces z ! bank] desir$ him to put 94 ,! puzzle is to f9d h[ _m sixp5ces ! bank] %\ld fir/ put 9 & h[ _m he %\ld ask ! cu/om] to transf]1 s t he may h ! be/ *ance ( w9n+4 #ace4--,,! ,,/onemason',s ,,problem4 ,a /onemason once _h a l>ge numb] ( cubic blocks ( /one 9 8 y>d1 all ( exactly ! same size4 ,he _h "s v fanciful little ways1 & "o ( 8 que] notions 0 to keep ^! blocks pil$ 9 cubical heaps1 no two heaps conta9+ ! same numb] ( blocks4 ,he _h discov]$ = himself 7a fact t is well kn[n to ma!maticians7 t if he took all ! blocks conta9$ 9 any numb] ( heaps 9 regul> ord]1 beg9n+ ) ! s+le cube1 he c\ld always arrange ^? on ! gr\nd s z to =m a p]fect squ>e4 ,?is w be cle> to ! read]1 because "o block is a squ>e1 #a #h #i is a squ>e1 #a #h #bg #cf is a squ>e1 #a #h #bg #fd #ajj is a squ>e1 & s on4 ,9 fact1 ! sum ( any numb] ( consecutive cubes1 beg9n+ always ) #a1 is 9 e case a squ>e numb]4 ,"o day a g5tleman ent]$ ! mason's y>d & (f]$ him a c]ta9 price if he w\ld supply him ) a consecutive numb] ( ^! cubical heaps :i* %\ld conta9 altoge!r a numb] ( blocks t c\ld be laid out to =m a squ>e1 b ! buy] insi/$ on m ?an ?ree heaps & decl9$ to take ! s+le block because x conta9$ a flaw4 ,:at 0 ! smalle/ possible numb] ( blocks ( /one t ! mason _h to supply8 #acf4--,,! ,,sultan',s ,,army4 ,a c]ta9 ,sultan wi%$ to s5d into battle an army t c\ld be =m$ into two p]fect squ>es 9 twelve diff]5t ways4 ,:at is ! smalle/ numb] ( m5 ( :i* t army c\ld be compos$8 ,to make x cle> to ! novice1 ,i w expla9 t if !re 7 #acj m51 !y c\ld be =m$ into two squ>es 9 only two diff]5t ways--#ha & #di1 or #aba & #i4 ,( c\rse1 all ! m5 mu/ be us$ on e occasion4 #acg4--;,a ,,/udy ,,9 ,,?rift4 ,c]ta9 numb]s are call$ triangul>1 because if !y are tak5 to repres5t c\nt]s or co9s !y may be laid out on ! table s z to =m triangles4 ,! numb] #a is always reg>d$ z triangul>1 j z #a is a squ>e & a cube numb]4 ,place "o c\nt] on ! table--t is1 ! fir/ triangul> numb]4 ,n[ place two m c\nt]s b5ea? x1 & y h a triangle ( ?ree c\nt]s2 !re=e #c is triangul>4 ,next place a r[ ( ?ree m c\nt]s1 & y h a triangle ( six c\nt]s2 !re=e #f is triangul>4 ,we see t e r[ ( c\nt]s t we add1 conta9+ j "o m c\nt] ?an ! r[ above x1 makes a l>g] triangle4 ,n[1 half ! sum ( any numb] & its squ>e is always a triangul> numb]4 ,?us half ( #b #b #c2 half ( #c #c #f2 half ( #d #d #aj2 half ( #e #e #ae2 & s on4 ,s if we want to =m a triangle ) #h c\nt]s on ea* side we %all require half ( #h #h1 or #cf c\nt]s4 ,?is is a pretty little prop]ty ( numb]s4 ,be=e go+ fur!r1 ,i w "h say t if ! read] ref]s to ! ,/onemason's ,problem 7,no4 #ace7 he w rememb] t ! sum ( any numb] ( consecutive cubes beg9n+ ) #a is always a squ>e1 & ^! =m ! s]ies #a1 #c1 #f1 #aj1 etc4 ,x w n[ be und]/ood :5 ,i say t "o ( ! keys to ! puzzle 0 ! fact t ^! are always ! squ>es ( triangul> numb]s--t is1 ! squ>es ( #a1 #c1 #f1 #aj1 #ae1 #ba1 #bh1 etc41 any ( :i* numb]s we h se5 w =m a triangle4 ,e :ole numb] is ei!r triangul>1 or ! sum ( two triangul> numb]s or ! sum ( ?ree triangul> numb]s4 ,t is1 if we take any numb] we *oose we c always =m "o1 two1 or ?ree triangles ) !m4 ,! numb] #a w obvi\sly1 & uniquely1 only =m "o triangle2 "s numb]s w only =m two triangles 7z #b1 #d1 #aa1 etc472 "s numb]s w only =m ?ree triangles 7z #e1 #h1 #ad1 etc474 ,!n1 aga91 "s numb]s w =m bo? "o & two triangles 7z #f71 o!rs bo? "o & ?ree triangles 7z #c & #aj71 o!rs bo? two & ?ree triangles 7z #g & #i71 :ile "s numb]s 7l #ba7 w =m "o1 two1 or ?ree triangles1 z we desire4 ,n[ = a little puzzle 9 triangul> numb]s4 ,s&y ,mc,allister1 ( ,ab]de51 practis$ /rict dome/ic economy1 & 0 anxi\s to tra9 8 good wife 9 8 own habits ( ?rift4 ,he told h] la/ ,new ,ye>'s ,eve t :5 %e _h sav$ s _m sov]eigns t %e c\ld lay !m all out on ! table s z to =m a p]fect squ>e1 or a p]fect triangle1 or two triangles1 or ?ree triangles1 j z he might *oose to ask he w\ld add five p\nds to h] treasure4 ,soon %e w5t to h] husb& ) a little bag ( #cf 9 sov]eigns & claim$ h] rew>d4 ,x w be f\nd t ! ?irty-six co9s w =m a squ>e 7) side #f71 t !y w =m a s+le triangle 7) side #h71 t !y w =m two triangles 7) sides #e & #f71 & t !y w =m ?ree triangles 7) sides #c1 #e1 & #e74 ,9 ea* ( ! f\r cases all ! ?irty-six co9s are us$1 z requir$1 & ,s&y !re=e made 8 wife ! promis$ pres5t l an hone/ man4 ,! ,scotsman !n und]took to ext5d 8 promise = five m ye>s1 s t if next ye> ! increas$ numb] ( sov]eigns t %e has sav$ c be laid out 9 ! same f\r diff]5t ways %e w receive a second pres5t2 if %e succe$s 9 ! foll[+ ye> %e w get a ?ird pres5t1 & s on until %e has e>n$ six pres5ts 9 all4 ,n[1 h[ _m sov]eigns mu/ %e put toge!r be=e %e c w9 ! six? pres5t8 ,:at y h to d is to f9d five numb]s1 ! smalle/ possible1 high] ?an #cf1 t c be display$ 9 ! f\r ways--to =m a squ>e1 to =m a triangle1 to =m two triangles1 & to =m ?ree triangles4 ,! highe/ ( y\r five numb]s w be y\r answ]4 #ach4--,,! ,,artill]ym5',s ,,dilemma4 ,illu/ration3 ,pyramid ( cannon-balls ,m,m,m,m,m,m,mr ,,mm ,,mm3 ,m #j r,w,z,x ,m 3 ,,mwm a,x 1,,bm ,m #j,m ,m a,m,m,m,m#b,,mw #jb ,m,m,w,m,mr ,,zm4 ,m,m #h,,mm #g ,,xm#b ,,ms#b ,m4,,mmmwmmmm ,,mm ,m ,,mx i,m,m ,m#g,w #h 4 ,m r ,w ,m ,z2,m ,m #jr 2 ,m ,m ,m ,w #bb ,w ,m ,m ,m ,m4,m#b,,wmmmmz 2,,mm,x3#g,,mmmb2 ,,mmm ,,zm ,m3,,mm#j2h3 1,,ms ,ma #h ,m,m,m,m,m,m,mi r,m #b,,mmmmmm ,,mb ,m #g ,,xm1 13 ,,bmm3 r#g,s 41,,mm ,,mm ,,mb ,m i 1,m 1 #b 2 a,m,m,m,m,m,m,m,m,m ,,xm2 ,,mzm ,m 4 ,m #g ,m 4 ,z ,m ,m ,m#h ,m ,m ,s ,m 4#j ,m #h,,mm a,mi3 ,,mmmm#g,m 1#g 4i,m ,x ,m a,z ,m ,m #h 1,m,m,m,m,b,m,m,ma ,,smw #g,m1,,xz,,mm ,m #h,m ,m 4,m ,,mmmm,x ,m,mr ,ma ,,mmmmmmmmm ,m 4,,wm ,m ,,wm#g,,wmm 4,,wx ,,mzs ,m #h,m 3,,mmmwmmmm #h,x ,,mmmbmmm#g #g;a,m #b,,mm r1 #hr ,,zm#b ,mr#b a,m,m2 ,mai 3,,ms 3i,m ,zi,m ,,mx ,m ,m 4 ,m ,wr4,m,m,ma,b,m,m,m,b ,m ,m ,,mz4 1,m ,,mmz ,mr ,m ,m ,b#j ,z #b,s i,m ,s ,,xm #g ,,wmm ,,mm 4,m ,m ,m ,w ,m4 ,m ,m #j2,m#b,m2,,mmmm3 ,,ww#h;a,m,m ,m ,s ,m ,m ,m 3 ,ma,m,m,m,m,m,m ,,mm#j,w2,,mzm3 ,m i ,m ,m ,,mm ,m,m,m,z,m,b,za#j;ar ,b#bjr,m,m,m ,si i ,,bw ,,mmm#jb #g,,mm#j #b,,mmm ,,mmmmmmmm 4 ,m ,,smai,m,m#bj;,,bwm ,,xm #j,,zmmm3,,mmmmw2 r4 #j,,wmbm#b,xr,b3 4 ,all cannon-balls are to be pil$ 9 squ>e pyramids1 0 ! ord] issu$ to ! regim5t4 ,?is 0 done4 ,!n came ! fur!r ord]1 ,all pyramids are to conta9 a squ>e numb] ( balls4 ,:]eupon ! tr\ble arose4 ,x c't be done1 said ! major4 ,look at ?is pyramid1 = example2 !re are sixte5 balls at ! base1 !n n9e1 !n f\r1 !n "o at ! top1 mak+ ?irty balls 9 all4 ,b !re mu/ be six m balls1 or five few]1 to make a squ>e numb]4 ,x mu/ be done1 insi/$ ! g5]al4 ,all y h to d is to put ! "r numb] ( balls 9 y\r pyramids4 ,i've got x6 said a lieut5ant1 ! ma!matical g5ius ( ! regim5t4 ,lay ! balls out s+ly4 ,bo%6 exclaim$ ! g5]al4 ,y c't pile "o ball into a pyramid6 ,is x really possible to obey bo? ord]s8 #aci4--,,! ,,dut*m5',s ,,wives4 ,i wond] h[ _m ( my read]s are acqua9t$ ) ! puzzle ( ! ,dut*m5's ,wives--9 :i* y h to det]m9e ! names ( ?ree m5's wives1 or1 r1 :i* wife belongs to ea* husb&4 ,"s ?irty ye>s ago x 0 go+ ! r\nds1 z some?+ q new1 b ,i rec5tly discov]$ x 9 ! ,ladies' ,di>y = #agci-dj1 s x 0 cle>ly famili> to ! fair sex ov] "o hundr$ & sev5ty ye>s ago4 ,h[ _m ( our mo!rs1 wives1 si/]s1 daught]s1 & aunts c\ld solve ! puzzle to-day8 ,a f> great] proportion ?an !n1 let u hope4 ,?ree ,dut*m51 nam$ ,h5drick1 ,elas1 & ,cornelius1 & _! wives1 ,gurtrn1 ,katrn1 & ,anna1 pur*ase hogs4 ,ea* buys z _m z he 7or %e7 gives %ill+s = "o4 ,ea* husb& pays altoge!r ?ree gu9eas m ?an 8 wife4 ,h5drick buys tw5ty-?ree m hogs ?an ,katrn1 & ,elas elev5 m ?an ,gurtrn4 ,n[1 :at 0 ! "n ( ea* man's wife8 ,illu/ration #adj4--;,,f9d ,,ada',s ,,surname4 ,?is puzzle closely resembles ! la/ "o1 my rem>ks on ! solution ( :i* ! read] may l to apply 9 ano!r case4 ,x 0 rec5tly submitt$ to a ,sydney ev5+ newspap] t indulges 9 intellect %>p5]s1 b 0 reject$ ) ! rem>k t x is *ildi% & t !y only publi%$ problems capable ( solution6 ,five ladies1 accompani$ by _! daught]s1 b\ght clo? at ! same %op4 ,ea* ( ! t5 paid z _m f>?+s p] foot z %e b\ght feet1 & ea* "m sp5t #hs4 #e;d4 m ?an h] daught]4 ,mrs4 ,rob9son sp5t #fs4 m ?an ,mrs4 ,evans1 :o sp5t ab\t a qu>t] z mu* z ,mrs4 ,jones4 ,mrs4 ,smi? sp5t mo/ ( all4 ,mrs4 ,br[n b\ght #ba y>ds m ?an ,bessie--"o ( ! girls4 ,annie b\ght #af y>ds m ?an ,m>y & sp5t #c1 #js4 #h;d4 m ?an ,emily4 ,! ,*ri/ian "n ( ! o!r girl 0 ,ada4 ,n[1 :at 0 h] surname8 #ada4--,,saturday ,,m>ket+4 ,"h is an amus+ little case ( m>ket+ :i*1 al?\gh x deals ) a good _m items ( money1 leads up to a "q ( a totally diff]5t "*4 ,f\r m>ri$ c\ples w5t into _! village on a rec5t ,saturday night to d a little m>ket+4 ,!y _h to be v economical1 = among !m !y only possess$ =ty %ill+ co9s4 ,! fact is1 ,ann sp5t #as41 ,m>y sp5t #bs41 ,jane sp5t #cs41 & ,kate sp5t #ds4 ,! m5 7 r m extravagant ?an _! wives1 = ,n$ ,smi? sp5t z mu* z 8 wife1 ,tom ,br[n twice z mu* z 8 wife1 ,bill ,jones ?ree times z mu* z 8 wife1 & ,jack ,rob9son f\r times z mu* z 8 wife4 ,on ! way home somebody sugge/$ t !y %\ld divide :at co9 !y _h left equally among !m4 ,?is 0 done1 & ! puzzl+ "q is simply ?is3 ,:at 0 ! surname ( ea* woman8 ,c y pair (f ! f\r c\ples8 ,,geometrical ,,problems4 ,god geometrizes cont9ually4 ,,plato4 ,!re is no /udy1 said ,augu/us de ,morgan1 :i* pres5ts s simple a beg9n+ z t ( geometry2 !re is none 9 :i* difficulties gr[ m rapidly z we proce$4 ,?is w be f\nd :5 ! read] comes to consid] ! foll[+ puzzles1 ?\gh !y are n arrang$ 9 /rict ord] ( difficulty4 ,& ! fact t !y h int]e/$ & giv5 pleasure to man = untold ages is no d\bt due 9 "s measure to ! appeal !y make to ! eye z well z to ! bra94 ,sometimes an algebraical =mula or !orem seems to give pleasure to ! ma!matician's eye1 b x is probably only an intellectual pleasure4 ,b !re c be no d\bt t 9 ! case ( c]ta9 geometrical problems1 notably dissection or sup]position puzzles1 ! /hetic faculty 9 man contributes to ! delight4 ,= example1 !re are probably few read]s :o w exam9e ! v>i\s cutt+s ( ! ,greek cross 9 ! foll[+ pages )\t be+ 9 "s degree /irr$ by a s5se ( beauty4 ,law & ord] 9 ,nature are always pleas+ to contemplate1 b :5 !y come "u ! v eye !y seem to make a specially /rong appeal4 ,ev5 ! p]son ) no geometrical k :atev] is induc$ aft] ! inspection ( su* ?+s to exclaim1 ,h[ v pretty6 ,9 fact1 ,i h kn[n m ?an "o p]son l$ on to a /udy ( geometry by ! fasc9ation ( cutt+-out puzzles4 ,i h1 !re=e1 ?\ght x well to keep ^! dissection puzzles di/9ct f ! geometrical problems on m g5]al l9es4 ,,dissection ,,puzzles4 ,take him & cut him out 9 little />s4 ,romeo & ,juliet1 iii4 #b4 ,puzzles h inf9ite v>iety1 b p]haps !re is no class m anci5t ?an dissection1 cutt+-out1 or sup]position puzzles4 ,!y 7 c]ta9ly kn[n to ! ,*9ese sev]al ?\s& ye>s be=e ! ,*ri/ian era4 ,& !y are j z fasc9at+ to-day z !y c h be5 at any p]iod ( _! hi/ory4 ,x is suppos$ by ^? :o h inve/igat$ ! matt] t ! anci5t ,*9ese philosoph]s us$ ^! puzzles z a sort ( k9d]g>t5 me?od ( imp>t+ ! pr9ciples ( geometry4 ,:e!r ?is 0 s or n1 x is c]ta9 t all good dissection puzzles 7= ! nurs]y type ( jig-saw puzzle1 :i* m]ely consi/s 9 cutt+ up a picture into pieces to be put toge!r aga91 is n wor?y ( s]i\s consid]ation7 are really bas$ on geometrical laws4 ,?is /atem5t ne$ n1 h[ev]1 fright5 (f ! novice1 = x means little m ?an ?is1 t geometry w give u ! reason :y1 if we are int]e/$ 9 kn[+ x1 ?\gh ! solutions may (t5 be discov]$ by any intellig5t p]son aft] ! ex]cise ( pati5ce1 +5uity1 & common sagacity4 ,if we want to cut "o plane figure into p>ts t by readju/m5t w =m ano!r figure1 ! fir/ ?+ is to f9d a way ( do+ x at all1 & !n to discov] h[ to d x 9 ! fewe/ possible pieces4 ,(t5 a dissection problem is q easy ap>t f ?is limitation ( pieces4 ,at ! "t ( ! publication 9 ! ,weekly ,dispat*1 9 #aijb1 ( a me?od ( cutt+ an equilat]al triangle into f\r p>ts t w =m a squ>e 7see ,no4 #bf1 ,cant]bury ,puzzles71 no geometrician w\ld h _h any difficulty 9 do+ :at is requir$ 9 five pieces3 ! :ole po9t ( ! discov]y lay 9 p]=m+ ! little feat 9 f\r pieces only4 ,m]e approximations 9 ! case ( ^! problems are valueless2 ! solution mu/ be geometrically exact1 or x is n a solution at all4 ,fallacies are cropp+ up n[ & aga91 & ,i %all h occasion to ref] to "o or two ( ^!4 ,!y are int]e/+ m]ely z fallacies4 ,b ,i want to say some?+ on two little po9ts t are always aris+ 9 cutt+-out puzzles--! que/ions ( hang+ by a ?read & turn+ ov]4 ,^! po9ts c be/ be illu/rat$ by a puzzle t is frequ5tly to be f\nd 9 ! old books1 b inv>iably ) a false solution4 ,! puzzle is to cut ! figure %[n 9 ,fig4 #a into ?ree pieces t w fit toge!r & =m a half-squ>e triangle4 ,! answ] t is inv>iably giv5 is t %[n 9 ,figs4 #a & #b4 ,n[1 x is claim$ t ! f\r pieces m>k$ ,c are really only "o piece1 because !y may be s cut t !y are left hang+ toge!r by a m]e ?read4 ,b no s]i\s puzzle lov] w ev] admit ?is4 ,if ! cut is made s z to leave ! f\r pieces jo9$ 9 "o1 !n x _c result 9 a p]fectly exact solution4 ,if1 on ! o!r h&1 ! solution is to be exact1 !n !re w be f\r pieces--or six pieces 9 all4 ,x is1 !re=e1 n a solution 9 ?ree pieces4 ,illu/ration3 ,fig4 #a ,illu/ration3 ,fig4 #b ,if1 h[ev]1 ! read] w look at ! solution 9 ,figs4 #c & #d1 he w see t no su* fault c be f\nd ) x4 ,!re is no "q :atev] t !re are ?ree pieces1 & ! solution is 9 ?is respect q satisfactory4 ,b ano!r "q arises4 ,x w be f\nd on inspection t ! piece m>k$ ,f1 9 ,fig4 #c1 is turn$ ov] 9 ,fig4 #d--t is to say1 a diff]5t side has necess>ily to be pres5t$4 ,if ! puzzle 7 m]ely to be cut out ( c>dbo>d or wood1 !re might be no objection to ?is rev]sal1 b x is q possible t ! mat]ial w\ld n admit ( be+ rev]s$4 ,!re might be a patt]n1 a poli%1 a diff]5ce ( texture1 t prev5ts x4 ,b x is g5]ally und]/ood t 9 dissection puzzles y are all[$ to turn pieces ov] unless x is di/9ctly /at$ t y may n d s4 ,& v (t5 a puzzle is greatly improv$ by ! add$ condition1 no piece may be turn$ ov]4 ,i h (t5 made puzzles1 too1 9 :i* ! diagram has a small repeat$ patt]n1 & ! pieces h !n s to be cut t n only is !re no turn+ ov]1 b ! patt]n has to be mat*$1 :i* _c be done if ! pieces are turn$ r\nd1 ev5 ) ! prop] side upp]mo/4 ,illu/ration3 ,fig4 #c ,illu/ration3 ,fig4 #d ,be=e pres5t+ a v>i$ s]ies ( cutt+-out puzzles1 "s v easy & o!rs difficult1 ,i propose to consid] "o family alone--^? problems involv+ :at is kn[n z ! ,greek cross ) ! squ>e4 ,?is w exhibit a great v>iety ( curi\s transpositions1 &1 by hav+ ! solutions z we g along1 ! read] w be sav$ ! tr\ble ( p]petually turn+ to ano!r "p ( ! book1 & w h ev]y?+ "u 8 eye4 ,x is hop$ t 9 ?is way ! article may prove some:at in/ructive to ! novice & int]e/+ to o!rs4 ,,greek ,,cross ,,puzzles4 ,to fret ?y s\l ) crosses4 ,,sp5s]4 ,b1 = my "p1 x 0 ,greek to me4 ,julius ,cs>1 i4 #b4 ,_m p are accu/om$ to consid] ! cross z a :olly ,*ri/ian symbol4 ,?is is errone\s3 x is ( v great antiquity4 ,! anci5t ,egyptians employ$ x z a sacr$ symbol1 & on ,greek sculptures we f9d repres5tations ( a cake 7! suppos$ real orig9 ( our hot cross buns7 be>+ a cross4 ,two su* cakes 7 discov]$ at ,h]culaneum4 ,cecrops (f]$ to ,jupit] ,olympus a sacr$ cake or b\n ( ?is k9d4 ,! cross & ball1 s frequ5tly f\nd on ,egyptian figures1 is a circle & ! tau cross4 ,! circle signifi$ ! et]nal pres]v] ( ! _w1 & ! ,t1 nam$ f ! ,greek lett] tau1 is ! monogram ( ,?o?1 ! ,egyptian ,m]cury1 mean+ wisdom4 ,?is tau cross is also call$ by ,*ri/ians ! cross ( ,/4 ,an?ony1 & is borne on a badge 9 ! bi%op's palace at ,exet]4 ,z = ! ,greek or mundane cross1 ! cross ) f\r equal arms1 we are told by compet5t antiqu>ies t x 0 reg>d$ by anci5t occulti/s = ?\s&s ( ye>s z a sign ( ! dual =ces ( ,nature--! male & female _s ( ev]y?+ t 0 ev]la/+4 ,illu/ration3 ,fig4 #e4 ,! ,greek cross1 z %[n 9 ,fig4 #e1 is =m$ by ! assembl+ toge!r ( five equal squ>es4 ,we w />t ) :at is kn[n z ! ,h9du problem1 suppos$ to be upw>ds ( ?ree ?\s& ye>s old4 ,x appe>s 9 ! seal ( ,h>v>d ,college1 & is (t5 giv5 9 old works z symbolical ( ma!matical sci5ce & exactitude4 ,cut ! cross into five pieces to =m a squ>e4 ,figs4 #f & #g %[ h[ ?is is done4 ,x 0 n until ! middle ( ! n9ete5? c5tury t we f\nd t ! cross might be trans=m$ into a squ>e 9 only f\r pieces4 ,figs4 #h & #i w %[ h[ to d x1 if we fur!r require ! f\r pieces to be all ( ! same size & %ape4 ,?is ,fig4 #i is rem>kable because1 accord+ to ,dr4 ,le ,plongeon & o!rs1 z exp\nd$ 9 a "w by ,pr(essor ,wilson ( ! ,smi?sonian ,in/itute1 "h we h ! great ,swa/ika1 or sign1 ( good luck to y --! mo/ anci5t symbol ( ! human race ( :i* !re is any record4 ,pr(essor ,wilson's "w gives "s f\r hundr$ illu/rations ( ?is curi\s sign z f\nd 9 ! ,aztec m\nds ( ,mexico1 ! pyramids ( ,egypt1 ! ru9s ( ,troy1 & ! anci5t lore ( ,india & ,*9a4 ,"o might almo/ say !re is a curi\s aff9ity betwe5 ! ,greek cross & ,swa/ika6 ,if1 h[ev]1 we require t ! f\r pieces %all be produc$ by only two clips ( ! scissors 7assum+ ! puzzle is 9 pap] =m71 !n we mu/ cut z 9 ,fig4 #aj to =m ,fig4 #aa1 ! fir/ clip ( ! scissors be+ f a to b4 ,( c\rse fold+ ! pap]1 or hold+ ! pieces toge!r aft] ! fir/ cut1 w\ld n 9 ?is case be all[$4 ,b !re is an inf9ite numb] ( diff]5t ways ( mak+ ! cuts to solve ! puzzle 9 f\r pieces4 ,to ?is po9t ,i propose to return4 ,illu/ration3 ,fig4 #f ,illu/ration3 ,fig4 #g ,illu/ration3 ,fig4 #h ,illu/ration3 ,fig4 #i ,illu/ration3 ,fig4 #aj ,illu/ration3 ,fig4 #aa ,x w be se5 t e "o ( ^! puzzles has its rev]se puzzle--to cut a squ>e into pieces to =m a ,greek cross4 ,b z a squ>e has n s _m angles z ! cross1 x is n always equally easy to discov] ! true directions ( ! cuts4 ,yet 9 ! case ( ! examples giv51 ,i w leave ! read] to det]m9e _! direction = himself1 z !y are r obvi\s f ! diagrams4 ,cut a squ>e into five pieces t w =m two sep>ate ,greek crosses ( diff]5t sizes4 ,?is is q an easy puzzle4 ,z w be se5 9 ,fig4 #ab1 we h only to divide our squ>e into #be little squ>es & !n cut z %[n4 ,! cross ,a is cut out entire1 & ! pieces ,b1 ,c1 ,d1 & ,e =m ! l>g] cross 9 ,fig4 #ac4 ,! read] may "h l to cut ! s+le piece1 ,b1 into f\r pieces all simil> 9 %ape to itself1 & =m a cross ) !m 9 ! mann] %[n 9 ,fig4 #ac4 ,i h>dly ne$ give ! solution4 ,illu/ration3 ,,fig4 #ab4 ,illu/ration3 ,,fig4 #ac4 ,cut a squ>e into five pieces t w =m two sep>ate ,greek crosses ( exactly ! same size4 ,?is is m difficult4 ,we make ! cuts z 9 ,fig4 #ad1 :]e ! cross ,a comes out entire & ! o!r f\r pieces =m ! cross 9 ,fig4 #ae4 ,! direction ( ! cuts is pretty obvi\s4 ,x w be se5 t ! sides ( ! squ>e 9 ,fig4 #ad are m>k$ (f into six equal p>ts4 ,! sides ( ! cross are f\nd by rul+ l9es f c]ta9 ( ^! po9ts to o!rs4 ,illu/ration3 ,,fig4 #ad4 ,illu/ration3 ,,fig4 #ae4 ,i w n[ expla91 z ,i promis$1 :y a ,greek cross may be cut into f\r pieces 9 an inf9ite numb] ( diff]5t ways to make a squ>e4 ,draw a cross1 z 9 ,fig4 #af4 ,!n draw on transp>5t pap] ! squ>e %[n 9 ,fig4 #ag1 tak+ c>e t ! di/ance c to d is exactly ! same z ! di/ance a to b 9 ! cross4 ,n[ place ! transp>5t pap] ov] ! cross & slide x ab\t into diff]5t positions1 only be v c>eful always to keep ! squ>e at ! same angle to ! cross z %[n1 :]e a b is p>allel to c d4 ,if y place ! po9t c exactly ov] a ! l9es w indicate ! solution 7,figs4 #aj & #aa74 ,if y place c 9 ! v c5tre ( ! dott$ squ>e1 x w give ! solution 9 ,figs4 #h & #i4 ,y w n[ see t by slid+ ! squ>e ab\t s t ! po9t c is always )9 ! dott$ squ>e y may get z _m diff]5t solutions z y l2 because1 s9ce an inf9ite numb] ( diff]5t po9ts may !oretically be plac$ )9 ?is squ>e1 !re mu/ be an inf9ite numb] ( diff]5t solutions4 ,b ! po9t c ne$ n necess>ily be plac$ )9 ! dott$ squ>e4 ,x may be plac$1 = example1 at po9t e to give a solution 9 f\r pieces4 ,"h ! jo9s at a & f may be z sl5d] z y l4 ,yet if y once get ov] ! edge at a or f y no long] h a solution 9 f\r pieces4 ,?is pro( w be f\nd bo? ent]ta9+ & in/ructive4 ,if y d n happ5 to h any transp>5t pap] at h&1 any ?9 pap] w ( c\rse d if y hold ! two %eets aga9/ a pane ( glass 9 ! w9d[4 ,illu/ration3 ,,fig4 #af4 ,illu/ration3 ,,fig4 #ag4 ,x may h be5 notic$ f ! solutions ( ! puzzles t ,i h giv5 t ! side ( ! squ>e =m$ f ! cross is always equal to ! di/ance a to b 9 ,fig4 #af4 ,?is mu/ necess>ily be s1 & ,i w pres5tly try to make ! po9t q cle>4 ,we w n[ g "o /ep fur!r4 ,i h already said t ! ideal solution to a cutt+-out puzzle is always t :i* requires ! fewe/ possible pieces4 ,we h j se5 t two crosses ( ! same size may be cut out ( a squ>e 9 five pieces4 ,! read] :o succe$$ 9 solv+ ?is p]haps ask$ himself3 ,c x be done 9 few] pieces8 ,?is is j ! sort ( "q t ! true puzzle lov] is always ask+1 & x is ! "r attitude = him to adopt4 ,! answ] to ! "q is t ! puzzle may be solv$ 9 f\r pieces--! fewe/ possible4 ,?is1 !n1 is a new puzzle4 ,cut a squ>e into f\r pieces t w =m two ,greek crosses ( ! same size4 ,illu/ration3 ,,fig4 #ah4 ,illu/ration3 ,,fig4 #ai4 ,illu/ration3 ,,fig4 #bj4 ,! solution is v beautiful4 ,if y divide by po9ts ! sides ( ! squ>e into ?ree equal p>ts1 ! directions ( ! l9es 9 ,fig4 #ah w be q obvi\s4 ,if y cut along ^! l9es1 ! pieces ,a & ,b w =m ! cross 9 ,fig4 #ai & ! pieces ,c & ,d ! simil> cross 9 ,fig4 #bj4 ,9 ?is squ>e we h ano!r =m ( ,swa/ika4 ,! read] w "h appreciate ! tru? ( my rem>k to ! effect t x is easi] to f9d ! directions ( ! cuts :5 trans=m+ a cross to a squ>e ?an :5 conv]t+ a squ>e into a cross4 ,?us1 9 ,figs4 #f1 #h1 & #aj ! directions ( ! cuts are m obvi\s ?an 9 ,fig4 #ad1 :]e we _h fir/ to divide ! sides ( ! squ>e into six equal p>ts1 & 9 ,fig4 #ah1 :]e we divide !m into ?ree equal p>ts4 ,!n1 suppos+ y 7 requir$ to cut two equal ,greek crosses1 ea* into two pieces1 to =m a squ>e1 a glance at ,figs4 #ai & #bj w %[ h[ absurdly m easy ?is is ?an ! rev]se puzzle ( cutt+ ! squ>e to make two crosses4 ,ref]r+ to my rem>ks on fallacies1 ,i w n[ give a little example ( ^! solutions t are n solutions4 ,"s ye>s ago a "y correspond5t s5t me :at he evid5tly ?\ght 0 a brilliant new discov]y--! trans=m+ ( a squ>e into a ,greek cross 9 f\r pieces by cuts all p>allel to ! sides ( ! squ>e4 ,i give 8 attempt 9 ,figs4 #ba & #bb1 :]e x w be se5 t ! f\r pieces d n =m a symmetrical ,greek cross1 because ! f\r arms are n really squ>es b oblongs4 ,to make x a true ,greek cross we %\ld require ! additions t ,i h indicat$ ) dott$ l9es4 ,( c\rse 8 solution produces a cross1 b x is n ! symmetrical ,greek v>iety requir$ by ! conditions ( ! puzzle4 ,my "y fri5d ?\ght 8 attempt 0 ne> en\gh to be correct2 b if he b\ght a p5ny apple ) a sixp5ce he probably w\ld n h ?\ght x ne> en\gh if he _h be5 giv5 only f\rp5ce *ange4 ,z ! read] advances he w realize ! importance ( ?is "q ( exactitude4 ,illu/ration3 ,,fig4 #ba4 ,illu/ration3 ,,fig4 #bb4 ,9 ^! cutt+-out puzzles x is necess>y n only to get ! directions ( ! cutt+ l9es z correct z possible1 b to rememb] t ^! l9es h no wid?4 ,if aft] cutt+ up "o ( ! crosses 9 a mann] indicat$ 9 ^! articles y f9d t ! pieces d n exactly fit to =m a squ>e1 y may be c]ta9 t ! fault is entirely y\r own4 ,ei!r y\r cross 0 n exactly drawn1 or y\r cuts 7 n made q 9 ! "r directions1 or 7if y us$ wood & a fret-saw7 y\r saw 0 n suffici5tly f9e4 ,if y cut out ! puzzles 9 pap] ) scissors1 or 9 c>dbo>d ) a p5knife1 no mat]ial is lo/2 b ) a saw1 h[ev] f9e1 !re is a c]ta9 loss4 ,9 ! case ( mo/ puzzles ?is slight loss is n suffici5t to be appreciable1 if ! puzzle is cut out on a l>ge scale1 b !re h be5 in/ances :]e ,i h f\nd x desirable to draw & cut out ea* "p sep>ately--n f "o diagram--9 ord] to produce a p]fect result4 ,illu/ration3 ,,fig4 #bc4 ,illu/ration3 ,,fig4 #bd4 ,n[ = ano!r puzzle4 ,if y h cut out ! five pieces indicat$ 9 ,fig4 #ad1 y w f9d t ^! c be put toge!r s z to =m ! curi\s cross %[n 9 ,fig4 #bc4 ,s if ,i ask$ y to cut ,fig4 #bd into five pieces to =m ei!r a squ>e or two equal ,greek crosses y w\ld "k h[ to d x4 ,y w\ld make ! cuts z 9 ,fig4 #bc1 & place !m toge!r z 9 ,figs4 #ad & #ae4 ,b ,i want some?+ bett] ?an t1 & x is ?is4 ,cut ,fig4 #bd into only f\r pieces t w fit toge!r & =m a squ>e4 ,illu/ration3 ,,fig4 #be4 ,illu/ration3 ,,fig4 #bf4 ,! solution to ! puzzle is %[n 9 ,figs4 #be & #bf4 ,! direction ( ! cut divid+ ,a & ,c 9 ! fir/ diagram is v obvi\s1 & ! second cut is made at "r angles to x4 ,t ! f\r pieces %\ld fit toge!r & =m a squ>e w surprise ! novice1 :o w d well to /udy ! puzzle ) "s c>e1 z x is mo/ in/ructive4 ,i w n[ expla9 ! beautiful rule by :i* we det]m9e ! size ( a squ>e t %all h ! same area z a ,greek cross1 = x is applicable1 & necess>y1 to ! solution ( almo/ e dissection puzzle t we meet )4 ,x 0 fir/ discov]$ by ! philosoph] ,py?agoras1 :o di$ #ejj ,b4,c41 & is ! #dg? proposition ( ,euclid4 ,! "y read] :o kn[s no?+ ( ! elem5ts ( geometry w get "s idea ( ! fasc9at+ "* ( t sci5ce4 ,! triangle ,,abc 9 ,fig4 #bg is :at we call a "r-angl$ triangle1 because ! side ,,bc is at "r angles to ! side ,,ab4 ,n[ if we build up a squ>e on ea* side ( ! triangle1 ! squ>es on ,,ab & ,,bc w toge!r be exactly equal to ! squ>e on ! long side ,,ac1 :i* we call ! hypot5use4 ,?is is prov$ 9 ! case ,i h giv5 by subdivid+ ! ?ree squ>es into cells ( equal dim5sions4 ,illu/ration3 ,,fig4 #bg4 ,illu/ration3 ,,fig4 #bh4 ,x w be se5 t #i add$ to #af equals #be1 ! numb] ( cells 9 ! l>ge squ>e4 ,if y make triangles ) ! sides #e1 #ab & #ac1 or ) #h1 #ae & #ag1 y w get simil> ari?metical pro(s1 = ^! are all rational "r-angl$ triangles1 b ! law is equally true = all cases4 ,suppos+ we cut (f ! l[] arm ( a ,greek cross & place x to ! left ( ! upp] arm1 z 9 ,fig4 #bh1 !n ! squ>e on ,,ef add$ to ! squ>e on ,,de exactly equals a squ>e on ,,df4 ,!re=e we "k t ! squ>e ( ,,df w conta9 ! same area z ! cross4 ,?is fact we h prov$ practically by ! solutions ( ! e>li] puzzles ( ?is s]ies4 ,b :atev] l5g? we give to ,,de & ,,ef1 we c nev] give ! exact l5g? ( ,,df 9 numb]s1 because ! triangle is n a rational "o4 ,b ! law is none ! less geometrically true4 ,illu/ration3 ,,fig4 #bi4 ,illu/ration3 ,,fig4 #cj4 ,n[ look at ,fig4 #bi1 & y w see an elegant me?od = cutt+ a piece ( wood ( ! %ape ( two squ>es 7( any relative dim5sions7 into ?ree pieces t w fit toge!r & =m a s+le squ>e4 ,if y m>k (f ! di/ance ab equal to ! side cd ! directions ( ! cuts are v evid5t4 ,f :at we h j be5 consid]+1 y w at once see :y bc mu/ be ! l5g? ( ! side ( ! new squ>e4 ,make ! exp]im5t z (t5 z y l1 tak+ diff]5t relative proportions = ! two squ>es1 & y w f9d ! rule always come true4 ,if y make ! two squ>es ( exactly ! same size1 y w see t ! diagonal ( any squ>e is always ! side ( a squ>e t is twice ! size4 ,all ?is1 :i* is s simple t anybody c und]/& x1 is v ess5tial to ! solv+ ( cutt+-out puzzles4 ,x is 9 fact ! key to mo/ ( !m4 ,& x is all s beautiful t x seems a pity t x %\ld n be famili> to ev]ybody4 ,we w n[ g "o /ep fur!r & deal ) ! half-squ>e4 ,take a squ>e & cut x 9 half diagonally4 ,n[ try to discov] h[ to cut ?is triangle into f\r pieces t w =m a ,greek cross4 ,! solution is %[n 9 ,figs4 #ca & #cb4 ,9 ?is case x w be se5 t we divide two ( ! sides ( ! triangle into ?ree equal p>ts & ! long side into f\r equal p>ts4 ,!n ! direction ( ! cuts w be easily f\nd4 ,x is a pretty puzzle1 & a little m difficult ?an "s ( ! o!rs t ,i h giv54 ,x %\ld be not$ aga9 t x w\ld h be5 mu* easi] to locate ! cuts 9 ! rev]se puzzle ( cutt+ ! cross to =m a half-squ>e triangle4 ,illu/ration3 ,,fig4 #ca4 ,illu/ration3 ,,fig4 #cb4 ,illu/ration3 ,,fig4 #cc4 ,illu/ration3 ,,fig4 #cd4 ,ano!r ideal t ! puzzle mak] always keeps 9 m9d is to contrive t !re %all1 if possible1 be only "o correct solution4 ,?us1 9 ! case ( ! fir/ puzzle1 if we only require t a ,greek cross %all be cut into f\r pieces to =m a squ>e1 !re is1 z ,i h %[n1 an inf9ite numb] ( diff]5t solutions4 ,x makes a bett] puzzle to add ! condition t all ! f\r pieces %all be ( ! same size & %ape1 because x c !n be solv$ 9 only "o way1 z 9 ,figs4 #h & #i4 ,9 ?is way1 too1 a puzzle t is too easy to be int]e/+ may be improv$ by su* an addition4 ,let u take an example4 ,we h se5 9 ,fig4 #bh t ,fig4 #cc c be cut into two pieces to =m a ,greek cross4 ,i suppose an intellig5t *ild w\ld d x 9 five m9utes4 ,b suppose we say t ! puzzle has to be solv$ ) a piece ( wood t has a bad knot 9 ! position %[n 9 ,fig4 #cc--;a knot t we mu/ n attempt to cut ?r\gh--!n a solution 9 two pieces is b>r$ out1 & x becomes a m int]e/+ puzzle to solve x 9 ?ree pieces4 ,i h %[n 9 ,figs4 #cc & #cd "o way ( do+ ?is1 & x w be f\nd ent]ta9+ to discov] o!r ways ( do+ x4 ,( c\rse ,i c\ld b> out all ^! o!r ways by introduc+ m knots1 & s r$uce ! puzzle to a s+le solution1 b x w\ld !n be ov]load$ ) conditions4 ,& ?is br+s u to ano!r po9t 9 seek+ ! ideal4 ,d n ov]load y\r conditions1 or y w make y\r puzzle too complex to be int]e/+4 ,! simpl] ! conditions ( a puzzle are1 ! bett]4 ,! solution may be z complex & difficult z y l1 or z happ5s1 b ! conditions ought to be easily und]/ood1 or p w n attempt a solution4 ,if ! read] 7 n[ ask$ to cut a half-squ>e into z few pieces z possible to =m a ,greek cross1 he w\ld probably produce our solution1 ,figs4 #ca-cb1 & confid5tly claim t he _h solv$ ! puzzle correctly4 ,9 ?is way he w\ld be wrong1 because x is n n[ /at$ t ! squ>e is to be divid$ diagonally4 ,al?\gh we %\ld always obs]ve ! exact conditions ( a puzzle we mu/ n read into x conditions t are n !re4 ,_m puzzles are bas$ entirely on ! t5d5cy t p h to d ?is4 ,! v fir/ ess5tial 9 solv+ a puzzle is to be sure t y und]/& ! exact conditions4 ,n[1 if y divid$ y\r squ>e 9 half s z to produce ,fig4 #ce x is possible to cut x into z few z ?ree pieces to =m a ,greek cross4 ,we ?us save a piece4 ,i give ano!r puzzle 9 ,fig4 #cf4 ,! dott$ l9es are add$ m]ely to %[ ! correct proportions ( ! figure--a squ>e ( #be cells ) ! f\r corn] cells cut out4 ,! puzzle is to cut ?is figure into five pieces t w =m a ,greek cross 7entire7 & a squ>e4 ,illu/ration3 ,,fig4 #ce4 ,illu/ration3 ,,fig4 #cf4 ,! solution to ! fir/ ( ! two puzzles la/ giv5--to cut a rectangle ( ! %ape ( a half-squ>e into ?ree pieces t w =m a ,greek cross--is %[n 9 ,figs4 #cg & #ch4 ,x w be se5 t we divide ! long sides ( ! oblong into six equal p>ts & ! %ort sides into ?ree equal p>ts1 9 ord] to get ! po9ts t w indicate ! direction ( ! cuts4 ,! read] %\ld comp>e ?is solution ) "s ( ! previ\s illu/rations4 ,he w see1 = example1 t if we cont9ue ! cut t divides ,b & ,c 9 ! cross1 we get ,fig4 #ae4 ,illu/ration3 ,,fig4 #cg4 ,illu/ration3 ,,fig4 #ch4 ,! o!r puzzle1 l ! "o illu/rat$ 9 ,figs4 #ab & #ac1 w %[ h[ useful a little ari?metic may sometimes prove to be 9 ! solution ( dissection puzzles4 ,!re are tw5ty-"o ( ^? little squ>e cells into :i* our figure is subdivid$1 f :i* we h to =m bo? a squ>e & a ,greek cross4 ,n[1 z ! cross is built up ( five squ>es1 & #e f #ba leaves #af--;a squ>e numb]--we ought easily to be l$ to ! solution %[n 9 ,fig4 #ci4 ,x w be se5 t ! cross is cut out entire1 :ile ! f\r rema9+ pieces =m ! squ>e 9 ,fig4 #dj4 ,illu/ration3 ,,fig4 #ci ,illu/ration3 ,,fig4 #dj ,( c\rse a half-squ>e rectangle is ! same z a d\ble squ>e1 or two equal squ>es jo9$ toge!r4 ,!re=e1 if y want to solve ! puzzle ( cutt+ a ,greek cross into f\r pieces to =m two sep>ate squ>es ( ! same size1 all y h to d is to cont9ue ! %ort cut 9 ,fig4 #ch "r across ! cross1 & y w h f\r pieces ( ! same size & %ape4 ,n[ divide ,fig4 #cg into two equal squ>es by a horizontal cut midway & y w see ! f\r pieces =m+ ! two squ>es4 ,illu/ration3 ,,fig4 #da ,cut a ,greek cross into five pieces t w =m two sep>ate squ>es1 "o ( :i* %all conta9 half ! area ( "o ( ! arms ( ! cross4 ,9 fur!r illu/ration ( :at ,i h already writt51 if ! two squ>es ( ! same size ,a ,b ,c ,d & ,b ,c ,f ,e1 9 ,fig4 #da1 are cut 9 ! mann] indicat$ by ! dott$ l9es1 ! f\r pieces w =m ! l>ge squ>e ,a ,g ,e ,c4 ,we ?us see t ! diagonal ,a ,c is ! side ( a squ>e twice ! size ( ,a ,b ,c ,d4 ,x is also cle> t half ! diagonal ( any squ>e is equal to ! side ( a squ>e ( half ! area4 ,!re=e1 if ! l>ge squ>e 9 ! diagram is "o ( ! arms ( y\r cross1 ! small squ>e is ! size ( "o ( ! squ>es requir$ 9 ! puzzle4 ,! solution is %[n 9 ,figs4 #db & #dc4 ,x w be se5 t ! small squ>e is cut out :ole & ! l>ge squ>e compos$ ( ! f\r pieces ,b1 ,c1 ,d1 & ,e4 ,aft] :at ,i h writt51 ! read] w h no difficulty 9 see+ t ! squ>e ,a is half ! size ( "o ( ! arms ( ! cross1 because ! l5g? ( ! diagonal ( ! =m] is cle>ly ! same z ! side ( ! latt]4 ,! ?+ is n[ self-evid5t4 ,i h ?us tri$ to %[ t "s ( ^! puzzles t _m p are apt to reg>d z q wond]ful & bewild]+1 are really n difficult if only we use a little ?\ght & judgm5t4 ,9 conclusion ( ?is p>ticul> subject ,i w give f\r ,greek cross puzzles1 ) deta*$ solutions4 #adb4--,,! ,,silk ,,pat*work4 ,! lady memb]s ( ! ,wilk9son family _h made a simple pat*work quilt1 z a small ,*ri/mas pres5t1 all compos$ ( squ>e pieces ( ! same size1 z %[n 9 ! illu/ration4 ,x only lack$ ! f\r corn] pieces to make x complete4 ,somebody po9t$ out to !m t if y unpick$ ! ,greek cross 9 ! middle & !n cut ! /it*es along ! d>k jo9s1 ! f\r pieces all ( ! same size & %ape w\ld fit toge!r & =m a squ>e4 ,?is ! read] kn[s1 f ! solution 9 ,fig4 #ci1 is q easily done4 ,b ,george ,wilk9son sudd5ly sugge/$ to !m ?is pos]4 ,he said1 ,in/ead ( pick+ out ! cross entire1 & =m+ ! squ>e f f\r equal pieces1 c y cut out a squ>e entire & f\r equal pieces t w =m a p]fect ,greek cross8 ,! puzzle is1 ( c\rse1 n[ q easy4 #adc4--,,two ,,crosses ,,f ,,"o4 ,cut a ,greek cross into five pieces t w =m two su* crosses1 bo? ( ! same size4 ,! solution ( ?is puzzle is v beautiful4 #add4--,,! ,,cross ,,& ,,! ,,triangle4 ,cut a ,greek cross into six pieces t w =m an equilat]al triangle4 ,?is is ano!r h>d problem1 & ,i w /ate "h t a solution is practically impossible )\t a previ\s k ( my me?od ( trans=m+ an equilat]al triangle into a squ>e 7see ,no4 #bf1 ,cant]bury ,puzzles74 #ade4--,,! ,,fold$ ,,cross4 ,cut out ( pap] a ,greek cross2 !n s fold x t ) a s+le /raight cut ( ! scissors ! f\r pieces produc$ w =m a squ>e4 ,,v>i\s ,,dissection ,,puzzles4 ,we w n[ consid] a small miscellane\s selection ( cutt+-out puzzles1 v>y+ 9 degrees ( difficulty4 #adf4--;,,an ,,easy ,,dissection ,,puzzle4 ,fir/1 cut out a piece ( pap] or c>dbo>d ( ! %ape %[n 9 ! illu/ration4 ,x w be se5 at once t ! proportions are simply ^? ( a squ>e atta*$ to half ( ano!r simil> squ>e1 divid$ diagonally4 ,! puzzle is to cut x into f\r pieces all ( precisely ! same size & %ape4 #adg4--;,,an ,,easy ,,squ>e ,,puzzle4 ,if y take a rectangul> piece ( c>dbo>d1 twice z long z x is broad1 & cut x 9 half diagonally1 y w get two ( ! pieces %[n 9 ! illu/ration4 ,! puzzle is ) five su* pieces ( equal size to =m a squ>e4 ,"o ( ! pieces may be cut 9 two1 b ! o!rs mu/ be us$ intact4 #adh4--,,! ,,bun ,,puzzle4 ,,! ?ree circles repres5t ?ree buns1 & x is simply requir$ to %[ h[ ^! may be equally divid$ among f\r boys4 ,! buns mu/ be reg>d$ z ( equal ?ickness ?r\gh\t & ( equal ?ickness to ea* o!r4 ,( c\rse1 !y mu/ be cut into z few pieces z possible4 ,to simplify x ,i w /ate ! r surpris+ fact t only five pieces are necess>y1 f :i* x w be se5 t "o boy gets 8 %>e 9 two pieces & ! o!r ?ree receive !irs 9 a s+le piece4 ,i am aw>e t ?is /atem5t gives away ! puzzle1 b x %\ld n de/roy its int]e/ to ^? :o l to discov] ! reason :y4 #adi4--,,! ,,*ocolate ,,squ>es4 ,"h is a slab ( *ocolate1 ind5t$ at ! dott$ l9es s t ! tw5ty squ>es c be easily sep>at$4 ,make a copy ( ! slab 9 pap] or c>dbo>d & !n try to cut x into n9e pieces s t !y w =m f\r p]fect squ>es all ( exactly ! same size4 #aej4--;,,dissect+ ,a ,,mitre4 ,! figure t is p]plex+ ! c>p5t] 9 ! illu/ration repres5ts a mitre4 ,x w be se5 t its proportions are ^? ( a squ>e ) "o qu>t] remov$4 ,! puzzle is to cut x into five pieces t w fit toge!r & =m a p]fect squ>e4 ,i %[ an attempt1 publi%$ 9 ,am]ica1 to p]=m ! feat 9 f\r pieces1 bas$ on :at is kn[n z ! /ep pr9ciple1 b x is a fallacy4 ,illu/ration ,we are told fir/ to cut (t ! pieces #a & #b & pack !m into ! triangul> space m>k$ (f by ! dott$ l9e1 & s =m a rectangle4 ,s f>1 s good4 ,n[1 we are direct$ to apply ! old /ep pr9ciple1 z %[n1 &1 by mov+ d[n ! piece #d "o /ep1 =m ! requir$ squ>e4 ,b1 un=tunately1 x does n produce a squ>e3 only an oblong4 ,call ! ?ree long sides ( ! mitre #hd 94 ea*4 ,!n1 be=e cutt+ ! /eps1 our rectangle 9 ?ree pieces w be #hd #fc4 ,! /eps mu/ be #aj 94 9 height & #ab 94 9 bread?4 ,!re=e1 by mov+ d[n a /ep we r$uce by #ab 94 ! side #hd 94 & increase by #aj 94 ! side #fc 94 ,h5ce our f9al rectangle mu/ be #gb 94 #gc 941 :i* c]ta9ly is n a squ>e6 ,! fact is1 ! /ep pr9ciple c only be appli$ to rectangles ) sides ( p>ticul> relative l5g?s4 ,= example1 if ! %ort] side 9 ?is case 7 #fae_/g 7in/ead ( #fc71 !n ! /ep me?od w\ld apply4 ,= ! /eps w\ld !n be #ajb_/g 94 9 height & #ab 94 9 bread?4 ,note t #fae_/g #hd ! squ>e ( #gb4 ,at pres5t no solution has be5 f\nd 9 f\r pieces1 & ,i d n believe "o possible4 #aea4--,,! ,,jo9]',s ,,problem4 ,i h (t5 _h occasion to rem>k on ! practical utility ( puzzles1 aris+ out ( an application to ! ord9>y affairs ( life ( ! little tricks & wr9kles t we le>n :ile solv+ recreation problems4 ,illu/ration ,! jo9]1 9 ! illu/ration1 wants to cut ! piece ( wood into z few pieces z possible to =m a squ>e table-top1 )\t any wa/e ( mat]ial4 ,h[ %\ld he g to "w8 ,h[ _m pieces w\ld y require8 #aeb4--;,,ano!r ,,jo9]',s ,,problem4 ,illu/ration ,a jo9] _h two pieces ( wood ( ! %apes & relative proportions %[n 9 ! diagram4 ,he wi%$ to cut !m into z few pieces z possible s t !y c\ld be fitt$ toge!r1 )\t wa/e1 to =m a p]fectly squ>e table-top4 ,h[ %\ld he h done x8 ,!re is no necessity to give measurem5ts1 = if ! small] piece 7:i* is half a squ>e7 be made a little too l>ge or a little too small x w n affect ! me?od ( solution4 #aec--;,a ,,cutt+-,,out ,,puzzle4 ,"h is a little cutt+-out pos]4 ,i take a /rip ( pap]1 measur+ five in*es by "o in*1 &1 by cutt+ x into five pieces1 ! p>ts fit toge!r & =m a squ>e1 z %[n 9 ! illu/ration4 ,n[1 x is q an int]e/+ puzzle to discov] h[ we c d ?is 9 only f\r pieces4 ,illu/ration #aed4--,,mrs4 ,,hobson',s ,,he>?rug4 ,illu/ration ,mrs4 ,hobson's boy _h an accid5t :5 play+ ) ! fire1 & burnt two ( ! corn]s ( a pretty he>?rug4 ,! damag$ corn]s h be5 cut away1 & x n[ has ! appe>ance & proportions %[n 9 my diagram4 ,h[ is ,mrs4 ,hobson to cut ! rug into ! fewe/ possible pieces t w fit toge!r & =m a p]fectly squ>e rug8 ,x w be se5 t ! rug is 9 ! proportions #cf #bg 7x does n matt] :e!r we say in*es or y>ds71 & ea* piece cut away measur$ #ab & #f on ! outside4 #aee4--,,! ,,p5tagon ,,& ,,squ>e4 ,i wond] h[ _m ( my read]s1 among/ ^? :o h n giv5 any close att5tion to ! elem5ts ( geometry1 c\ld draw a regul> p5tagon1 or five-sid$ figure1 if !y sudd5ly requir$ to d s4 ,a regul> hexagon1 or six-sid$ figure1 is easy en\gh1 = ev]ybody kn[s t all y h to d is to describe a circle & !n1 tak+ ! radius z ! l5g? ( "o ( ! sides1 m>k (f ! six po9ts r\nd ! circumf]5ce4 ,b a p5tagon is q ano!r matt]4 ,s1 z my puzzle has to d ) ! cutt+ up ( a regul> p5tagon1 x w p]haps be well if ,i fir/ %[ my less exp]i5c$ read]s h[ ?is figure is to be correctly drawn4 ,describe a circle & draw ! two l9es ,h ,b & ,d ,g1 9 ! diagram1 ?r\gh ! c5tre at "r angles4 ,n[ f9d ! po9t ,a1 midway betwe5 ,c & ,b4 ,next place ! po9t ( y\r compasses at ,a & ) ! di/ance ,a ,d describe ! arc cutt+ ,h ,b at ,e4 ,!n place ! po9t ( y\r compasses at ,d & ) ! di/ance ,d ,e describe ! arc cutt+ ! circumf]5ce at ,f4 ,n[1 ,d ,f is "o ( ! sides ( y\r p5tagon1 & y h simply to m>k (f ! o!r sides r\nd ! circle4 ,q simple :5 y "k h[1 b o!rwise some:at ( a pos]4 ,illu/ration ,hav+ =m$ y\r p5tagon1 ! puzzle is to cut x into ! fewe/ possible pieces t w fit toge!r & =m a p]fect squ>e4 ,illu/ration #aef4--,,! ,,dissect$ ,,triangle4 ,a good puzzle is t :i* ! g5tleman 9 ! illu/ration is %[+ to 8 fri5ds4 ,he has simply cut out ( pap] an equilat]al triangle--t is1 a triangle ) all its ?ree sides ( ! same l5g?4 ,he proposes t x %all be cut into five pieces 9 su* a way t !y w fit toge!r & =m ei!r two or ?ree small] equilat]al triangles1 us+ all ! mat]ial 9 ea* case4 ,c y discov] h[ ! cuts %\ld be made8 ,rememb] t :5 y h made y\r five pieces1 y mu/ be able1 z desir$1 to put !m toge!r to =m ei!r ! s+le orig9al triangle or to =m two triangles or to =m ?ree triangles--all equilat]al4 #aeg4--,,! ,,table-,,top ,,& ,,/ools4 ,i h frequ5tly _h occasion to %[ t ! publi%$ answ]s to a great _m ( ! olde/ & mo/ widely kn[n puzzles are ei!r q incorrect or capable ( improvem5t4 ,i propose to consid] ! old pos] ( ! table-top & /ools t mo/ ( my read]s h probably se5 9 "s =m or ano!r 9 books compil$ = ! recreation ( *ildhood4 ,! /ory is told t an economical & +5i\s s*oolma/] once wi%$ to conv]t a circul> table-top1 = :i* he _h no use1 into seats = two oval /ools1 ea* ) a h&-hole 9 ! c5tre4 ,he in/ruct$ ! c>p5t] to make ! cuts z 9 ! illu/ration & !n jo9 ! eight pieces toge!r 9 ! mann] %[n4 ,s impress$ 0 he ) ! +5uity ( 8 p]=mance t he set ! puzzle to 8 geometry class z a little /udy 9 dissection4 ,b ! rema9d] ( ! /ory has nev] be5 publi%$1 because1 s x is said1 x 0 a *>act]i/ic ( ! pr9cipals ( academies t !y w\ld nev] admit t !y c\ld err4 ,i get my in=mation f a desc5dant ( ! orig9al boy :o _h mo/ reason to be int]e/$ 9 ! matt]4 ,! clev] y\? sugge/$ mode/ly to ! ma/] t ! h&-holes 7 too big1 & t a small boy might p]haps fall ?r\gh !m4 ,he !re=e propos$ ano!r way ( mak+ ! cuts t w\ld get ov] ?is objection4 ,= 8 imp]t95ce he receiv$ su* sev]e *a/isem5t t he became conv9c$ t ! l>g] ! h&-hole 9 ! /ools ! m com=table might !y be4 ,illu/ration ,n[ :at 0 ! me?od ! boy propos$8 ,c y %[ h[ ! circul> table-top may be cut into eight pieces t w fit toge!r & =m two oval seats = /ools 7ea* ( exactly ! same size & %ape7 & ea* hav+ simil> h&-holes ( small] dim5sions ?an 9 ! case %[n above8 ,( c\rse1 all ! wood mu/ be us$4 #aeh4--,,! ,,great ,,monad4 ,illu/ration ,"h is a symbol ( trem5d\s antiquity :i* is wor?y ( notice4 ,x is borne on ! ,korean ensign & m]*ant flag1 & has be5 adopt$ z a trade sign by ! ,nor!rn ,pacific ,railroad ,company1 ?\gh probably few are aw>e t x is ! ,great ,monad1 z %[n 9 ! sket* bel[4 ,?is sign is to ! ,*9aman :at ! cross is to ! ,*ri/ian4 ,x is ! sign ( ,deity & et]nity1 :ile ! two p>ts into :i* ! circle is divid$ are call$ ! ,y9 & ! ,yan--! male & female =ces ( nature4 ,a writ] on ! subject m ?an ?ree ?\s& ye>s ago is report$ to h said 9 ref]5ce to x3 ,! illimitable produces ! great extreme4 ,! great extreme produces ! two pr9ciples4 ,! two pr9ciples produce ! f\r qu>t]s1 & f ! f\r qu>t]s we develop ! quadrature ( ! eight diagrams ( ,feuh-hi4 ,i hope read]s w n ask me to expla9 ?is1 = ,i h n ! slighte/ idea :at x means4 ,yet ,i am p]suad$ t = ages ! symbol has _h occult & probably ma!matical mean+s = ! esot]ic /ud5t4 ,i w introduce ! ,monad 9 its elem5t>y =m4 ,"h are ?ree easy que/ions respect+ ?is great symbol3-- 7,i47 ,:i* has ! great] area1 ! inn] circle conta9+ ! ,y9 & ! ,yan1 or ! out] r+8 7,,ii47 ,divide ! ,y9 & ! ,yan into f\r pieces ( ! same size & %ape by "o cut4 7,,iii47 ,divide ! ,y9 & ! ,yan into f\r pieces ( ! same size1 b diff]5t %ape1 by "o /raight cut4 #aei4--,,! ,,squ>e ,,( ,,v5e]4 ,! foll[+ repres5ts a piece ( wood 9 my possession1 #e 94 squ>e4 ,by m>k+s on ! surface x is divid$ into tw5ty-five squ>e in*es4 ,i want to discov] a way ( cutt+ ?is piece ( wood into ! fewe/ possible pieces t w fit toge!r & =m two p]fect squ>es ( diff]5t sizes & ( kn[n dim5sions4 ,b1 un=tunately1 at e "o ( ! sixte5 int]sections ( ! cross l9es a small nail has be5 driv5 9 at "s "t or o!r1 & my fret-saw w be injur$ if x comes 9 contact ) any ( ^!4 ,i h !re=e to f9d a me?od ( do+ ! "w t w n necessitate my cutt+ ?r\gh any ( ^? sixte5 po9ts4 ,h[ is x to be done8 ,rememb]1 ! exact dim5sions ( ! two squ>es mu/ be giv54 ,illu/ration #afj4--,,! ,,two ,,horse%oes4 ,illu/ration ,:y horse%oes %\ld be consid]$ lucky is "o ( ^? ?+s :i* no man c und]/&4 ,x is a v old sup]/ition1 & ,john ,aubrey 7#afbf-agjj7 says1 ,mo/ h\ses at ! ,we/ ,end ( ,london h a horse%oe on ! ?re%old4 ,9 ,monm\? ,/reet !re 7 sev5te5 9 #ahac & sev5 s late z #ahee4 ,ev5 ,"l ,nelson _h "o nail$ to ! ma/ ( ! %ip ,victory4 ,to-day we f9d x m conducive to good luck to see t !y are securely nail$ on ! feet ( ! horse we are ab\t to drive4 ,nev]!less1 s f> z ! horse%oe1 l ! ,swa/ika & o!r emblems t ,i h _h occasion at times to deal )1 has s]v$ to symbolize heal?1 prosp]ity1 & goodwill t[>ds m51 we may well treat x ) a c]ta9 am\nt ( respectful int]e/4 ,may !re n1 moreov]1 be "s esot]ic or lo/ ma!matical my/]y conceal$ 9 ! =m ( a horse%oe8 ,i h be5 look+ into ?is matt]1 & ,i wi% to draw my read]s' att5tion to ! v rem>kable fact t ! pair ( horse%oes %[n 9 my illu/ration are relat$ 9 a /rik+ & beautiful mann] to ! circle1 :i* is ! symbol ( et]nity4 ,i pres5t ?is fact 9 ! =m ( a simple problem1 s t x may be se5 h[ subtly ?is relation has be5 conceal$ = ages & ages4 ,my read]s w1 ,i "k1 be pleas$ :5 !y f9d ! key to ! my/]y4 ,cut out ! two horse%oes c>efully r\nd ! outl9e & !n cut !m into f\r pieces1 all diff]5t 9 %ape1 t w fit toge!r & =m a p]fect circle4 ,ea* %oe mu/ be cut into two pieces & all ! "p ( ! horse's ho( conta9$ )9 ! outl9e is to be us$ & reg>d$ z "p ( ! area4 #afa4--,,! ,,betsy ,,ross ,,puzzle4 ,a correspond5t ask$ me to supply him ) ! solution to an old puzzle t is attribut$ to a c]ta9 ,betsy ,ross1 ( ,philadelphia1 :o %[$ x to ,george ,wa%+ton4 ,x consi/s 9 s fold+ a piece ( pap] t ) "o clip ( ! scissors a five-po9t$ /> ( ,fre$om may be produc$4 ,:e!r ! /ory ( ! puzzle's orig9 is a true "o or n ,i _c say1 b ,i h a pr9t ( ! old h\se 9 ,philadelphia :]e ! lady is said to h liv$1 & ,i believe x /ill /&s !re4 ,b my read]s w d\btless be int]e/$ 9 ! little pos]4 ,take a circul> piece ( pap] & s fold x t ) "o cut ( ! scissors y c produce a p]fect five-po9t$ />4 #afb4--,,! ,,c>dbo>d ,,*a94 ,illu/ration ,c y cut ?is *a9 out ( a piece ( c>dbo>d )\t any jo9 :atev]8 ,e l9k is solid2 )\t its hav+ be5 split & aft]w>ds jo9$ at any place4 ,x is an int]e/+ old puzzle t ,i le>nt z a *ild1 b ,i h no k z to its inv5tor4 #afc4--,,! ,,pap] ,,box4 ,x may be int]e/+ to introduce "h1 ?\gh x is n /rictly a puzzle1 an +5i\s me?od = mak+ a pap] box4 ,take a squ>e ( /\t pap] & by successive fold+s make all ! creases indicat$ by ! dott$ l9es 9 ! illu/ration4 ,!n cut away ! eight little triangul> pieces t are %ad$1 & cut ?r\gh ! pap] along ! d>k l9es4 ,! second illu/ration %[s ! box half fold$ up1 & ! read] w h no difficulty 9 effect+ its completion4 ,be=e fold+ up1 ! read] might cut out ! circul> piece indicat$ 9 ! diagram1 = a purpose ,i w n[ expla94 ,?is box w be f\nd to s]ve excell5tly = ! production ( vortex r+s4 ,^! r+s1 :i* 7 discuss$ by ,von ,helmholtz 9 #aheh1 are mo/ int]e/+1 & ! box 7) ! hole cut out7 w produce !m to p]fection4 ,fill ! box ) tobacco smoke by bl[+ x g5tly ?r\gh ! hole4 ,n[1 if y hold x horizontally1 & s(tly tap ! side t is opposite to ! hole1 an imm5se numb] ( p]fect r+s c be produc$ f "o m\?ful ( smoke4 ,x is be/ t !re %\ld be no curr5ts ( air 9 ! room4 ,p (t5 d n realise t ^! r+s are =m$ 9 ! air :5 no smoke is us$4 ,! smoke only makes !m visible4 ,n[1 "o ( ^! r+s1 if prop]ly direct$ on its c\rse1 w travel across ! room & put out ! flame ( a c&le1 & ?is feat is mu* m /rik+ if y c manage to d x )\t ! smoke4 ,( c\rse1 ) a little practice1 ! r+s may be bl[n f ! m\?1 b ! box produces !m 9 mu* great] p]fection1 & no skill :atev] is requir$4 ,"l ,kelv9 prop\nd$ ! !ory t matt] may consi/ ( vortex r+s 9 a fluid t fills all space1 & by a developm5t ( ! hypo!sis he 0 able to expla9 *emical comb9ation4 ,illu/ration3 ---------------------------------------- ---- _/ _/ ---------------------------------------- ---- ,illu/ration #afd4--,,! ,,potato ,,puzzle4 ,take a circul> slice ( potato1 place x on ! table1 & see into h[ l>ge a numb] ( pieces y c divide x ) six cuts ( a knife4 ,( c\rse y mu/ n readju/ ! pieces or pile !m aft] a cut4 ,:at is ! greate/ numb] ( pieces y c make8 ,illu/ration3 -------- _/ #a_/ _/ #b _/ #c _/ _/ _/ _/ _/ #d _/ #e_/ #f _/ _/ _/ _/ #g_/ #h_/ #i_/aj _/ _/ _/ _/ _/ _/#aa_/ab_/ac_/ _/ _/ _/ _/#ad_/ae_/ _/ _/ _/#af_/ ----- ,! illu/ration %[s h[ to make sixte5 pieces4 ,?is c1 ( c\rse1 be easily beat54 #afe4--,,! ,,sev5 ,,pigs4 ,illu/ration ------------------------------ ,p ,p ,p ,p ,p ,p ,p ------------------------------ ,"h is a little puzzle t 0 put to "o ( ! sons ( ,er9 ! o!r day & p]plex$ him unduly1 = x is really q easy4 ,x w be se5 f ! illu/ration t he 0 %[n a sket* ( a squ>e p5 conta9+ sev5 pigs4 ,he 0 ask$ h[ he w\ld int]sect ! p5 ) ?ree /raight f5ces s z to enclose e pig 9 a sep>ate /y4 ,9 o!r words1 all y h to d is to take y\r p5cil &1 ) ?ree /raight /rokes across ! squ>e1 enclose ea* pig sep>ately4 ,no?+ c\ld be simpl]4 ,illu/ration ,! ,iri%man compla9$ t ! pigs w\ld n keep /ill :ile he 0 putt+ up ! f5ces4 ,he said t !y w\ld all flock toge!r1 or "o ob/9ate bea/ w\ld g into a corn] & flock all by himself4 ,x 0 po9t$ out to him t = ! purposes ( ! puzzle ! pigs 7 /ation>y4 ,he answ]$ t ,iri% pigs are n /ation]y--!y are pork4 ,be+ p]suad$ to make ! attempt1 he drew ?ree l9es1 "o ( :i* cut ?r\gh a pig4 ,:5 x 0 expla9$ t ?is is n all[$1 he prote/$ t a pig 0 no use until y cut its ?roat4 ,begorra1 if x's bacon ye want )\t cutt+ y\r pig1 x w be all gammon4 ,we w n d ! ,iri%man ! inju/ice ( sugge/+ t ! mis]able pun 0 int5tional4 ,h[ev]1 he fail$ to solve ! puzzle4 ,c y d x8 #aff4--,,! ,,l&[n]',s ,,f5ces4 ,! l&[n] 9 ! illu/ration is consult+ ) 8 bailiff ov] a r puzzl+ little "q4 ,he has a l>ge plan ( "o ( 8 fields1 9 :i* !re are elev5 trees4 ,n[1 he wants to divide ! field into j elev5 enclosures by means ( /raight f5ces1 s t e enclosure %all conta9 "o tree z a %elt] = 8 cattle4 ,h[ is he to d x ) z few f5ces z possible8 ,take y\r p5cil & draw /raight l9es across ! field until y h m>k$ (f ! elev5 enclosures 7& no m71 & !n see h[ _m f5ces y require4 ,( c\rse ! f5ces may cross "o ano!r4 #afg4--,,! ,,wiz>d',s ,,cats4 ,illu/ration ,a wiz>d plac$ t5 cats inside a magic circle z %[n 9 our illu/ration1 & hypnotiz$ !m s t !y %\ld rema9 /ation>y dur+ 8 pleasure4 ,he !n propos$ to draw ?ree circles inside ! l>ge "o1 s t no cat c\ld approa* ano!r cat )\t cross+ a magic circle4 ,try to draw ! ?ree circles s t e cat has its own enclosure & _c rea* ano!r cat )\t cross+ a l9e4 #afh4--,,! ,,*ri/mas ,,pudd+4 ,illu/ration ,speak+ ( ,*ri/mas pudd+s1 said ! ho/1 z he glanc$ at ! impos+ delicacy at ! o!r end ( ! table4 ,i am rem9d$ ( ! fact t a fri5d gave me a new puzzle ! o!r day respect+ "o4 ,"h x is1 he add$1 div+ into 8 brea/ pocket4 ',problem3 ,to f9d ! cont5ts1' ,i suppose1 said ! ,eton boy4 ,no2 ! pro( ( t is 9 ! eat+4 ,i w read y ! conditions4 ',cut ! pudd+ into two p>ts1 ea* ( exactly ! same size & %ape1 )\t t\*+ any ( ! plums4 ,! pudd+ is to be reg>d$ z a flat disc1 n z a sph]e4' ,:y %\ld y reg>d a ,*ri/mas pudd+ z a disc8 ,& :y %\ld any reasonable p]son ev] wi% to make su* an accurate division8 ask$ ! cynic4 ,x is j a puzzle--a problem 9 dissection4 ,all 9 turn _h a look at ! puzzle1 b nobody succe$$ 9 solv+ x4 ,x is a little difficult unless y are acqua9t$ ) ! pr9ciple involv$ 9 ! mak+ ( su* pudd+s1 b easy en\gh :5 y "k h[ x is done4 #afi4--;,a ,,tangram ,,p>adox4 ,_m pa/imes ( great antiquity1 su* z *ess1 h s develop$ & *ang$ d[n ! c5turies t _! orig9al inv5tors w\ld sc>cely recognize !m4 ,?is is n ! case ) ,tangrams1 a recreation t appe>s to be at lea/ f\r ?\s& ye>s old1 t has app>5tly nev] be5 dormant1 & t has n be5 alt]$ or improv$ ^u s9ce ! leg5d>y ,*9aman ,tan fir/ cut out ! sev5 pieces %[n 9 ,diagram ,i4 ,if y m>k ! po9t ,b1 midway betwe5 ,a & ,c1 on "o side ( a squ>e ( any size1 & ,d1 midway betwe5 ,c & ,e1 on an adjo9+ side1 ! direction ( ! cuts is too obvi\s to ne$ fur!r explanation4 ,e design 9 ?is article is built up f ! sev5 pieces ( black5$ c>dbo>d4 ,x w at once be und]/ood t ! possible comb9ations are inf9ite4 ,illu/ration ,! late ,mr4 ,sam ,loyd1 ( ,new ,york1 :o publi%$ a small book ( v +5i\s designs1 possess$ ! manuscripts ( ! late ,mr4 ,*all5or1 :o made a long & close /udy ( ,tangrams4 ,?is g5tleman1 x is said1 records t !re 7 orig9ally sev5 books ( ,tangrams1 compil$ 9 ,*9a two ?\s& ye>s be=e ! ,*ri/ian era4 ,^! books are s r>e t1 aft] =ty ye>s' resid5ce 9 ! c\ntry1 he only succe$$ 9 see+ p]fect copies ( ! fir/ & sev5? volumes ) fragm5ts ( ! second4 ,portions ( "o ( ! books1 pr9t$ 9 gold leaf ^u p>*m5t1 7 f\nd 9 ,pek+ by an ,engli% soldi] & sold = ?ree hundr$ p\nds4 ,a few ye>s ago a little book came into my possession1 f ! libr>y ( ! late ,lewis ,c>roll1 entitl$ ,! ,fa%ionable ,*9ese ,puzzle4 ,x conta9s ?ree hundr$ & tw5ty-?ree ,tangram designs1 mo/ly nondescript geometrical figures1 to be con/ruct$ f ! sev5 pieces4 ,x 0 ,publi%$ by ,j4 & ,e4 ,wallis1 #db ,sk9n] ,/reet1 & ,j4 ,wallis1 ,jun41 ,m>9e ,libr>y1 ,sidm\? 7,s\? ,devon74 ,!re is no date1 b ! foll[+ note fixes ! "t ( publication pretty closely3 ,?is +5i\s contrivance has = "s "t pa/ be5 ! fav\rite amusem5t ( ! ex-,emp]or ,napoleon1 :o1 be+ n[ 9 a debilitat$ /ate & liv+ v retir$1 passes _m h\rs a day 9 ?us ex]cis+ 8 pati5ce & +5uity4 ,! read] w f9d1 z did ! great exile1 t mu* amusem5t1 n :olly un9/ructive1 may be d]iv$ f =m+ ! designs ( o!rs4 ,he w f9d _m ( ! illu/rations to ?is article q easy to build up1 & "s r difficult4 ,e picture may ?us be reg>d$ z a puzzle4 ,b x is ano!r pa/ime altoge!r to create new & orig9al designs ( a pictorial "*1 & x is surpris+ :at extraord9>y scope ! ,tangrams af=d = produc+ pictures ( real life--angul> & (t5 grotesque1 x is true1 b full ( "*4 ,i give an example ( a recumb5t figure 7#b7 t is p>ticul>ly graceful1 & only ne$s "s slight r$uction ( its angul>ities to produce an entirely satisfactory outl9e4 ,z ,i h ref]r$ to ! au?or ( ,alice 9 ,wond]l&1 ,i give also my designs ( ! ,m>* ,h>e 7#c7 & ! ,hatt] 7#d74 ,i also give an attempt at ,napoleon 7#e71 & a v excell5t ,r$ ,indian ) 8 ,squaw by ,mr4 ,loyd 7#f & #g74 ,a l>ge numb] ( o!r designs w be f\nd 9 an article by me 9 ,! ,/r& ,magaz9e = ,novemb]1 #aijh4 ,illu/ration3 #b ,illu/ration3 #c ,illu/ration3 #d ,on ! appe>ance ( ?is magaz9e article1 ! late ,sir ,james ,murray1 ! em95t philologi/1 tri$1 ) t amaz+ indu/ry t *>act]iz$ all 8 "w1 to trace ! ^w tangram to its s\rce4 ,at l5g? he wrote z foll[s3--,"o ( my sons is a pr(essor 9 ! ,anglo-,*9ese college at ,ti5ts94 ,?r\gh him1 8 colleagues1 & 8 /ud5ts1 ,i 0 able to make inquiries z to ! alleg$ ,tan among ,*9ese s*ol>s4 ,our ,*9ese pr(essor "h 7,ox=d7 also took an int]e/ 9 ! matt] & obta9$ in=mation f ! secret>y ( ! ,*9ese ,legation 9 ,london1 :o is a v em95t repres5tative ( ! ,*9ese lit]ati4 ,illu/ration3 #e ,! result has be5 to %[ t ! man ,tan1 ! god ,tan1 & ! ',book ( ,tan' are entirely unkn[n to ,*9ese lit]ature1 hi/ory1 or tradition4 ,by mo/ ( ! le>n$ m5 ! "n1 or allegation ( ! exi/5ce1 ( ^! _h nev] be5 he>d (4 ,! puzzle is1 ( c\rse1 well kn[n4 ,x is call$ 9 ,*9ese *'i *'iao t'u2 lit]ally1 'sev5-+5i\s-plan' or '+5i\s-puzzle figure ( sev5 pieces4' ,no "n approa*+ 'tangram1' or ev5 'tan1' occurs 9 ,*9ese1 & ! only sugge/ions = ! latt] 7 ! ,*9ese t'an1 'to ext5d'2 or t'ang1 ,cantonese dialect = ',*9ese4' ,x 0 sugge/$ t probably "s ,am]ican or ,engli%man :o knew a little ,*9ese or ,cantonese1 want+ a "n = ! puzzle1 might concoct "o out ( "o ( ^! words & ! ,european end+ 'gram4' ,i %\ld say ! "n 'tangram' 0 probably inv5t$ by an ,am]ican "s little "t be=e #ahfd & aft] #ahdg1 b ,i _c f9d x 9 pr9t be=e ! #ahfd edition ( ,web/]4 ,i h !re=e _h to deal v %ortly ) ! ^w 9 ! diction>y1 tell+ :at x is appli$ to & :at conjectures or guesses h be5 made at ! "n1 & giv+ a few quotations1 "o f y\r own article1 :i* has enabl$ me to make m ( ! subject ?an ,i c\ld o!rwise h done4 ,illu/ration3 #f ,illu/ration3 #g ,sev]al correspond5ts h in=m$ me t !y possess1 or _h possess$1 specim5s ( ! old ,*9ese books4 ,an ,am]ican g5tleman writes to me z foll[s3--,i h 9 my possession a book made ( tissue pap]1 pr9t$ 9 black 7) a ,*9ese inscription on ! front page71 conta9+ ov] ?ree hundr$ designs1 :i* belongs to ! box ( 'tangrams1' :i* ,i also own4 ,! blocks are sev5 9 numb]1 made ( "m-(-pe>l1 highly poli%$ & f9ely engrav$ on ei!r side4 ,^! are conta9$ 9 a rosewood box #ba_/h 94 squ>e4 ,my great uncle1 ----1 0 "o ( ! fir/ mission>ies to visit ,*9a4 ,?is box & book1 along ) q a collection ( o!r relics1 7 s5t to my gr&fa!r & desc5d$ to myself4 ,my correspond5t k9dly suppli$ me ) rubb+s ( ! ,tangrams1 f :i* x is cle> t !y are cut 9 ! exact proportions t ,i h indicat$4 ,i reproduce ! ,*9ese inscription 7#h7 = ?is reason4 ,! own] ( ! book in=ms me t he has submitt$ x to a numb] ( ,*9am5 9 ! ,unit$ ,/ates & (f]$ z mu* z a doll> = a translation4 ,b !y all /eadfa/ly refus$ to read ! words1 (f]+ ! lame excuse t ! inscription is ,japanese4 ,natives ( ,japan1 h[ev]1 insi/ t x is ,*9ese4 ,is !re some?+ occult & esot]ic ab\t ,tangrams1 t x is s difficult to lift ! veil8 ,p]haps ?is page w come "u ! eye ( "s read] acqua9t$ ) ! ,*9ese language1 :o w supply ! requir$ translation1 :i* may1 or may n1 ?r[ a little light on ?is curi\s "q4 ,illu/ration3 #h ,by us+ sev]al sets ( ,tangrams at ! same "t we may con/ruct m ambiti\s pictures4 ,i 0 advis$ by a fri5d n to s5d my picture1 ,a ,game ( ,billi>ds 7#i71 to ! ,academy4 ,he assur$ me t x w\ld n be accept$ because ! judges are s hide-b\nd by conv5tion4 ,p]haps he 0 "r1 & x w be m appreciat$ by ,po/-impressioni/s & ,cubi/s4 ,! play]s are consid]+ a v delicate /roke at ! top ( ! table4 ,( c\rse1 ! two m51 ! table1 & ! clock are =m$ f f\r sets ( ,tangrams4 ,my second picture is nam$ ,! ,or*e/ra 7#aj71 & x 0 design$ = ! decoration ( a l>ge hall ( music4 ,"h we h ! conductor1 ! piani/1 ! fat little cornet-play]1 ! left-h&$ play] ( ! d\ble-bass1 ^: attitude is life-l1 ?\gh he does /& at an unusual di/ance f 8 in/rum5t1 & ! drumm]-boy1 ) 8 impos+ music-/&4 ,! dog at ! back ( ! pian(orte is n h[l+3 he is an appreciative li/5]4 ,illu/ration3 #i ,illu/ration3 #aj ,"o rem>kable ?+ ab\t ^! ,tangram pictures is t !y sugge/ to ! imag9ation su* a lot t is n really !re4 ,:o1 = example1 c look = a few m9utes at ,lady ,bel9da 7#aa7 & ! ,dut* girl 7#ab7 )\t soon feel+ ! haughty expression 9 ! "o case & ! ar* look 9 ! o!r8 ,!n look aga9 at ! /ork 7#ac71 & see h[ x is sugge/$ to ! m9d t ! leg is actually mu* m sl5d] ?an any "o ( ! pieces employ$4 ,x is really an optical illusion4 ,aga91 notice 9 ! case ( ! ya*t 7#ad7 h[1 by leav+ t little angul> po9t at ! top1 a complete ma/ is sugge/$4 ,if y place y\r ,tangrams toge!r on :ite pap] s t !y d n q t\* "o ano!r1 9 "s cases ! effect is improv$ by ! :ite l9es2 9 o!r cases x is almo/ de/roy$4 ,illu/ration3 #aa ,illu/ration3 #ab ,f9ally1 ,i give an example f ! _m curi\s p>adoxes t "o happ5s ^u 9 manipulat+ ,tangrams4 ,i %[ designs ( two dignifi$ individuals 7#ae & #af7 :o appe> to be exactly alike1 except = ! fact t "o has a foot & ! o!r has n4 ,n[1 bo? ( ^! figures are made f ! same sev5 ,tangrams4 ,:]e does ! second man get 8 foot f8 ,illu/ration3 #ac ,illu/ration3 #ad ,illu/ration3 #ae ,illu/ration3 #af ,,pat*work ,,puzzles4 ,( %r$s & pat*es4--,hamlet1 iii4 #d4 #agj4--,,! ,,cu%ion ,,cov]s4 ,illu/ration ,! above repres5ts a squ>e ( brocade4 ,a lady wi%es to cut x 9 f\r pieces s t two pieces w =m "o p]fectly squ>e cu%ion top1 & ! rema9+ two pieces ano!r squ>e cu%ion top4 ,h[ is %e to d x8 ,( c\rse1 %e c only cut along ! l9es t divide ! tw5ty-five squ>es1 & ! patt]n mu/ mat* prop]ly )\t any irregul>ity :atev] 9 ! design ( ! mat]ial4 ,!re is only "o way ( do+ x4 ,c y f9d x8 #aga4--,,! ,,bann] ,,puzzle4 ,illu/ration ,a ,lady _h a squ>e piece ( bunt+ ) two lions on x1 ( :i* ! illu/ration is an exactly reproduc$ r$uction4 ,%e wi%$ to cut ! /uff into pieces t w\ld fit toge!r & =m two squ>e bann]s ) a lion on ea* bann]4 ,%e discov]$ t ?is c\ld be done 9 z few z f\r pieces4 ,h[ did %e manage x8 ,( c\rse1 to cut ! ,briti% ,lion w\ld be an unp>donable (f5ce1 s y mu/ be c>eful t no cut passes ?r\gh any portion ( ei!r ( !m4 ,ladies are in=m$ t no all[ance :atev] has to be made = turn+s1 & no "p ( ! mat]ial may be wa/$4 ,x is q a simple little dissection puzzle if rightly attack$4 ,rememb] t ! bann]s h to be p]fect squ>es1 ?\gh !y ne$ n be bo? ( ! same size4 #agb4--,,mrs4 ,,smiley',s ,,*ri/mas ,,pres5t4 ,mrs4 ,smiley's expression ( pleasure 0 s9c]e :5 h] six gr&daught]s s5t to h]1 z a ,*ri/mas pres5t1 a v pretty pat*work quilt1 :i* !y _h made ) _! own h&s4 ,x 0 con/ruct$ ( squ>e pieces ( silk mat]ial1 all ( "o size1 & z !y made a l>ge quilt ) f\rte5 ( ^! little squ>es on ea* side1 x is obvi\s t j #aif pieces _h be5 /it*$ into x4 ,n[1 ! six gr&daught]s ea* contribut$ a "p ( ! "w 9 ! =m ( a p]fect squ>e 7all six portions be+ diff]5t 9 size71 b 9 ord] to jo9 !m up to =m ! squ>e quilt x 0 necess>y t ! "w ( "o girl %\ld be unpick$ into ?ree sep>ate pieces4 ,c y %[ h[ ! jo9s might h be5 made8 ,( c\rse1 no portion c be turn$ ov]4 ,illu/ration #agc4--,,mrs4 ,,p]k9s',s ,,quilt4 ,illu/ration ,x w be se5 t 9 ?is case ! squ>e pat*work quilt is built up ( #afi pieces4 ,! puzzle is to f9d ! smalle/ possible numb] ( squ>e portions ( :i* ! quilt c\ld be compos$ & %[ h[ !y might be jo9$ toge!r4 ,or1 to put x ! rev]se way1 divide ! quilt into z few squ>e portions z possible by m]ely cutt+ ! /it*es4 #agd4--,,! ,,squ>es ,,( ,,brocade4 ,illu/ration ,i happ5$ to be pay+ a call at ! h\se ( a lady1 :5 ,i took up f a table two lovely squ>es ( brocade4 ,!y 7 beautiful specim5s ( ,ea/]n workman%ip--bo? ( ! same design1 a delicate *equ]$ patt]n4 ,are !y n exquisite8 said my fri5d4 ,!y 7 br\ght to me by a c\s9 :o has j return$ f ,india4 ,n[1 ,i want y to give me a little assi/ance4 ,y see1 ,i h decid$ to jo9 !m toge!r s z to make "o l>ge squ>e cu%ion-cov]4 ,h[ %\ld ,i d ?is s z to mutilate ! mat]ial z little z possible8 ,( c\rse ,i propose to make my cuts only along ! l9es t divide ! little *equ]s4 ,illu/ration ,i cut ! two squ>es 9 ! mann] desir$ into f\r pieces t w\ld fit toge!r & =m ano!r l>g] squ>e1 tak+ c>e t ! patt]n %\ld mat* prop]ly1 & :5 ,i _h f9i%$ ,i notic$ t two ( ! pieces 7 ( exactly ! same area2 t is1 ea* ( ! two conta9$ ! same numb] ( *equ]s4 ,c y %[ h[ ! cuts 7 made 9 accordance ) ^! conditions8 #age--;,,ano!r ,,pat*work ,,puzzle4 ,illu/ration ,a lady 0 pres5t$1 by two ( h] girl fri5ds1 ) ! pretty pieces ( silk pat*work %[n 9 our illu/ration4 ,x w be se5 t bo? pieces are made up ( squ>es all ( ! same size--"o #ab #ab & ! o!r #e #e4 ,%e proposes to jo9 !m toge!r & make "o squ>e pat*work quilt1 #ac #ac1 b1 ( c\rse1 %e w n cut any ( ! mat]ial--m]ely cut ! /it*es :]e necess>y & jo9 toge!r aga94 ,:at p]plexes h] is ?is4 ,a fri5d assures h] t !re ne$ be no m ?an f\r pieces 9 all to jo9 up = ! new quilt4 ,c\ld y %[ h] h[ ?is little ne$lework puzzle is to be solv$ 9 s few pieces8 #agf4--,,l9oleum ,,cutt+4 ,illu/ration ,! diagram h]e) repres5ts two sep>ate pieces ( l9oleum4 ,! *equ]$ patt]n is n repeat$ at ! back1 s t ! pieces _c be turn$ ov]4 ,! puzzle is to cut ! two squ>es into f\r pieces s t !y %all fit toge!r & =m "o p]fect squ>e #aj #aj1 s t ! patt]n %all prop]ly mat*1 & s t ! l>g] piece %all h z small a portion z possible cut f x4 #agg4--;,,ano!r ,,l9oleum ,,puzzle4 ,illu/ration ,c y cut ?is piece ( l9oleum into f\r pieces t w fit toge!r & =m a p]fect squ>e8 ,( c\rse ! cuts may only be made along ! l9es4 ,,v>i\s ,,geometrical ,,puzzles4 ,s v>i\s are ! ta/es ( m54 ,,m>k ,,ak5side4 #agh4--,,! ,,c>dbo>d ,,box4 ,?is puzzle is n difficult1 b x w be f\nd ent]ta9+ to discov] ! simple rule = its solution4 ,i h a rectangul> c>dbo>d box4 ,! top has an area ( #abj squ>e in*es1 ! side #if squ>e in*es1 & ! end #hj squ>e in*es4 ,:at are ! exact dim5sions ( ! box8 #agi4--,,/eal+ ,,! ,,bell-,,ropes4 ,two m5 broke into a *ur* t[] "o night to /eal ! bell-ropes4 ,! two ropes pass$ ?r\gh holes 9 ! wood5 ceil+ high above !m1 & !y lo/ no "t 9 climb+ to ! top4 ,!n "o man drew 8 knife & cut ! rope above 8 head1 9 consequ5ce ( :i* he fell to ! floor & 0 badly injur$4 ,8 fell[-?ief call$ out t x s]v$ him "r = be+ su* a fool4 ,he said t he %\ld h done z he 0 do+1 ^u :i* he cut ! rope bel[ ! place at :i* he held on4 ,!n1 to 8 dismay1 he f\nd t he 0 9 no bett] plight1 =1 aft] hang+ on z long z 8 /r5g? la/$1 he 0 compell$ to let g & fall beside 8 comrade4 ,"h !y 7 bo? f\nd ! next morn+ ) _! limbs brok54 ,h[ f> did !y fall8 ,"o ( ! ropes :5 !y f\nd x 0 j t\*+ ! floor1 & :5 y pull$ ! end to ! wall1 keep+ ! rope taut1 x t\*$ a po9t j ?ree in*es above ! floor1 & ! wall 0 f\r feet f ! rope :5 x hung at re/4 ,h[ long 0 ! rope f floor to ceil+8 #ahj4--,,! ,,f\r ,,sons4 ,read]s w recognize ! diagram z a famili> fri5d ( _! y\?4 ,a man possess$ a squ>e-%ap$ e/ate4 ,he bequea!d to 8 wid[ ! qu>t] ( x t is %ad$ (f4 ,! rema9d] 0 to be divid$ equitably among/ 8 f\r sons1 s t ea* %\ld receive l& ( exactly ! same area & exactly simil> 9 %ape4 ,we are %[n h[ ?is 0 done4 ,b ! rema9d] ( ! /ory is n s g5]ally kn[n4 ,9 ! c5tre ( ! e/ate 0 a well1 indicat$ by ! d>k spot1 & ,b5jam91 ,*>les1 & ,david compla9$ t ! division 0 n equitable1 s9ce ,alfr$ _h access to ?is well1 :ile !y c\ld n rea* x )\t trespass+ on somebody else's l&4 ,! puzzle is to %[ h[ ! e/ate is to be apportion$ s t ea* son %all h l& ( ! same %ape & area1 & ea* h access to ! well )\t go+ (f 8 own l&4 ,illu/ration #aha4--,,! ,,?ree ,,railway ,,/ations4 ,z ,i sat 9 a railway c>riage ,i notic$ at ! o!r end ( ! comp>tm5t a wor?y squire1 :om ,i knew by sight1 engag$ 9 conv]sation ) ano!r pass5g]1 :o 0 evid5tly a fri5d ( 84 ,h[ f> h y to drive to y\r place f ! railway /ation8 ask$ ! /rang]4 ,well1 repli$ ! squire1 if ,i get out at ,apple=d1 x is j ! same di/ance z if ,i g to ,bridgefield1 ano!r fifte5 miles f>!r on2 & if ,i *ang$ at ,apple=d & w5t ?irte5 miles f !re to ,c>t]ton1 x w\ld /ill be ! same di/ance4 ,y see1 ,i am equidi/ant f ! ?ree /ations1 s ,i get a good *oice ( tra9s4 ,n[ ,i happ5$ to "k t ,bridgefield is j f\rte5 miles f ,c>t]ton1 s ,i amus$ myself 9 work+ out ! exact di/ance t ! squire _h to drive home :i*ev] /ation he got out at4 ,:at 0 ! di/ance8 #ahb4--,,! ,,g>d5 ,,puzzle4 ,pr(essor ,rackbra9 tells me t he 0 rec5tly smok+ a fri5dly pipe "u a tree 9 ! g>d5 ( a c\ntry acqua9tance4 ,! g>d5 0 enclos$ by f\r /raight walls1 & 8 fri5d in=m$ him t he _h measur$ ^! & f\nd ! l5g?s to be #hj1 #de1 #ajj1 & #fc y>ds respectively4 ,!n1 said ! pr(essor1 we c calculate ! exact area ( ! g>d54 ,impossible1 8 ho/ repli$1 because y c get an inf9ite numb] ( diff]5t %apes ) ^? f\r sides4 ,b y =get1 ,rackbrane said1 ) a tw9kle 9 8 eye1 t y told me once y _h plant$ ?is tree equidi/ant f all ! f\r corn]s ( ! g>d54 ,c y "w out ! g>d5's area8 #ahc4--;,,draw+ ,a ,,spiral4 ,if y hold ! page horizontally & give x a quick rot>y motion :ile look+ at ! c5tre ( ! spiral1 x w appe> to revolve4 ,p]haps a good _m read]s are acqua9t$ ) ?is little optical illusion4 ,b ! puzzle is to %[ h[ ,i 0 able to draw ?is spiral ) s mu* exactitude )\t us+ any?+ b a pair ( compasses & ! %eet ( pap] on :i* ! diagram 0 made4 ,h[ w\ld y proce$ 9 su* circum/ances8 ,illu/ration #ahd4--;,,h[ ,,to ,,draw ,,an ,,oval4 ,c y draw a p]fect oval on a %eet ( pap] ) "o sweep ( ! compasses8 ,x is "o ( ! easie/ ?+s 9 ! _w :5 y "k h[4 #ahe4--,,/4 ,,george',s ,,bann]4 ,at a celebration ( ! national fe/ival ( ,/4 ,george's ,day ,i 0 contemplat+ ! famili> bann] ( ! patron sa9t ( our c\ntry4 ,we all "k ! r$ cross on a :ite gr\nd1 %[n 9 our illu/ration4 ,?is is ! bann] ( ,/4 ,george4 ,! bann] ( ,/4 ,&rew 7,scotl&7 is a :ite ,/4 ,&rew's ,cross on a blue gr\nd4 ,t ( ,/4 ,patrick 7,irel&7 is a simil> cross 9 r$ on a :ite gr\nd4 ,^! ?ree are unit$ 9 "o to =m our ,union ,jack4 ,n[ on look+ at ,/4 ,george's bann] x occurr$ to me t ! foll[+ "q w\ld make a simple b pretty little puzzle4 ,suppos+ ! flag measures f\r feet by ?ree feet1 h[ wide mu/ ! arm ( ! cross be if x is requir$ t !re %all be us$ j ! same quantity ( r$ & ( :ite bunt+8 ,illu/ration #ahf4--,,! ,,clo!s ,,l9e ,,puzzle4 ,a boy ti$ a clo!s l9e f ! top ( ea* ( two poles to ! base ( ! o!r4 ,he !n propos$ to 8 fa!r ! foll[+ "q4 ,z "o pole 0 exactly sev5 feet above ! gr\nd & ! o!r exactly five feet1 :at 0 ! height f ! gr\nd :]e ! two cords cross$ "o ano!r8 #ahg4--,,! ,,milkmaid ,,puzzle4 ,illu/ration ,"h is a little pa/oral puzzle t ! read] may1 at fir/ sight1 be l$ into suppos+ is v pr(\nd1 involv+ deep calculations4 ,he may ev5 say t x is q impossible to give any answ] unless we are told some?+ def9ite z to ! di/ances4 ,& yet x is really q *ildlike & bl&4 ,9 ! corn] ( a field is se5 a milkmaid milk+ a c[1 & on ! o!r side ( ! field is ! dairy :]e ! extract has to be deposit$4 ,b x has be5 notic$ t ! "y woman always goes d[n to ! riv] ) h] pail be=e return+ to ! dairy4 ,"h ! suspici\s read] w p]haps ask :y %e pays ^! visits to ! riv]4 ,i c only reply t x is no bus9ess ( ours4 ,! alleg$ milk is entirely = local consumption4 ,:]e are y go+ to1 my pretty maid8 ,d[n to ! riv]1 sir1 %e said4 ,i'll n *oose y\r dairy1 my pretty maid4 ,nobody ax$ y1 sir1 %e said4 ,if "o _h any curiosity 9 ! matt]1 su* an indep5d5t _s w\ld entirely dis>m "o4 ,s we w pass f ! po9t ( comm]cial morality to ! subject ( ! puzzle4 ,draw a l9e f ! milk+-/ool d[n to ! riv] & !nce to ! door ( ! dairy1 :i* %all indicate ! %orte/ possible r\te = ! milkmaid4 ,t is all4 ,x is q easy to indicate ! exact spot on ! bank ( ! riv] to :i* %e %\ld direct h] /eps if %e wants z %ort a walk z possible4 ,c y f9d t spot8 #ahh4--,,! ,,ball ,,problem4 ,illu/ration ,a /onemason 0 engag$ ! o!r day 9 cutt+ out a r\nd ball = ! purpose ( "s ar*itectural decoration1 :5 a sm>t s*oolboy came ^u ! sc5e4 ,look "h1 said ! mason1 y seem to be a %>p y\ng/]1 c y tell me ?is8 ,if ,i plac$ ?is ball on ! level gr\nd1 h[ _m o!r balls ( ! same size c\ld ,i lay ar\nd x 7also on ! gr\nd7 s t e ball %\ld t\* ?is "o8 ,! boy at once gave ! correct answ]1 & !n put ?is little "q to ! mason3-- ,if ! surface ( t ball conta9$ j z _m squ>e feet z its volume conta9$ cubic feet1 :at w\ld be ! l5g? ( its diamet]8 ,! /onemason c\ld n give an answ]4 ,c\ld y h repli$ correctly to ! mason's & ! boy's que/ions8 #ahi4--,,! ,,york%ire ,,e/ates4 ,illu/ration ,i 0 on a visit to "o ( ! l>ge t[ns ( ,york%ire4 ,:ile walk+ to ! railway /ation on ! day ( my dep>ture a man ?ru/ a h&-bill ^u me1 & ,i took ?is into ! railway c>riage & read x at my leisure4 ,x in=m$ me t ?ree ,york%ire neighb\r+ e/ates 7 to be (f]$ = sale4 ,ea* e/ate 0 squ>e 9 %ape1 & !y jo9$ "o ano!r at _! corn]s1 j z %[n 9 ! diagram4 ,e/ate ,a conta9s exactly #cgj acres1 ,b conta9s #aaf acres1 & ,c #gd acres4 ,n[1 ! little triangul> bit ( l& enclos$ by ! ?ree squ>e e/ates 0 n (f]$ = sale1 &1 = no reason 9 p>ticul>1 ,i became curi\s z to ! area ( t piece4 ,h[ _m acres did x conta98 #aij4--;,,f>m] ,,wurzel',s ,,e/ate4 ,illu/ration ,i w n[ pres5t ano!r l& problem4 ,! demon/ration ( ! answ] t ,i %all give w1 ,i ?9k1 be f\nd bo? int]e/+ & easy ( compreh5sion4 ,f>m] ,wurzel own$ ! ?ree squ>e fields %[n 9 ! annex$ plan1 conta9+ respectively #ah1 #bj1 & #bf acres4 ,9 ord] to get a r+-f5ce r\nd 8 prop]ty he b\ght ! f\r int]v5+ triangul> fields4 ,! puzzle is to discov] :at 0 !n ! :ole area ( 8 e/ate4 #aia4--,,! ,,cresc5t ,,puzzle4 ,illu/ration ,"h is an easy geometrical puzzle4 ,! cresc5t is =m$ by two circles1 & ,c is ! c5tre ( ! l>g] circle4 ,! wid? ( ! cresc5t betwe5 ,b & ,d is #i in*es1 & betwe5 ,e & ,f #e in*es4 ,:at are ! diamet]s ( ! two circles8 #aib4--,,! ,,puzzle ,,wall4 ,illu/ration ,!re 0 a small lake1 ar\nd :i* f\r poor m5 built _! cottages4 ,f\r ri* m5 aft]w>ds built _! mansions1 z %[n 9 ! illu/ration1 & !y wi%$ to h ! lake to !mselves1 s !y in/ruct$ a build] to put up ! %orte/ possible wall t w\ld exclude ! cottag]s1 b give !mselves free access to ! lake4 ,h[ 0 ! wall to be built8 #aic4--,,! ,,%eepfold4 ,x is a curi\s fact t ! answ]s always giv5 to "s ( ! be/-kn[n puzzles t appe> 9 e little book ( fireside recreations t has be5 publi%$ = ! la/ fifty or a hundr$ ye>s are ei!r q unsatisfactory or cle>ly wrong4 ,yet nobody ev] seems to detect _! faults4 ,"h is an example3--,a f>m] _h a p5 made ( fifty hurdles1 capable ( hold+ a hundr$ %eep only4 ,suppos+ he want$ to make x suffici5tly l>ge to hold d\ble t numb]1 h[ _m additional hurdles mu/ he h8 #aid4--,,! ,,g>d5 ,,walls4 ,illu/ration ,a speculative c\ntry build] has a circul> field1 on :i* he has erect$ f\r cottages1 z %[n 9 ! illu/ration4 ,! field is surr\nd$ by a brick wall1 & ! own] und]took to put up ?ree o!r brick walls1 s t ! neighb\rs %\ld n be ov]look$ by ea* o!r1 b ! f\r t5ants insi/ t !re %all be no fav\ritism1 & t ea* %all h exactly ! same l5g? ( wall space = 8 wall fruit trees4 ,! puzzle is to %[ h[ ! ?ree walls may be built s t ea* t5ant %all h ! same area ( gr\nd1 & precisely ! same l5g? ( wall4 ,( c\rse1 ea* g>d5 mu/ be entirely enclos$ by its walls1 & x mu/ be possible to prove t ea* g>d5 has exactly ! same l5g? ( wall4 ,if ! puzzle is prop]ly solv$ no figures are necess>y4 #aie4--,,lady ,,bel9da',s ,,g>d54 ,lady ,bel9da is an en?usia/ic g>d5]4 ,9 ! illu/ration %e is depict$ 9 ! act ( worry+ out a pleasant little problem :i* ,i w relate4 ,"o ( h] g>d5s is oblong 9 %ape1 enclos$ by a high holly h$ge1 & %e is turn+ x into a ros>y = ! cultivation ( "s ( h] *oice/ roses4 ,%e wants to devote exactly half ( ! area ( ! g>d5 to ! fl[]s1 9 "o l>ge b$1 & ! o!r half to be a pa? go+ all r\nd x ( equal bread? ?r\gh\t4 ,su* a g>d5 is %[n 9 ! diagram at ! foot ( ! picture4 ,h[ is %e to m>k out ! g>d5 "u ^! simple conditions8 ,%e has only a tape1 ! l5g? ( ! g>d51 to d x )1 &1 z ! holly h$ge is s ?ick & d5se1 %e mu/ make all h] measurem5ts inside4 ,lady ,bel9da did n "k ! exact dim5sions ( ! g>d51 &1 z x 0 n necess>y = h] to "k1 ,i also give no dim5sions4 ,x is q a simple task no matt] :at ! size or proportions ( ! g>d5 may be4 ,yet h[ _m lady g>d5]s w\ld "k j h[ to proce$8 ,! tape may be q pla9--t is1 x ne$ n be a graduat$ measure4 ,illu/ration #aif4--,,! ,,te!r$ ,,goat4 ,illu/ration ,"h is a little problem t ev]ybody %\ld "k h[ to solve4 ,! goat is plac$ 9 a half-acre mead[1 t is 9 %ape an equilat]al triangle4 ,x is te!r$ to a po/ at "o corn] ( ! field4 ,:at %\ld be ! l5g? ( ! te!r 7to ! ne>e/ in*7 9 ord] t ! goat %all be able to eat j half ! grass 9 ! field8 ,x is assum$ t ! goat c fe$ to ! end ( ! te!r4 #aig4--,,! ,,compasses ,,puzzle4 ,x is curi\s h[ an add$ condition or re/riction w sometimes conv]t an absurdly easy puzzle into an int]e/+ & p]haps difficult "o4 ,i rememb] buy+ 9 ! /reet _m ye>s ago a little me*anical puzzle t _h a trem5d\s sale at ! "t4 ,x consi/$ ( a m$al ) holes 9 x1 & ! puzzle 0 to "w a r+ ) a gap 9 x f hole to hole until x 0 f9ally deta*$4 ,z ,i 0 walk+ along ! /reet ,i v soon acquir$ ! trick ( tak+ (f ! r+ ) "o h& :ile hold+ ! puzzle 9 my pocket4 ,a fri5d to :om ,i %[$ ! little feat set ab\t accompli%+ x himself1 & :5 ,i met him "s days aft]w>ds he exhibit$ 8 pr(ici5cy 9 ! art4 ,b he 0 a little tak5 aback :5 ,i !n took ! puzzle f him &1 :ile simply hold+ ! m$al betwe5 ! f+] & ?umb ( "o h&1 by a s]ies ( little %akes & j]ks caus$ ! r+1 )\t my ev5 t\*+ x1 to fall (f ^u ! floor4 ,! foll[+ little pos] w probably prove a r t\gh nut = a great _m read]s1 simply on acc\nt ( ! re/rict$ conditions3-- ,%[ h[ to f9d exactly ! middle ( any /raight l9e by means ( ! compasses only4 ,y are n all[$ to use any rul]1 p5cil1 or o!r article--only ! compasses2 & no trick or dodge1 su* z fold+ ! pap]1 w be p]mitt$4 ,y mu/ simply use ! compasses 9 ! ord9>y legitimate way4 #aih4--,,! ,,eight ,,/icks4 ,i h eight /icks1 f\r ( !m be+ exactly half ! l5g? ( ! o!rs4 ,i lay e "o ( ^! on ! table1 s t !y enclose ?ree squ>es1 all ( ! same size4 ,h[ d ,i d x8 ,!re mu/ be no loose ends hang+ ov]4 #aii4--,,papa',s ,,puzzle4 ,"h is a puzzle by ,pappus1 :o liv$ at ,alex&ria ab\t ! end ( ! ?ird c5tury4 ,x is ! fif? proposition 9 ! eigh? book ( 8 ,ma!matical ,collections4 ,i give x 9 ! =m t ,i pres5t$ x "s ye>s ago "u ! title ,papa's ,puzzle1 j to see h[ _m read]s w\ld discov] t x 0 by ,pappus himself4 ,! little maid's papa has tak5 two diff]5t-siz$ rectangul> pieces ( c>dbo>d1 & has clipp$ (f a triangul> piece f "o ( !m1 s t :5 x is susp5d$ by a ?read f ! po9t ,a x hangs ) ! long side p]fectly horizontal1 z %[n 9 ! illu/ration4 ,he has p]plex$ ! *ild by ask+ h] to f9d ! po9t ,a on ! o!r c>d1 s z to produce a simil> result :5 cut & susp5d$ by a ?read4 ,( c\rse1 ! po9t mu/ n be f\nd by trial clipp+s4 ,a curi\s & pretty po9t is involv$ 9 ?is sett+ ( ! puzzle4 ,c ! read] discov] x8 ,illu/ration #bjj4--;,a ,,kite-,,fly+ ,,puzzle4 ,:ile accompany+ my fri5d ,pr(essor ,highflite dur+ a sci5tific kite-fly+ competition on ! ,s\? ,d[ns ( ,sussex ,i 0 l$ into a little calculation t ought to int]e/ my read]s4 ,! ,pr(essor 0 pay+ out ! wire to :i* 8 kite 0 atta*$ f a w9* on :i* x _h be5 roll$ into a p]fectly sph]ical =m4 ,?is ball ( wire 0 j two feet 9 diamet]1 & ! wire _h a diamet] ( "o-hundr$? ( an in*4 ,:at 0 ! l5g? ( ! wire8 ,n[1 a simple little "q l ?is t ev]ybody c p]fectly und]/& w puzzle _m p to answ] 9 any way4 ,let u see :e!r1 )\t go+ into any pr(\nd ma!matical calculations1 we c get ! answ] r\ghly--say1 )9 a mile ( :at is correct6 ,we w assume t :5 ! wire is all w\nd up ! ball is p]fectly solid ?r\gh\t1 & t no all[ance has to be made = ! axle t passes ?r\gh x4 ,) t simplification1 ,i wond] h[ _m read]s c /ate )9 ev5 a mile ( ! correct answ] ! l5g? ( t wire4 #bja4--;,,h[ ,,to ,,make ,,ci/]ns4 ,illu/ration ,our fri5d 9 ! illu/ration has a l>ge %eet ( z9c1 measur+ 7be=e cutt+7 eight feet by ?ree feet1 & he has cut out squ>e pieces 7all ( ! same size7 f ! f\r corn]s & n[ proposes to fold up ! sides1 sold] ! edges1 & make a ci/]n4 ,b ! po9t t puzzles him is ?is3 ,has he cut out ^? squ>e pieces ( ! correct size 9 ord] t ! ci/]n may hold ! greate/ possible quantity ( wat]8 ,y see1 if y cut !m v small y get a v %all[ ci/]n2 if y cut !m l>ge y get a tall & sl5d] "o4 ,x is all a "q ( f9d+ a way ( cutt+ put ^! f\r squ>e pieces exactly ! "r size4 ,h[ are we to avoid mak+ !m too small or too l>ge8 #bjb4--,,! ,,cone ,,puzzle4 ,illu/ration ,i h a wood5 cone1 z %[n 9 ,fig4 #a4 ,h[ am ,i to cut out ( x ! greate/ possible cyl9d]8 ,x w be se5 t ,i c cut out "o t is long & sl5d]1 l ,fig4 #b1 or %ort & ?ick1 l ,fig4 #c4 ,b nei!r is ! l>ge/ possible4 ,a *ild c\ld tell y :]e to cut1 if he knew ! rule4 ,c y f9d ?is simple rule8 #bjc4--;,,conc]n+ ,,:eels4 ,illu/ration ,!re are "s curi\s facts conc]n+ ! movem5ts ( :eels t are apt to p]plex ! novice4 ,= example3 :5 a railway tra9 is travell+ f ,london to ,crewe c]ta9 p>ts ( ! tra9 at any giv5 mom5t are actually mov+ f ,crewe t[>ds ,london4 ,c y indicate ^? p>ts8 ,x seems absurd t p>ts ( ! same tra9 c at any "t travel 9 opposite directions1 b su* is ! case4 ,9 ! accompany+ illu/ration we h two :eels4 ,! l[] "o is suppos$ to be fix$ & ! upp] "o runn+ r\nd x 9 ! direction ( ! arr[s4 ,n[1 h[ _m times does ! upp] :eel turn on its own axis 9 mak+ a complete revolution ( ! o!r :eel8 ,d n be 9 a hurry ) y\r answ]1 or y are almo/ c]ta9 to be wrong4 ,exp]im5t ) two p5nies on ! table & ! correct answ] w surprise y1 :5 y succe$ 9 see+ x4 #bjd4--;,a ,,new ,,mat* ,,puzzle4 ,illu/ration ,9 ! illu/ration eighte5 mat*es are %[n arrang$ s t !y enclose two spaces1 "o j twice z l>ge z ! o!r4 ,c y re>range !m 7#a7 s z to enclose two f\r-sid$ spaces1 "o exactly ?ree times z l>ge z ! o!r1 & 7#b7 s z to enclose two five-sid$ spaces1 "o exactly ?ree times z l>ge z ! o!r8 ,all ! eighte5 mat*es mu/ be fairly us$ 9 ea* case2 ! two spaces mu/ be q deta*$1 & !re mu/ be no loose ends or duplicat$ mat*es4 #bje4--,,! ,,six ,,%eep-,,p5s4 ,illu/ration ,"h is a new little puzzle ) mat*es4 ,x w be se5 9 ! illu/ration t ?irte5 mat*es1 repres5t+ a f>m]'s hurdles1 h be5 s plac$ t !y enclose six %eep-p5s all ( ! same size4 ,n[1 "o ( ^! hurdles 0 /ol51 & ! f>m] want$ /ill to enclose six p5s ( equal size ) ! rema9+ twelve4 ,h[ 0 he to d x8 ,all ! twelve mat*es mu/ be fairly us$1 & !re mu/ be no duplicat$ mat*es or loose ends4 ,,po9ts ,,& ,,l9es ,,problems4 ,l9e ^u l9e1 l9e ^u l9e2 "h a little & !re a little4--,isa4 xxviii4 #aj4 ,:at are kn[n z ,po9ts & ,l9es puzzles are f\nd v int]e/+ by _m p4 ,! mo/ famili> example1 "h giv51 to plant n9e trees s t !y %all =m t5 /raight r[s ) ?ree trees 9 e r[1 is attribut$ to ,sir ,isaac ,newton1 b ! e>lie/ collection ( su* puzzles is1 ,i believe1 9 a r>e little book t ,i possess--publi%$ 9 #ahba--,rational ,amusem5t = ,w9t] ,ev5+s1 by ,john ,jackson4 ,! au?or gives t5 examples ( ,trees plant$ 9 ,r[s4 ,^! tree-plant+ puzzles h always be5 a matt] ( great p]plexity4 ,!y are real puzzles1 9 ! true/ s5se ( ! ^w1 because nobody has yet succe$$ 9 f9d+ a direct & c]ta9 way ( solv+ !m4 ,!y dem& ! ex]cise ( sagacity1 +5uity1 & pati5ce1 & :at we call luck is also sometimes ( s]vice4 ,p]haps "s day a g5ius w discov] ! key to ! :ole my/]y4 ,rememb] t ! trees mu/ be reg>d$ z m]e po9ts1 = if we 7 all[$ to make our trees big en\gh we might easily fudge our diagrams & get 9 a few extra /raight r[s t 7 m app>5t ?an real4 ,illu/ration #bjf4--,,! ,,k+ ,,& ,,! ,,ca/les4 ,!re 0 once1 9 anci5t times1 a p[]ful k+1 :o _h ecc5tric ideas on ! subject ( milit>y ar*itecture4 ,he held t !re 0 great /r5g? & economy 9 symmetrical =ms1 & always cit$ ! example ( ! bees1 :o con/ruct _! combs 9 p]fect hexagonal cells1 to prove t he _h nature to support him4 ,he resolv$ to build t5 new ca/les 9 8 c\ntry all to be connect$ by =tifi$ walls1 :i* %\ld =m five l9es ) f\r ca/les 9 e l9e4 ,! royal ar*itect pres5t$ 8 prelim9>y plan 9 ! =m ,i h %[n4 ,b ! mon>* po9t$ out t e ca/le c\ld be approa*$ f ! outside1 & comm&$ t ! plan %\ld be s modifi$ t z _m ca/les z possible %\ld be free f attack f ! outside1 & c\ld only be rea*$ by cross+ ! =tifi$ walls4 ,! ar*itect repli$ t he ?\ght x impossible s to arrange !m t ev5 "o ca/le1 :i* ! k+ propos$ to use z a royal resid5ce1 c\ld be s protect$1 b 8 maje/y soon enlight5$ him by po9t+ out h[ x might be done4 ,h[ w\ld y h built ! t5 ca/les & =tifications s z be/ to fulfil ! k+'s requirem5ts8 ,rememb] t !y mu/ =m five /raight l9es ) f\r ca/les 9 e l9e4 ,illu/ration #bjg4--;,,*]ries ,,& ,,plums4 ,illu/ration ,! illu/ration is a plan ( a cottage z x /&s surr\nd$ by an or*>d ( fifty-five trees4 ,t5 ( ^! trees are *]ries1 t5 are plums1 & ! rema9d] apples4 ,! *]ries are s plant$ z to =m five /raight l9es1 ) f\r *]ry trees 9 e l9e4 ,! plum trees are also plant$ s z to =m five /raight l9es ) f\r plum trees 9 e l9e4 ,! puzzle is to %[ :i* are ! t5 *]ry trees & :i* are ! t5 plums4 ,9 ord] t ! *]ries & plums %\ld h ! mo/ fav\rable aspect1 z few z possible 7"u ! conditions7 are plant$ on ! nor? & ea/ sides ( ! or*>d4 ,( c\rse 9 pick+ out a gr\p ( t5 trees 7*]ry or plum1 z ! case may be7 y ignore all int]v5+ trees4 ,t is to say1 f\r trees may be 9 a /raight l9e irrespective ( o!r trees 7or ! h\se7 be+ 9 betwe54 ,aft] ! la/ puzzle ?is w be q easy4 #bjh4--;,a ,,plantation ,,puzzle4 ,illu/ration ,a man _h a squ>e plantation ( =ty-n9e trees1 b1 z w be se5 by ! omissions 9 ! illu/ration1 f\r trees 7 bl[n d[n & remov$4 ,he n[ wants to cut d[n all ! rema9d] except t5 trees1 :i* are to be s left t !y %all =m five /raight r[s ) f\r trees 9 e r[4 ,:i* are ! t5 trees t he mu/ leave8 #bji4--,,! ,,tw5ty-,,"o ,,trees4 ,a g5tleman wi%$ to plant tw5ty-"o trees 9 8 p>k s t !y %\ld =m twelve /raight r[s ) five trees 9 e r[4 ,c\ld y h suppli$ him ) a pretty symmetrical arrangem5t t w\ld satisfy ^! conditions8 #baj4--,,! ,,t5 ,,co9s4 ,place t5 p5nies on a l>ge %eet ( pap] or c>dbo>d1 z %[n 9 ! diagram1 five on ea* edge4 ,n[ remove f\r ( ! co9s1 )\t di/urb+ ! o!rs1 & replace !m on ! pap] s t ! t5 %all =m five /raight l9es ) f\r co9s 9 e l9e4 ,?is 9 itself is n difficult1 b y %\ld try to discov] 9 h[ _m diff]5t ways ! puzzle may be solv$1 assum+ t 9 e case ! two r[s at />t+ are exactly ! same4 ,illu/ration #baa4--,,! ,,twelve ,,m9ce-,,pies4 ,x w be se5 9 our illu/ration h[ twelve m9ce-pies may be plac$ on ! table s z to =m six /raight r[s ) f\r pies 9 e r[4 ,! puzzle is to remove only f\r ( !m to new positions s t !re %all be sev5 /raight r[s ) f\r 9 e r[4 ,:i* f\r w\ld y remove1 & :]e w\ld y replace !m8 ,illu/ration #bab4--,,! ,,burmese ,,plantation4 ,illu/ration ,a %ort "t ago ,i receiv$ an int]e/+ communication f ! ,briti% *apla9 at ,meiktila1 ,upp] ,burma1 9 :i* my correspond5t in=m$ me t he _h f\nd "s amusem5t on bo>d %ip on 8 way out 9 try+ to solve ?is little pos]4 ,if he has a plantation ( =ty-n9e trees1 plant$ 9 ! =m ( a squ>e z %[n 9 ! accompany+ illu/ration1 he wi%es to "k h[ he may cut d[n tw5ty-sev5 ( ! trees s t ! tw5ty-two left /&+ %all =m z _m r[s z possible ) f\r trees 9 e r[4 ,( c\rse !re may n be m ?an f\r trees 9 any r[4 #bac4--,,turks ,,& ,,russians4 ,?is puzzle is on ! l9es ( ! ,afridi problem publi%$ by me 9 ,tit-,bits "s ye>s ago4 ,on an op5 level tract ( c\ntry a p>ty ( ,russian infantry1 no two ( :om 7 /ation$ at ! same spot1 7 sudd5ly surpris$ by ?irty-two ,turks1 :o op5$ fire on ! ,russians f all directions4 ,ea* ( ! ,turks simultane\sly fir$ a bullet1 & ea* bullet pass$ imm$iately ov] ! heads ( ?ree ,russian soldi]s4 ,z ea* ( ^! bullets :5 fir$ kill$ a diff]5t man1 ! puzzle is to discov] :at is ! smalle/ possible numb] ( soldi]s ( :i* ! ,russian p>ty c\ld h consi/$ & :at 7 ! casualties on ea* side4 ,,mov+ ,,c\nt] ,,problems4 ,i _c d't )\t c\nt]s4 ,w9t]'s ,tale1 iv4 #c4 ,puzzles ( ?is class1 except s f> z !y occur 9 connection ) actual games1 su* z *ess1 seem to be a comp>atively mod]n introduction4 ,ma!maticians 9 rec5t times1 notably ,v&]monde & ,reiss1 h devot$ "s att5tion to !m1 b !y d n appe> to h be5 consid]$ by ! old writ]s4 ,s f> z games ) c\nt]s are conc]n$1 p]haps ! mo/ anci5t & widely kn[n 9 old times is ,n9e ,m5's ,morris 7kn[n also1 z ,i %all %[1 "u a great _m o!r names71 unless ! simpl] game1 di/9ctly m5tion$ 9 ! works ( ,ovid 7,no4 #aaj1 ,ovid's ,game1 9 ,! ,cant]bury ,puzzles71 f :i* ,n\ghts & ,crosses seems to be d]iv$1 is /ill m anci5t4 ,9 ,france ! game is call$ ,m>elle1 9 ,pol& ,sieg5 ,wulf ,myll 7,%e-goat ,wolf ,mill1 or ,fight71 9 ,g]many & ,au/ria x is call$ ,muhle 7! ,mill71 9 ,icel& x goes by ! "n ( ,mylla1 :ile ! ,bogas 7or native b>gees7 ( ,s\? ,am]ica are said to play x1 & on ! ,amazon x is call$ ,trique1 & held to be ( ,indian orig94 ,9 our own c\ntry x has diff]5t names 9 diff]5t di/ricts1 su* z ,meg ,m]rylegs1 ,peg ,m]yll1 ,n9e ,peg o',m]ryal1 ,n9e-,p9 ,miracle1 ,m]ry ,peg1 & ,m]ry ,hole4 ,%akespe>e ref]s to x 9 ,midsumm] ,night's ,dream 7,act ii41 sc5e #a73-- ,! n9e-m5's morris is fill$ up ) mud2 ,& ! qua9t mazes 9 ! wanton gre51 ,= lack ( tread1 are undi/+ui%able4 ,x 0 play$ by ! %eph]ds ) /ones 9 holes cut 9 ! turf4 ,john ,cl>e1 ! peasant poet ( ,nor?ampton%ire1 9 ,! ,%eph]d ,boy 7#ahce7 says3--,(t we track 8 haunts 4444 ,by n9e-peg-morris nick$ ^u ! gre54 ,x is also m5tion$ by ,drayton 9 8 ,polyolbion4 ,x 0 f\nd on an old ,roman tile discov]$ dur+ ! excavations at ,sil*e/]1 & cut ^u ! /eps ( ! ,acropolis at ,a!ns4 ,:5 visit+ ! ,*ri/iania ,museum a few ye>s ago ,i 0 %[n ! great ,vik+ %ip t 0 discov]$ at ,gok/ad 9 #ahhj4 ,on ! oak planks =m+ ! deck ( ! vessel 7 f\nd boles & l9es m>k+ out ! game1 ! holes be+ made to receive pegs4 ,:ile inspect+ ! anci5t oak furniture 9 ! ,rijks ,museum at ,am/]dam ,i became int]e/$ 9 an old cate*um5's settle1 & 0 surpris$ to f9d ! game diagram cut 9 ! c5tre ( ! seat--q conv5i5tly = surreptiti\s play4 ,x has be5 discov]$ cut 9 ! *oir /alls ( sev]al ( our ,engli% ca!drals4 ,9 ! e>ly eighties x 0 f\nd scrat*$ ^u a /one built into a wall 7probably ab\t ! date #abjj71 dur+ ! re/oration ( ,h>grave *ur* 9 ,nor?ampton%ire4 ,?is /one is n[ 9 ! ,nor?ampton ,museum4 ,a simil> /one has s9ce be5 f\nd at ,sempr+ham1 ,l9coln%ire4 ,x is to be se5 on an anci5t tomb/one 9 ! ,isle ( ,man1 & pa9t$ on old ,dut* tiles4 ,& 9 #aija a /one 0 dug out ( a gravel pit ne> ,oswe/ry be>+ an und\bt$ diagram ( ! game4 ,! game has be5 play$ ) diff]5t rules at diff]5t p]iods & places4 ,i give a copy ( ! bo>d4 ,sometimes ! diagonal l9es are omitt$1 b ?is evid5tly 0 n int5d$ to affect ! play3 x simply meant t ! angles alone 7 ?\ght suffici5t to indicate ! po9ts4 ,?is is h[ ,/rutt1 9 ,sports & ,pa/imes1 describes ! game1 & x agrees ) ! way ,i play$ x z a boy3--,two p]sons1 hav+ ea* ( !m n9e pieces1 or m51 lay !m d[n alt]nately1 "o by "o1 ^u ! spots2 & ! bus9ess ( ei!r p>ty is to prev5t 8 antagoni/ f plac+ ?ree ( 8 pieces s z to =m a r[ ( ?ree1 )\t ! int]v5tion ( an oppon5t piece4 ,if a r[ be =m$1 he t made x is at lib]ty to take up "o ( 8 competitor's pieces f any "p he ?9ks mo/ to 8 advantage2 except+ he has made a r[1 :i* mu/ n be t\*$ if he h ano!r piece ^u ! bo>d t is n a compon5t "p ( t r[4 ,:5 all ! pieces are laid d[n1 !y are play$ backw>ds & =w>ds1 9 any direction t ! l9es run1 b only c move f "o spot to ano!r 7next to x7 at "o "t4 ,he t takes (f all 8 antagoni/'s pieces is ! conqu]or4 ,illu/ration #bad4--,,! ,,six ,,frogs4 ,illu/ration ,! six educat$ frogs 9 ! illu/ration are tra9$ to rev]se _! ord]1 s t _! numb]s %all read #f1 #e1 #d1 #c1 #b1 #a1 ) ! blank squ>e 9 its pres5t position4 ,!y c jump to ! next squ>e 7if vacant7 or leap ov] "o frog to ! next squ>e beyond 7if vacant71 j z we move 9 ! game ( draughts1 & c g backw>ds or =w>ds at pleasure4 ,c y %[ h[ !y p]=m _! feat 9 ! fewe/ possible moves8 ,x is q easy1 s :5 y h done x add a sev5? frog to ! "r & try aga94 ,!n add m frogs until y are able to give ! %orte/ solution = any numb]4 ,= x c always be done1 ) t s+le vacant squ>e1 no matt] h[ _m frogs !re are4 #bae4--,,! ,,gras%opp] ,,puzzle4 ,x has be5 sugge/$ t ?is puzzle 0 a great fav\rite among ! "y appr5tices ( ! ,city ( ,london 9 ! sixte5? & sev5te5? c5turies4 ,read]s w h notic$ ! curi\s brass gras%opp] on ! ,royal ,ex*ange4 ,?is long-liv$ creature escap$ ! fires ( #afff & #ahch4 ,! gras%opp]1 aft] 8 k9d1 0 ! cre/ ( ,sir ,?omas ,gre%am1 m]*ant groc]1 :o di$ 9 #aegi1 & f ?is cause x has be5 us$ z a sign by groc]s 9 g5]al4 ,un=tunately = ! leg5d z to its orig91 ! puzzle 0 only produc$ by myself s late z ! ye> #aijj4 ,on twelve ( ! ?irte5 black discs are plac$ numb]$ c\nt]s or gras%opp]s4 ,! puzzle is to rev]se _! ord]1 s t !y %all read1 #a1 #b1 #c1 #d1 etc41 9 ! opposite direction1 ) ! vacant disc left 9 ! same position z at pres5t4 ,move "o at a "t 9 any ord]1 ei!r to ! adjo9+ vacant disc or by jump+ ov] "o gras%opp]1 l ! moves 9 draughts4 ,! moves or leaps may be made 9 ei!r direction t is at any "t possible4 ,:at are ! fewe/ possible moves 9 :i* x c be done8 ,illu/ration #baf4--,,! ,,educat$ ,,frogs4 ,illu/ration ,our six educat$ frogs h le>nt a new & pretty feat4 ,:5 plac$ on glass tumbl]s1 z %[n 9 ! illu/ration1 !y *ange sides s t ! ?ree black ones are to ! left & ! :ite frogs to ! "r1 ) ! unoccupi$ tumbl] at ! opposite end--,no4 #g4 ,!y c jump to ! next tumbl] 7if unoccupi$71 or ov] "o1 or two1 frogs to an unoccupi$ tumbl]4 ,! jumps c be made 9 ei!r direction1 & a frog may jump ov] 8 own or ! opposite col\r1 or bo? col\rs4 ,f\r successive specim5 jumps w make ev]y?+ q pla93 #d to #a1 #e to #d1 #c to #e1 #f to #c4 ,c y %[ h[ !y d x 9 t5 jumps8 #bag4--,,! ,,twick5ham ,,puzzle4 ,illu/ration3 7 ,i 7 77,n77 7 ,m 7 77,a77 7 ,h 7 77,t77 7 ,e 7 77,w77 7 ,c 7 77,k77 7 7 ,9 ! illu/ration we h elev5 discs 9 a circle4 ,on five ( ! discs we place :ite c\nt]s ) black lett]s--z %[n--& on five o!r discs ! black c\nt]s ) :ite lett]s4 ,! bottom disc is left vacant4 ,/>t+ ?us1 x is requir$ to get ! c\nt]s into ord] s t !y spell ! ^w ,twick5ham 9 a clockwise direction1 leav+ ! vacant disc 9 ! orig9al position4 ,! black c\nt]s move 9 ! direction t a clock-h& revolves1 & ! :ite c\nt]s g ! opposite way4 ,a c\nt] may jump ov] "o ( ! opposite col\r if ! vacant disc is next beyond4 ,?us1 if y\r fir/ move is ) ,k1 !n ,c c jump ov] ,k4 ,if !n ,k moves t[>ds ,e1 y may next jump ,w ov] ,c1 & s on4 ,! puzzle may be solv$ 9 tw5ty-six moves4 ,rememb] a c\nt] _c jump ov] "o ( its own col\r4 #bah4--,,! ,,victoria ,,cross ,,puzzle4 ,illu/ration3 --------------------- 444 ,a 444_/ 7,i7 4444444 7,v7 _/ 444444 ------ 44 ,r 4 4 ,i 44 444444 444444 _/ 4444444 7,o7 44 ,t 44 7,c7 _/444444444 --------------------- ,! puzzle-mak] is peculi>ly a snapp]-up ( unconsid]$ trifles1 & 8 productions are (t5 built up ) ! sl5d]e/ mat]ials4 ,trivialities t might entirely escape ! obs]vation ( o!rs1 or1 if !y 7 obs]v$1 w\ld be reg>d$ z ( no possible mom5t1 (t5 supply ! man :o is 9 que/ ( pos]s ) a pretty !me or an idea t he ?9ks possesses "s basal value4 ,:5 seat$ opposite to a lady 9 a railway c>riage at ! "t ( ,que5 ,victoria's ,diamond ,jubilee1 my att5tion 0 attract$ to a broo* t %e 0 we>+4 ,x 0 9 ! =m ( a ,maltese or ,victoria ,cross1 & bore ! lett]s ( ! ^w ,,victoria4 ,! numb] & arrangem5t ( ! lett]s imm$iately gave me ! sugge/ion = ! puzzle :i* ,i n[ pres5t4 ,! diagram1 x w be se51 is compos$ ( n9e divisions4 ,! puzzle is to place eight c\nt]s1 be>+ ! lett]s ( ! ^w ,,victoria1 exactly 9 ! mann] %[n1 & !n slide "o lett] at a "t f black to :ite & :ite to black alt]nately1 until ! ^w reads r\nd 9 ! same direction1 only ) ! initial lett] ,v on "o ( ! black arms ( ! cross4 ,at no "t may two lett]s be 9 ! same division4 ,x is requir$ to f9d ! %orte/ me?od4 ,leap+ moves are1 ( c\rse1 n p]mitt$4 ,! fir/ move mu/ obvi\sly be made ) ,a1 ,i1 ,t1 or ,r4 ,suppos+ y move ,t to ! c5tre1 ! next c\nt] play$ w be ,o or ,c1 s9ce ,i or ,r _c be mov$4 ,!re is some?+ a little rem>kable 9 ! solution ( ?is puzzle :i* ,i w expla94 #bai4--,,! ,,lett] ,,block ,,puzzle4 ,illu/ration3 --------------- ,g ,e ,f --------------- ,h ,c ,b --------------- ,d ,a --------------- ,"h is a little rem9isc5ce ( our old fri5d ! ,fifte5 ,block ,puzzle4 ,eight wood5 blocks are lett]$1 & are plac$ 9 a box1 z %[n 9 ! illu/ration4 ,x w be se5 t y c only move "o block at a "t to ! place vacant = ! "t be+1 z no block may be lift$ out ( ! box4 ,! puzzle is to %ift !m ab\t until y get !m 9 ! ord]-- ,a ,b ,c ,d ,e ,f ,g ,h ,?is y w f9d by no means difficult if y are all[$ z _m moves z y l4 ,b ! puzzle is to d x 9 ! fewe/ possible moves4 ,i w n say :at ?is smalle/ numb] ( moves is1 because ! read] may l to discov] x = himself4 ,9 writ+ d[n y\r moves y w f9d x necess>y to record no m ?an ! lett]s 9 ! ord] t !y are %ift$4 ,?us1 y\r fir/ five moves might be ,c1 ,h1 ,g1 ,e1 ,f2 & ?is notation c h no possible ambiguity4 ,9 practice y only ne$ eight c\nt]s & a simple diagram on a %eet ( pap]4 #bbj4--;,a ,,lodg+-,,h\se ,,difficulty4 ,illu/ration ,! ,dobsons secur$ ap>tm5ts at ,slocomb-on-,sea4 ,!re 7 six rooms on ! same floor1 all communicat+1 z %[n 9 ! diagram4 ,! rooms !y took 7 numb]s #d1 #e1 & #f1 all fac+ ! sea4 ,b a little difficulty arose4 ,mr4 ,dobson insi/$ t ! piano & ! bookcase %\ld *ange rooms4 ,?is 0 wily1 = ! ,dobsons 7 n musical1 b !y want$ to prev5t any "o else play+ ! in/rum5t4 ,n[1 ! rooms 7 v small & ! pieces ( furniture indicat$ 7 v big1 s t no two ( ^! articles c\ld be got into any room at ! same "t4 ,h[ 0 ! ex*ange to be made ) ! lea/ possible lab\r8 ,suppose1 = example1 y fir/ move ! w>drobe into ,no4 #b2 !n y c move ! bookcase to ,no4 #e & ! piano to ,no4 #f1 & s on4 ,x is a fasc9at+ puzzle1 b ! l&lady _h reasons = n appreciat+ x4 ,try to solve h] difficulty 9 ! fewe/ possible removals ) c\nt]s on a %eet ( pap]4 #bba4--,,! ,,eight ,,eng9es4 ,! diagram repres5ts ! eng9e-y>d ( a railway company "u ecc5tric managem5t4 ,! eng9es are all[$ to be /ation>y only at ! n9e po9ts indicat$1 "o ( :i* is at pres5t vacant4 ,x is requir$ to move ! eng9es1 "o at a "t1 f po9t to po9t1 9 sev5te5 moves1 s t _! numb]s %all be 9 num]ical ord] r\nd ! circle1 ) ! c5tral po9t left vacant4 ,b "o ( ! eng9es has _h its fire drawn1 & !re=e _c move4 ,h[ is ! ?+ to be done8 ,& :i* eng9e rema9s /ation>y ?r\gh\t8 ,illu/ration #bbb4--;,a ,,railway ,,puzzle4 ,illu/ration ,make a diagram1 on a l>ge %eet ( pap]1 l ! illu/ration1 & h ?ree c\nt]s m>k$ ,a1 ?ree m>k$ ,b1 & ?ree m>k$ ,c4 ,x w be se5 t at ! int]section ( l9es !re are n9e /opp+-places1 & a t5? /opp+-place is atta*$ to ! out] circle l ! tail ( a ,q4 ,place ! ?ree c\nt]s or eng9es m>k$ ,a1 ! ?ree m>k$ ,b1 & ! ?ree m>k$ ,c at ! places indicat$4 ,! puzzle is to move ! eng9es1 "o at a "t1 along ! l9es1 f /opp+-place to /opp+-place1 until y succe$ 9 gett+ an ,a1 a ,b1 & a ,c on ea* circle1 & also ,a1 ,b1 & ,c on ea* /raight l9e4 ,y are requir$ to d ?is 9 z few moves z possible4 ,h[ _m moves d y ne$8 #bbc4--;,a ,,railway ,,muddle4 ,! plan repres5ts a portion ( ! l9e ( ! ,london1 ,clodville1 & ,mud=d ,railway ,company4 ,x is a s+le l9e ) a loop4 ,!re is only room = eight wagons1 or sev5 wagons & an eng9e1 betwe5 ,b & ,c on ei!r ! left l9e or ! "r l9e ( ! loop4 ,x happ5$ t two goods tra9s 7ea* consi/+ ( an eng9e & sixte5 wagons7 got into ! position %[n 9 ! illu/ration4 ,x look$ l a hopeless deadlock1 & ea* eng9e-driv] want$ ! o!r to g back to ! next /ation & take (f n9e wagons4 ,b an +5i\s /ok] und]took to pass ! tra9s & s5d !m on _! respective j\rneys ) _! eng9es prop]ly 9 front4 ,he also contriv$ to rev]se ! eng9es ! fewe/ times possible4 ,c\ld y h p]=m$ ! feat8 ,& h[ _m times w\ld y require to rev]se ! eng9es8 ,a rev]sal means a *ange ( direction1 backw>d or =w>d4 ,no rope-%unt+1 fly-%unt+1 or o!r trick is all[$4 ,all ! "w mu/ be done legitimately by ! two eng9es4 ,x is a simple b int]e/+ puzzle if attempt$ ) c\nt]s4 ,illu/ration #bbd4--,,! ,,motor-,,g>age ,,puzzle4 ,illu/ration ,! difficulties ( ! proprietor ( a motor g>age are conv]t$ into a little pa/ime ( a k9d t has a peculi> fasc9ation4 ,all y ne$ is to make a simple plan or diagram on a %eet ( pap] or c>dbo>d & numb] eight c\nt]s1 #a to #h4 ,!n a :ole family c ent] into an amus+ competition to f9d ! be/ possible solution ( ! difficulty4 ,! illu/ration repres5ts ! plan ( a motor g>age1 ) accommodation = twelve c>s4 ,b ! premises are s inconv5i5tly re/rict$ t ! proprietor is (t5 caus$ consid]able p]plexity4 ,suppose1 = example1 t ! eight c>s numb]$ #a to #h are 9 ! positions %[n1 h[ are !y to be %ift$ 9 ! quicke/ possible way s t #a1 #b1 #c1 & #d %all *ange places ) #e1 #f1 #g1 & #h--t is1 ) ! numb]s /ill runn+ f left to "r1 z at pres5t1 b ! top r[ ex*ang$ ) ! bottom r[8 ,:at are ! fewe/ possible moves8 ,"o c> moves at a "t1 & any di/ance c\nts z "o move4 ,to prev5t misund]/&+1 ! /opp+-places are m>k$ 9 squ>es1 & only "o c> c be 9 a squ>e at ! same "t4 #bbe4--,,! ,,t5 ,,prison]s4 ,if prisons _h no o!r use1 !y might /ill be pres]v$ = ! special b5efit ( puzzle-mak]s4 ,!y appe> to be an inexhau/ible m9e ( p]plex+ ideas4 ,"h is a little pos] t w p]haps int]e/ ! read] = a %ort p]iod4 ,we h 9 ! illu/ration a prison ( sixte5 cells4 ,! locations ( ! t5 prison]s w be se54 ,! jail] has que] sup]/itions ab\t odd & ev5 numb]s1 & he wants to re>range ! t5 prison]s s t !re %all be z _m ev5 r[s ( m51 v]tically1 horizontally1 & diagonally1 z possible4 ,at pres5t x w be se51 z indicat$ by ! arr[s1 t !re are only twelve su* r[s ( #b & #d4 ,i w /ate at once t ! greate/ numb] ( su* r[s t is possible is sixte54 ,b ! jail] only all[s f\r m5 to be remov$ to o!r cells1 & in=ms me t1 z ! man :o is seat$ 9 ! bottom "r-h& corn] is infirm1 he mu/ n be mov$4 ,n[1 h[ are we to get ^? sixte5 r[s ( ev5 numb]s "u su* conditions8 ,illu/ration #bbf4--,,r\nd ,,! ,,coa/4 ,illu/ration ,"h is a puzzle t w1 ,i ?9k1 be f\nd z amus+ z in/ructive4 ,we are giv5 a r+ ( eight circles4 ,leav+ circle #h blank1 we are requir$ to write 9 ! "n ( a sev5-lett]$ port 9 ! ,unit$ ,k+dom 9 ?is mann]4 ,t\* a blank circle ) y\r p5cil1 !n jump ov] two circles 9 ei!r direction r\nd ! r+1 & write d[n ! fir/ lett]4 ,!n t\* ano!r vacant circle1 jump ov] two circles1 & write d[n y\r second lett]4 ,proce$ simil>ly ) ! o!r lett]s 9 _! prop] ord] until y h complet$ ! ^w4 ,?us1 suppose we select ,glasg[1 & proce$ z foll[s3 #f--a1 #g--b1 #h--c1 #g--d1 #h--e1 :i* means t we t\* #f1 jump ov] #g & & write d[n ,g on #a2 !n t\* #g1 jump ov] #h & #a1 & write d[n l on #b2 & s on4 ,x w be f\nd t aft] we h writt5 d[n ! fir/ five lett]s--,glasg--z above1 we _c g any fur!r4 ,ei!r !re is some?+ wrong ) ,glasg[1 or we h n manag$ our jumps prop]ly4 ,c y get to ! bottom ( ! my/]y8 #bbg4--;,,c5tral ,,solitaire4 ,illu/ration ,?is anci5t puzzle 0 a great fav\rite ) our gr&mo!rs1 & mo/ ( u1 ,i imag9e1 h on occasions come across a ,solitaire bo>d--a r\nd poli%$ bo>d ) holes cut 9 x 9 a geometrical patt]n1 & a glass m>ble 9 e hole4 ,sometimes ,i h notic$ "o on a side table 9 a suburban front p>l\r1 or f\nd "o on a %elf 9 a c\ntry cottage1 or _h "o br\ght "u my notice at a wayside inn4 ,sometimes !y are ( ! =m %[n above1 b x is equally common = ! bo>d to h f\r m holes1 at ! po9ts indicat$ by dots4 ,i select ! simpl] =m4 ,?\gh ,solitaire bo>ds are /ill sold at ! toy %ops1 x w be suffici5t if ! read] w make an enl>g$ copy ( ! above on a %eet ( c>dbo>d or pap]1 numb] ! holes1 & provide himself ) #cc c\nt]s1 buttons1 or beans4 ,n[ place a c\nt] 9 e hole except ! c5tral "o1 ,no4 #ag1 & ! puzzle is to take (f all ! c\nt]s 9 a s]ies ( jumps1 except ! la/ c\nt]1 :i* mu/ be left 9 t c5tral hole4 ,y are all[$ to jump "o c\nt] ov] ! next "o to a vacant hole beyond1 j z 9 ! game ( draughts1 & ! c\nt] jump$ ov] is imm$iately tak5 (f ! bo>d4 ,only rememb] e move mu/ be a jump2 consequ5tly y w take (f a c\nt] at ea* move1 & ?irty-"o s+le jumps w ( c\rse remove all ! ?irty-"o c\nt]s4 ,b comp\nd moves are all[$ 7z 9 draughts1 aga971 = s long z "o c\nt] cont9ues to jump1 ! jumps all c\nt z "o move4 ,"h is ! beg9n+ ( an imag9>y solution :i* w s]ve to make ! mann] ( mov+ p]fectly pla91 & %[ h[ ! solv] %\ld write out 8 attempts3 #e-ag1 #ab-aj1 #bf-ab1 #bd-bf 7#ac-aa1 #aa-be71 #i-aa 7#bf-bd1 #bd-aj1 #aj-ab71 etc41 etc4 ,! jumps conta9$ )9 brackets c\nt z "o move1 because !y are made ) ! same c\nt]4 ,f9d ! fewe/ possible moves4 ,( c\rse1 no diagonal jumps are p]mitt$2 y c only jump 9 ! direction ( ! l9es4 #bbh4--,,! ,,t5 ,,apples4 ,illu/ration ,! family repres5t$ 9 ! illu/ration are amus+ !mselves ) ?is little puzzle1 :i* is n v difficult b q int]e/+4 ,!y h1 x w be se51 plac$ sixte5 plates on ! table 9 ! =m ( a squ>e1 & put an apple 9 ea* ( t5 plates4 ,!y want to f9d a way ( remov+ all ! apples except "o by jump+ ov] "o at a "t to ! next vacant squ>e1 z 9 draughts2 or1 bett]1 z 9 solitaire1 = y are n all[$ to make any diagonal moves--only moves p>allel to ! sides ( ! squ>e4 ,x is obvi\s t z ! apples /& no move c be made1 b y are p]mitt$ to transf] any s+le apple y l to a vacant plate be=e />t+4 ,!n ! moves mu/ be all leaps1 tak+ (f ! apples leap$ ov]4 #bbi4--,,! ,,n9e ,,almonds4 ,"h is a little puzzle1 said a ,p>son1 t ,i h f\nd peculi>ly fasc9at+4 ,x is s simple1 & yet x keeps y int]e/$ indef9itely4 ,! rev]5d g5tleman took a %eet ( pap] & divid$ x (f into tw5ty-five squ>es1 l a squ>e portion ( a *essbo>d4 ,!n he plac$ n9e almonds on ! c5tral squ>es1 z %[n 9 ! illu/ration1 :]e we h repres5t$ numb]$ c\nt]s = conv5i5ce 9 giv+ ! solution4 ,n[1 ! puzzle is1 cont9u$ ! ,p>son1 to remove eight ( ! almonds & leave ! n9? 9 ! c5tral squ>e4 ,y make ! removals by jump+ "o almond ov] ano!r to ! vacant squ>e beyond & tak+ (f ! "o jump$ ov]--j z 9 draughts1 only "h y c jump 9 any direction1 & n diagonally only4 ,! po9t is to d ! ?+ 9 ! fewe/ possible moves4 ,! foll[+ specim5 attempt w make ev]y?+ cle>4 ,jump #d ov] #a1 #e ov] #i1 #c ov] #f1 #e ov] #c1 #g ov] #e & #b1 #d ov] #g1 #h ov] #d4 ,b #h is n left 9 ! c5tral squ>e1 z x %\ld be4 ,rememb] to remove ^? y jump ov]4 ,any numb] ( jumps 9 succession ) ! same almond c\nt z "o move4 ,illu/ration #bcj4--,,! ,,twelve ,,p5nies4 ,"h is a pretty little puzzle t only requires twelve p5nies or c\nt]s4 ,arrange !m 9 a circle1 z %[n 9 ! illu/ration4 ,n[ take up "o p5ny at a "t &1 pass+ x ov] two p5nies1 place x on ! ?ird p5ny4 ,!n take up ano!r s+le p5ny & d ! same ?+1 & s on1 until1 9 six su* moves1 y h ! co9s 9 six pairs 9 ! positions #a1 #b1 #c1 #d1 #e1 #f4 ,y c move 9 ei!r direction r\nd ! circle at e play1 & x does n matt] :e!r ! two jump$ ov] are sep>ate or a pair4 ,?is is q easy if y use j a little ?\ght4 ,illu/ration #bca4--,,plates ,,& ,,co9s4 ,place twelve plates1 z %[n1 on a r\nd table1 ) a p5ny or orange 9 e plate4 ,/>t f any plate y l &1 always go+ 9 "o direction r\nd ! table1 take up "o p5ny1 pass x ov] two o!r p5nies1 & place x 9 ! next plate4 ,g on aga92 take up ano!r p5ny &1 hav+ pass$ x ov] two p5nies1 place x 9 a plate2 & s cont9ue y\r j\rney4 ,six co9s only are to be remov$1 & :5 ^! h be5 plac$ !re %\ld be two co9s 9 ea* ( six plates & six plates empty4 ,an important po9t ( ! puzzle is to g r\nd ! table z few times z possible4 ,x does n matt] :e!r ! two co9s pass$ ov] are 9 "o or two plates1 nor h[ _m empty plates y pass a co9 ov]4 ,b y mu/ always g 9 "o direction r\nd ! table & end at ! po9t f :i* y set out4 ,y\r h&1 t is to say1 goes /eadily =w>d 9 "o direction1 )\t ev] mov+ backw>ds4 ,illu/ration #bcb4--;,,cat*+ ,,! ,,mice4 ,illu/ration ,play fair6 said ! mice4 ,y "k ! rules ( ! game4 ,yes1 ,i "k ! rules1 said ! cat4 ,i've got to g r\nd & r\nd ! circle1 9 ! direction t y are look+1 & eat e ?irte5? m\se1 b ,i mu/ keep ! :ite m\se = a tit-bit at ! f9i%4 ,?irte5 is an unlucky numb]1 b ,i w d my be/ to oblige y4 ,hurry up1 !n6 %\t$ ! mice4 ,give a fell[ "t to ?9k1 said ! cat4 ,i don't "k :i* ( y to />t at4 ,i mu/ figure x out4 ,:ile ! cat 0 work+ out ! puzzle he fell asleep1 &1 ! spell be+ ?us brok51 ! mice return$ home 9 safety4 ,at :i* m\se %\ld ! cat h />t$ ! c\nt 9 ord] t ! :ite m\se %\ld be ! la/ eat58 ,:5 ! read] has solv$ t little puzzle1 "h is a second "o = him4 ,:at is ! smalle/ numb] t ! cat c c\nt r\nd & r\nd ! circle1 if he mu/ />t at ! :ite m\se 7call+ t "o 9 ! c\nt7 & /ill eat ! :ite m\se la/ ( all8 ,& z a ?ird puzzle try to discov] :at is ! smalle/ numb] t ! cat c c\nt r\nd & r\nd if %e mu/ />t at ! :ite m\se 7call+ t "o7 & make ! :ite m\se ! ?ird eat54 #bcc4--,,! ,,ecc5tric ,,*eesemong]4 ,illu/ration ,! *eesemong] depict$ 9 ! illu/ration is an invet]ate puzzle lov]4 ,"o ( 8 fav\rite puzzles is ! pil+ ( *eeses 9 8 w>eh\se1 an amusem5t t he f9ds good ex]cise = ! body z well z = ! m9d4 ,he places sixte5 *eeses on ! floor 9 a /raight r[ & !n makes !m into f\r piles1 ) f\r *eeses 9 e pile1 by always pass+ a *eese ov] f\r o!rs4 ,if y use sixte5 c\nt]s & numb] !m 9 ord] f #a to #af1 !n y may place #a on #f1 #aa on #a1 #g on #d1 & s on1 until !re are f\r 9 e pile4 ,x w be se5 t x does n matt] :e!r ! f\r pass$ ov] are /&+ alone or pil$2 !y c\nt j ! same1 & y c always c>ry a *eese 9 ei!r direction4 ,!re are a great _m diff]5t ways ( do+ x 9 twelve moves1 s x makes a good game ( pati5ce to try to solve x s t ! f\r piles %all be left 9 diff]5t /ipulat$ places4 ,= example1 try to leave ! piles at ! extreme ends ( ! r[1 on ,nos4 #a1 #b1 #ae & #af2 ?is is q easy4 ,!n try to leave ?ree piles toge!r1 on ,nos4 #ac1 #ad1 & #ae4 ,!n aga9 play s t !y %all be left on ,nos4 #c1 #e1 #ab1 & #ad4 #bcd4--,,! ,,ex*ange ,,puzzle4 ,"h is a r ent]ta9+ little puzzle ) mov+ c\nt]s4 ,y only ne$ twelve c\nt]s--six ( "o col\r1 m>k$ ,a1 ,c1 ,e1 ,g1 ,i1 & ,k1 & ! o!r six m>k$ ,b1 ,d1 ,f1 ,h1 ,j1 & ,l4 ,y fir/ place !m on ! diagram1 z %[n 9 ! illu/ration1 & ! puzzle is to get !m into regul> alphabetical ord]1 z foll[s3-- ,a ,b ,c ,d ,e ,f ,g ,h ,i ,j ,k ,l ,! moves are made by ex*anges ( opposite col\rs /&+ on ! same l9e4 ,?us1 ,g & ,j may ex*ange places1 or ,f & ,a1 b y _c ex*ange ,g & ,c1 or ,f & ,d1 because 9 "o case !y are bo? :ite & 9 ! o!r case bo? black4 ,c y br+ ab\t ! requir$ arrangem5t 9 sev5te5 ex*anges8 ,illu/ration ,x _c be done 9 few] moves4 ,! puzzle is really mu* easi] ?an x looks1 if prop]ly attack$4 #bce4--,,torp$o ,,practice4 ,illu/ration ,if a fleet ( sixte5 m5-(-w> 7 ly+ at an*or & surr\nd$ by ! enemy1 h[ _m %ips might be sunk if e torp$o1 project$ 9 a /raight l9e1 pass$ "u ?ree vessels & sank ! f\r?8 ,9 ! diagram we h arrang$ ! fleet 9 squ>e =mation1 :]e x w be se5 t z _m z sev5 %ips may be sunk 7^? 9 ! top r[ & fir/ column7 by fir+ ! torp$oes indicat$ by arr[s4 ,an*or+ ! fleet z we l1 to :at ext5t c we increase ?is numb]8 ,rememb] t ea* successive %ip is sunk be=e ano!r torp$o is laun*$1 & t e torp$o proce$s 9 a diff]5t direction2 o!rwise1 by plac+ ! %ips 9 a /raight l9e1 we might s9k z _m z ?irte56 ,x is an int]e/+ little /udy 9 naval w>f>e1 & em95tly practical--provid$ ! enemy w all[ y to arrange 8 fleet = y\r conv5i5ce & promise to lie /ill & d no?+6 #bcf4--,,! ,,hat ,,puzzle4 ,t5 hats 7 hung on pegs z %[n 9 ! illu/ration--five silk hats & five felt b[l]s1 alt]nately silk & felt4 ,! two pegs at ! end ( ! r[ 7 empty4 ,illu/ration ,! puzzle is to remove two contigu\s hats to ! vacant pegs1 !n two o!r adjo9+ hats to ! pegs n[ unoccupi$1 & s on until five pairs h be5 mov$ & ! hats aga9 hang 9 an unbrok5 r[1 b ) all ! silk ones toge!r & all ! felt hats toge!r4 ,rememb]1 ! two hats remov$ mu/ always be contigu\s ones1 & y mu/ take "o 9 ea* h& & place !m on _! new pegs )\t rev]s+ _! relative position4 ,y are n all[$ to cross y\r h&s1 nor to hang up "o at a "t4 ,c y solve ?is old puzzle1 :i* ,i give z introductory to ! next8 ,try x ) c\nt]s ( two col\rs or ) co9s1 & rememb] t ! two empty pegs mu/ be left at "o end ( ! r[4 #bcg4--;,,boys ,,& ,,girls4 ,if y m>k (f t5 divisions on a %eet ( pap] to repres5t ! *airs1 & use eight numb]$ c\nt]s = ! *ildr51 y w h a fasc9at+ pa/ime4 ,let ! odd numb]s repres5t boys & ev5 numb]s girls1 or y c use c\nt]s ( two col\rs1 or co9s4 ,! puzzle is to remove two *ildr5 :o are occupy+ adjo9+ *airs & place !m 9 two empty *airs1 mak+ !m fir/ *ange sides2 !n remove a second pair ( *ildr5 f adjo9+ *airs & place !m 9 ! two n[ vacant1 mak+ !m *ange sides2 & s on1 until all ! boys are toge!r & all ! girls toge!r1 ) ! two vacant *airs at "o end z at pres5t4 ,to solve ! puzzle y mu/ d ?is 9 five moves4 ,! two *ildr5 mu/ always be tak5 f *airs t are next to "o ano!r2 & rememb] ! important po9t ( mak+ ! two *ildr5 *ange sides1 z ?is latt] is ! di/9ctive feature ( ! puzzle4 ,by *ange sides ,i simply mean t if1 = example1 y fir/ move #a & #b to ! vacant *airs1 !n ! fir/ 7! outside7 *air w be occupi$ by #b & ! second "o by #a4 ,illu/ration #bch4--;,,arrang+ ,,! ,,jampots4 ,i happ5$ to see a little girl sort+ out "s jam 9 a cupbo>d = h] "m4 ,%e 0 putt+ ea* diff]5t k9d ( pres]ve ap>t on ! %elves4 ,i notic$ t %e took a pot ( damson 9 "o h& & a pot ( gooseb]ry 9 ! o!r & made !m *ange places2 !n %e *ang$ a /rawb]ry ) a raspb]ry1 & s on4 ,x 0 int]e/+ to obs]ve :at a lot ( unnecess>y tr\ble %e gave h]self by mak+ m int]*anges ?an !re 0 any ne$ =1 & ,i ?\ght x w\ld "w into a good puzzle4 ,x w be se5 9 ! illu/ration t little ,doro?y has to manipulate tw5ty-f\r l>ge jampots 9 z _m pigeon-holes4 ,%e wants to get !m 9 correct num]ical ord]--t is1 #a1 #b1 #c1 #d1 #e1 #f on ! top %elf1 #g1 #h1 #i1 #aj1 #aa1 #ab on ! next %elf1 & s on4 ,n[1 if %e always takes "o pot 9 ! "r h& & ano!r 9 ! left & makes !m *ange places1 h[ _m ( ^! int]*anges w be necess>y to get all ! jampots 9 prop] ord]8 ,%e w\ld naturally fir/ *ange ! #a & ! #c1 !n ! #b & ! #c1 :5 %e w\ld h ! fir/ ?ree pots 9 _! places4 ,h[ w\ld y advise h] to g on !n8 ,place "s numb]$ c\nt]s on a %eet ( pap] divid$ into squ>es = ! pigeon-holes1 & y w f9d x an amus+ puzzle4 ,illu/ration ,,unicursal ,,& ,,r\te ,,problems4 ,i see !m on _! w9d+ way4 ,,reg9ald ,,heb]4 ,x is reasonable to suppose t f ! e>lie/ ages "o man has ask$ ano!r su* que/ions z ^!3 ,:i* is ! ne>e/ way home8 ,:i* is ! easie/ or pleasante/ way8 ,h[ c we f9d a way t w enable u to dodge ! ma/odon & ! plesiosaurus8 ,h[ c we get !re )\t ev] cross+ ! track ( ! enemy8 ,all ^! are elem5t>y r\te problems1 & !y c be turn$ into good puzzles by ! introduction ( "s conditions t complicate matt]s4 ,a v>iety ( su* complications w be f\nd 9 ! foll[+ examples4 ,i h also includ$ "s enum]ations ( m or less difficulty4 ,^! af=d excell5t practice = ! reason+ faculties1 & enable "o to g5]alize 9 ! case ( symmetrical =ms 9 a mann] t is mo/ in/ructive4 #bci4--;,a ,,juv5ile ,,puzzle4 ,= ye>s ,i h be5 p]petually consult$ by my juv5ile fri5ds ab\t ?is little puzzle4 ,mo/ *ildr5 seem to "k x1 & yet1 curi\sly en\gh1 !y are inv>iably unacqua9t$ ) ! answ]4 ,! "q !y always ask is1 ,d1 please1 tell me :e!r x is really possible4 ,i believe ,h\d9 ! conjur] us$ to be v fond ( giv+ x to 8 *ild fri5ds1 b ,i _c say :e!r he inv5t$ ! little puzzle or n4 ,no d\bt a l>ge numb] ( my read]s w be glad to h ! my/]y ( ! solution cle>$ up1 s ,i make no apology = introduc+ ?is old teas]4 ,! puzzle is to draw ) ?ree /rokes ( ! p5cil ! diagram t ! little girl is exhibit+ 9 ! illu/ration4 ,( c\rse1 y mu/ n remove y\r p5cil f ! pap] dur+ a /roke or g ov] ! same l9e a second "t4 ,y w f9d t y c get 9 a good deal ( ! figure ) "o cont9u\s /roke1 b x w always appe> z if f\r /rokes are necess>y4 ,illu/ration ,ano!r =m ( ! puzzle is to draw ! diagram on a slate & !n rub x out 9 ?ree rubs4 #bdj4--,,! ,,union ,,jack4 ,illu/ration ,! illu/ration is a r\gh sket* some:at resembl+ ! ,briti% flag1 ! ,union ,jack4 ,x is n possible to draw ! :ole ( x )\t lift+ ! p5cil f ! pap] or go+ ov] ! same l9e twice4 ,! puzzle is to f9d out j h[ mu* ( ! draw+ x is possible to make )\t lift+ y\r p5cil or go+ twice ov] ! same l9e4 ,take y\r p5cil & see :at is ! be/ y c d4 #bda4--,,! ,,dissect$ ,,circle4 ,h[ _m cont9u\s /rokes1 )\t lift+ y\r p5cil f ! pap]1 d y require to draw ! design %[n 9 our illu/ration8 ,directly y *ange ! direction ( y\r p5cil x beg9s a new /roke4 ,y may g ov] ! same l9e m ?an once if y l4 ,x requires j a little c>e1 or y may f9d y\rself beat5 by "o /roke4 ,illu/ration #bdb4--,,! ,,tube ,,inspector',s ,,puzzle4 ,! man 9 our illu/ration is 9 a little dilemma4 ,he has j be5 appo9t$ inspector ( a c]ta9 sy/em ( tube railways1 & x is 8 duty to inspect regul>ly1 )9 a /at$ p]iod1 all ! company's sev5te5 l9es connect+ twelve /ations1 z %[n on ! big po/] plan t he is contemplat+4 ,n[ he wants to arrange 8 r\te s t x %all take him ov] all ! l9es ) z little travell+ z possible4 ,he may beg9 :]e he likes & end :]e he likes4 ,:at is 8 %orte/ r\te8 ,c\ld any?+ be simpl]8 ,b ! read] w soon f9d t1 h[ev] he decides to proce$1 ! inspector mu/ g ov] "s ( ! l9es m ?an once4 ,9 o!r words1 if we say t ! /ations are a mile ap>t1 he w h to travel m ?an sev5te5 miles to inspect e l9e4 ,!re is ! little difficulty4 ,h[ f> is he compell$ to travel1 & :i* r\te d y recomm5d8 ,illu/ration #bdc4--,,visit+ ,,! ,,t[ns4 ,illu/ration ,a travell]1 />t+ f t[n ,no4 #a1 wi%es to visit e "o ( ! t[ns once1 & once only1 go+ only by roads indicat$ by /raight l9es4 ,h[ _m diff]5t r\tes are !re f :i* he c select8 ,( c\rse1 he mu/ end 8 j\rney at ,no4 #a1 f :i* he />t$1 & mu/ take no notice ( cross roads1 b g /raight f t[n to t[n4 ,?is is an absurdly easy puzzle1 if y g ! "r way to "w4 #bdd4--,,! ,,fifte5 ,,turn+s4 ,"h is ano!r que] travell+ puzzle1 ! solution ( :i* calls = +5uity4 ,9 ?is case ! travell] />ts f ! black t[n & wi%es to g z f> z possible :ile mak+ only fifte5 turn+s & nev] go+ along ! same road twice4 ,! t[ns are suppos$ to be a mile ap>t4 ,suppos+1 = example1 t he w5t /raight to ,a1 !n /raight to ,b1 !n to ,c1 ,d1 ,e1 & ,f1 y w !n f9d t he has travell$ ?irty-sev5 miles 9 five turn+s4 ,n[1 h[ f> c he g 9 fifte5 turn+s8 ,illu/ration #bde4--,,! ,,fly ,,on ,,! ,,octah$ron4 ,look "h1 said ! pr(essor to 8 colleague1 ,i h be5 wat*+ t fly on ! octah$ron1 & x conf9es its walks entirely to ! edges4 ,:at c be its reason = avoid+ ! sides8 ,p]haps x is try+ to solve "s r\te problem1 sugge/$ ! o!r4 ,suppos+ x to />t f ! top po9t1 h[ _m diff]5t r\tes are !re by :i* x may walk ov] all ! edges1 )\t ev] go+ twice along ! same edge 9 any r\te8 ,illu/ration ,! problem 0 a h>d] "o ?an !y expect$1 & aft] work+ at x dur+ leisure mom5ts = sev]al days _! results did n agree--9 fact1 !y 7 bo? wrong4 ,if ! read] is surpris$ at _! failure1 let him attempt ! little puzzle himself4 ,i w j expla9 t ! octah$ron is "o ( ! five regul>1 or ,platonic1 bodies1 & is conta9$ "u eight equal & equilat]al triangles4 ,if y cut out ! two pieces ( c>dbo>d ( ! %ape %[n 9 ! m>g9 ( ! illu/ration1 cut half ?r\gh along ! dott$ l9es & !n b5d !m & put !m toge!r1 y w h a p]fect octah$ron4 ,9 any r\te ov] all ! edges x w be f\nd t ! fly mu/ end at ! po9t ( dep>ture at ! top4 #bdf4--,,! ,,icosah$ron ,,puzzle4 ,! icosah$ron is ano!r ( ! five regul>1 or ,platonic1 bodies hav+ all _! sides1 angles1 & planes simil> & equal4 ,x is b\nd$ by tw5ty simil> equilat]al triangles4 ,if y cut out a piece ( c>dbo>d ( ! =m %[n 9 ! small] diagram1 & cut half ?r\gh along ! dott$ l9es1 x w fold up & =m a p]fect icosah$ron4 ,n[1 a ,platonic body does n mean a heav5ly body2 b x w suit ! purpose ( our puzzle if we suppose !re to be a habitable planet ( ?is %ape4 ,we w also suppose t1 ow+ to a sup]fluity ( wat]1 ! only dry l& is along ! edges1 & t ! inhabitants h no k ( navigation4 ,if e "o ( ^? edges is #aj1jjj miles long & a solit>y travell] is plac$ at ! ,nor? ,pole 7! highe/ po9t %[n71 h[ f> w he h to travel be=e he w h visit$ e habitable "p ( ! planet--t is1 h trav]s$ e "o ( ! edges8 ,illu/ration #bdg4--;,,inspect+ ,a ,,m9e4 ,! diagram is suppos$ to repres5t ! passages or gall]ies 9 a m9e4 ,we w assume t e passage1 ,a to ,b1 ,b to ,c1 ,c to ,h1 ,h to ,i1 & s on1 is "o furlong 9 l5g?4 ,x w be se5 t !re are ?irty-"o ( ^! passages4 ,n[1 an (ficial has to inspect all ( !m1 & he desc5ds by ! %aft to ! po9t ,a4 ,h[ f> mu/ he travel1 & :at r\te d y recomm5d8 ,! read] may at fir/ say1 ,z !re are ?irty-"o passages1 ea* a furlong 9 l5g?1 he w h to travel j ?irty-"o furlongs4 ,b ?is is assum+ t he ne$ nev] g along a passage m ?an once1 :i* is n ! case4 ,take y\r p5cil & try to f9d ! %orte/ r\te4 ,y w soon discov] t !re is room = consid]able judgm5t4 ,9 fact1 x is a p]plex+ puzzle4 ,illu/ration #bdh4--,,! ,,cycli/s' ,,t\r4 ,two cycli/s 7 consult+ a road map 9 prep>ation = a little t\r toge!r4 ,! circles repres5t t[ns1 & all ! good roads are repres5t$ by l9es4 ,!y are />t+ f ! t[n ) a />1 & mu/ complete _! t\r at ,e4 ,b be=e arriv+ !re !y want to visit e o!r t[n once1 & only once4 ,t is ! difficulty4 ,mr4 ,spic] said1 ,i am c]ta9 we c f9d a way ( do+ x2 b ,mr4 ,maggs repli$1 ,no way1 ,i'm sure4 ,n[1 :i* ( !m 0 correct8 ,take y\r p5cil & see if y c f9d any way ( do+ x4 ,( c\rse y mu/ keep to ! roads indicat$4 ,illu/ration #bdi4--,,! ,,sailor',s ,,puzzle4 ,! sailor depict$ 9 ! illu/ration /at$ t he _h s9ce 8 boyhood be5 engag$ 9 trad+ ) a small vessel among "s tw5ty little isl&s 9 ! ,pacific4 ,he suppli$ ! r\gh *>t ( :i* ,i h giv5 a copy1 & expla9$ t ! l9es f isl& to isl& repres5t$ ! only r\tes t he ev] adopt$4 ,he always />t$ f isl& ,a at ! beg9n+ ( ! season1 & !n visit$ e isl& once1 & once only1 f9i%+ up 8 t\r at ! />t+-po9t ,a4 ,b he always put (f 8 visit to ,c z long z possible1 = trade reasons t ,i ne$ n ent] into4 ,! puzzle is to discov] 8 exact r\te1 & ?is c be done ) c]ta9ty4 ,take y\r p5cil &1 />t+ at ,a1 try to trace x out4 ,if y write d[n ! isl&s 9 ! ord] 9 :i* y visit !m--?us1 = example1 ,a1 ,i1 ,o1 ,l1 ,g1 etc4--y c at once see if y h visit$ an isl& twice or omitt$ any4 ,( c\rse1 ! cross+s ( ! l9es mu/ be ignor$--t is1 y mu/ cont9ue y\r r\te direct1 & y are n all[$ to swit* (f at a cross+ & proce$ 9 ano!r direction4 ,!re is no trick ( ?is k9d 9 ! puzzle4 ,! sailor knew ! be/ r\te4 ,c y f9d x8 ,illu/ration #bej4--,,! ,,gr& ,,t\r4 ,"o ( ! ev]yday puzzles ( life is ! work+ out ( r\tes4 ,if y are tak+ a holiday on y\r bicycle1 or a motor t\r1 !re always arises ! "q ( h[ y are to make ! be/ ( y\r "t & o!r res\rces4 ,y h det]m9$ to get z f> z "s p>ticul> place1 to include visits to su*-&-su* a t[n1 to try to see some?+ ( special int]e/ else:]e1 & p]haps to try to look up an old fri5d at a spot t w n take y mu* out ( y\r way4 ,!n y h to plan y\r r\te s z to avoid bad roads1 un9t]e/+ c\ntry1 &1 if possible1 ! necessity ( a return by ! same way t y w5t4 ,) a map be=e y1 ! int]e/+ puzzle is attack$ & solv$4 ,i w pres5t a little pos] bas$ on ^! l9es4 ,i give a r\gh map ( a c\ntry--x is n necess>y to say :at p>ticul> c\ntry--! circles repres5t+ t[ns & ! dott$ l9es ! railways connect+ !m4 ,n[ !re liv$ 9 ! t[n m>k$ ,a a man :o 0 born !re1 & dur+ ! :ole ( 8 life _h nev] once left 8 native place4 ,f 8 y\? upw>ds he _h be5 v indu/ri\s1 /ick+ incessantly to 8 trade1 & _h no desire :atev] to roam abroad4 ,h[ev]1 on atta9+ 8 fiftie? bir?day he decid$ to see some?+ ( 8 c\ntry1 & especially to pay a visit to a v old fri5d liv+ at ! t[n m>k$ ,z4 ,:at he propos$ 0 ?is3 t he w\ld />t f 8 home1 ent] e t[n once & only once1 & f9i% 8 j\rney at ,z4 ,z he made up 8 m9d to p]=m ?is gr& t\r by rail only1 he f\nd x r a puzzle to "w out 8 r\te1 b he at l5g? succe$$ 9 do+ s4 ,h[ did he manage x8 ,d n =get t e t[n has to be visit$ once1 & n m ?an once4 ,illu/ration #bea4--,,wat]1 ,,gas1 ,,& ,,electricity4 ,!re are "s half-doz5 puzzles1 z old z ! hills1 t are p]petually cropp+ up1 & !re is h>dly a mon? 9 ! ye> t does n br+ inquiries z to _! solution4 ,occasionally "o ( ^!1 t "o _h ?\ght 0 an ext9ct volcano1 bur/s into eruption 9 a surpris+ mann]4 ,i h receiv$ an extraord9>y numb] ( lett]s respect+ ! anci5t puzzle t ,i h call$ ,wat]1 ,gas1 & ,electricity4 ,x is mu* old] ?an electric light+1 or ev5 gas1 b ! new dress br+s x up to date4 ,! puzzle is to lay on wat]1 gas1 & electricity1 f ,w1 ,g1 & ,e1 to ea* ( ! ?ree h\ses1 ,a1 ,b1 & ,c1 )\t any pipe cross+ ano!r4 ,take y\r p5cil & draw l9es %[+ h[ ?is %\ld be done4 ,y w soon f9d y\rself l&$ 9 difficulties4 ,illu/ration #beb4--;,a ,,puzzle ,,= ,,motori/s4 ,illu/ration ,eight motori/s drove to *ur* "o morn+4 ,_! respective h\ses & *ur*es1 toge!r ) ! only roads available 7! dott$ l9es71 are %[n4 ,"o w5t f 8 h\se ,a to 8 *ur* ,a1 ano!r f 8 h\se ,b to 8 *ur* ,b1 ano!r f ,c to ,c1 & s on1 b x 0 aft]w>ds f\nd t no driv] ev] cross$ ! track ( ano!r c>4 ,take y\r p5cil & try to trace out _! v>i\s r\tes4 #bec4--;,a ,,bank ,,holiday ,,puzzle4 ,two fri5ds 7 sp5d+ _! bank holiday on a cycl+ trip4 ,/opp+ = a re/ at a village inn1 !y consult$ a r\te map1 :i* is repres5t$ 9 our illu/ration 9 an exce$+ly simplifi$ =m1 = ! puzzle is int]e/+ en\gh )\t all ! orig9al complexities4 ,!y />t$ f ! t[n 9 ! top left-h& corn] m>k$ ,a4 ,x w be se5 t !re are "o hundr$ & tw5ty su* t[ns1 all connect$ by /raight roads4 ,n[ !y discov]$ t !re are exactly #a1cfe diff]5t r\tes by :i* !y may rea* _! de/9ation1 always travell+ ei!r due s\? or due ea/4 ,! puzzle is to discov] :i* t[n is _! de/9ation4 ,illu/ration ,( c\rse1 if y f9d t !re are m ?an #a1cfe diff]5t r\tes to a t[n x _c be ! "r "o4 #bed4--,,! ,,motor-,,c> ,,t\r4 ,illu/ration ,9 ! above diagram ! circles repres5t t[ns & ! l9es good roads4 ,9 j h[ _m diff]5t ways c a motori/1 />t+ f ,london 7m>k$ ) an ,l71 make a t\r ( all ^! t[ns1 visit+ e t[n once1 & only once1 on a t\r1 & always com+ back to ,london on ! la/ ride8 ,! exact rev]se ( any r\te is n c\nt$ z diff]5t4 #bee4--,,! ,,level ,,puzzle4 ,illu/ration ,?is is a simple c\nt+ puzzle4 ,9 h[ _m diff]5t ways c y spell out ! ^w ,,level by plac+ ! po9t ( y\r p5cil on an ,l & !n pass+ along ! l9es f lett] to lett]4 ,y may g 9 any direction1 backw>ds or =w>ds4 ,( c\rse y are n all[$ to miss lett]s--t is to say1 if y come to a lett] y mu/ use x4 #bef4--,,! ,,diamond ,,puzzle4 ,,9 h[ _m diff]5t ways may ! ^w ,,diamond be read 9 ! arrangem5t %[n8 ,y may />t :]ev] y l at a ,d & g up or d[n1 backw>ds or =w>ds1 9 & out1 9 any direction y l1 s long z y always pass f "o lett] to ano!r t adjo9s x4 ,h[ _m ways are !re8 ,illu/ration #beg4--,,! ,,deifi$ ,,puzzle4 ,9 h[ _m diff]5t ways may ! ^w ,,deifi$ be read 9 ?is arrangem5t "u ! same conditions z 9 ! la/ puzzle1 ) ! addition t y c use any lett]s twice 9 ! same read+8 ,illu/ration #beh4--,,! ,,vot]s' ,,puzzle4 ,illu/ration ,"h we h1 p]haps1 ! mo/ int]e/+ =m ( ! puzzle4 ,9 h[ _m diff]5t ways c y read ! political injunction1 ,,rise ,,to ,,vote1 ,,sir1 "u ! same conditions z be=e8 ,9 ?is case e read+ ( ! pal9drome requires ! use ( ! c5tral ,v z ! middle lett]4 #bei4--;,,hannah',s ,,puzzle4 ,a man 0 9 love ) a "y lady ^: ,*ri/ian "n 0 ,hannah4 ,:5 he ask$ h] to be 8 wife %e wrote d[n ! lett]s ( h] "n 9 ?is mann]3-- ,h ,h ,h ,h ,h ,h ,h ,a ,a ,a ,a ,h ,h ,a ,n ,n ,a ,h ,h ,a ,n ,n ,a ,h ,h ,a ,a ,a ,a ,h ,h ,h ,h ,h ,h ,h & promis$ t %e w\ld be 8 if he c\ld tell h] correctly 9 h[ _m diff]5t ways x 0 possible to spell out h] "n1 always pass+ f "o lett] to ano!r t 0 adjac5t4 ,diagonal /eps are "h all[$4 ,:e!r %e did ?is m]ely to tease him or to te/ 8 clev]ness is n record$1 b x is satisfactory to "k t he succe$$4 ,w\ld y h be5 equally successful8 ,take y\r p5cil & try4 ,y may />t f any ( ! ,h's & g backw>ds or =w>ds & 9 any direction1 s long z all ! lett]s 9 a spell+ are adjo9+ "o ano!r4 ,h[ _m ways are !re1 no two exactly alike8 #bfj4--,,! ,,honeycomb ,,puzzle4 ,illu/ration ,"h is a little puzzle ) ! simple/ possible conditions4 ,place ! po9t ( y\r p5cil on a lett] 9 "o ( ! cells ( ! honeycomb1 & trace out a v famili> prov]b by pass+ always f a cell to "o t is contigu\s to x4 ,if y take ! "r r\te y w h visit$ e cell once1 & only once4 ,! puzzle is mu* easi] ?an x looks4 #bfa4--,,! ,,monk ,,& ,,! ,,bridges4 ,9 ?is case ,i give a r\gh plan ( a riv] ) an isl& & five bridges4 ,on "o side ( ! riv] is a mona/]y1 & on ! o!r side is se5 a monk 9 ! =egr\nd4 ,n[1 ! monk has decid$ t he w cross e bridge once1 & only once1 on 8 return to ! mona/]y4 ,?is is1 ( c\rse1 q easy to d1 b on ! way he ?\ght to himself1 ,i wond] h[ _m diff]5t r\tes !re are f :i* ,i might h select$4 ,c\ld y h told him8 ,t is ! puzzle4 ,take y\r p5cil & trace out a r\te t w take y once ov] all ! five bridges4 ,!n trace out a second r\te1 !n a ?ird1 & see if y c c\nt all ! v>iations4 ,y w f9d t ! difficulty is tw(old3 y h to avoid dropp+ r\tes on ! "o h& & c\nt+ ! same r\tes m ?an once on ! o!r4 ,illu/ration ,,comb9ation ,,& ,,gr\p ,,problems4 ,a comb9ation & a =m inde$4 ,hamlet1 iii4 #d4 ,v>i\s puzzles 9 ?is class might be t]m$ problems 9 ! geometry ( situation1 b _! solution really dep5ds on ! !ory ( comb9ations :i*1 9 its turn1 is d]iv$ directly f ! !ory ( p]mutations4 ,x has seem$ conv5i5t to include "h c]ta9 gr\p puzzles & enum]ations t might1 p]haps1 ) equal reason h be5 plac$ else:]e2 b read]s are aga9 ask$ n to be too critical ab\t ! classification1 :i* is v difficult & arbitr>y4 ,z ,i h includ$ my problem ( ,! ,r\nd ,table 7,no4 #bgc71 p]haps a few rem>ks on ano!r well-kn[n problem ( ! same class1 kn[n by ! ,fr5* z ,la ,problme des ,mnages1 may be int]e/+4 ,if n m>ri$ ladies are seat$ at a r\nd table 9 any det]m9$ ord]1 9 h[ _m diff]5t ways may _! n husb&s be plac$ s t e man is betwe5 two ladies b nev] next to 8 own wife8 ,?is difficult problem 0 fir/ solv$ by ,laisant1 & ! me?od %[n 9 ! foll[+ table is due to ,moreau3-- #d #j #b #e #c #ac #f #ac #hj #g #hc #egi #h #eib #dgch #i #dhba #dcchg #aj #dcigi #dcigib ,! fir/ column %[s ! numb] ( m>ri$ c\ples4 ,! numb]s 9 ! second column are obta9$ 9 ?is way3 #e #c #j - #b #ac2 #f #ac #c #b #hc2 #g #hc #ac - #b #eib2 #h #eib #hc #b #dhba2 & s on4 ,f9d all ! numb]s1 except #b1 9 ! table1 & ! me?od w be evid5t4 ,x w be not$ t ! #b is subtract$ :5 ! fir/ numb] 7! numb] ( c\ples7 is odd1 & add$ :5 t numb] is ev54 ,! numb]s 9 ! ?ird column are obta9$ ?us3 #ac - #j #ac2 #hc - #c #hj2 #eib - #ac #egi2 #dhba - #hc #dgch2 & s on4 ,! numb]s 9 ?is la/ column give ! requir$ solutions4 ,?us1 f\r husb&s may be seat$ 9 two ways1 five husb&s may be plac$ 9 ?irte5 ways1 & six husb&s 9 eighty ways4 ,! foll[+ me?od1 by ,lucas1 w %[ ! rem>kable way 9 :i* *essbo>d analysis may be appli$ to ! solution ( a circul> problem ( ?is k9d4 ,divide a squ>e into ?irty-six cells1 six by six1 & /rike out all ! cells 9 ! long diagonal f ! bottom left-h& corn] to ! top "r-h& corn]1 also ! five cells 9 ! diagonal next above x & ! cell 9 ! bottom "r-h& corn]4 ,! answ] = six c\ples w be ! same z ! numb] ( ways 9 :i* y c place six rooks 7n us+ ! cancell$ cells7 s t no rook %all ev] attack ano!r rook4 ,x w be f\nd t ! six rooks may be plac$ 9 eighty diff]5t ways1 :i* agrees ) ! above table4 #bfb4--,,^? ,,fifte5 ,,%eep4 ,a c]ta9 cyclopdia has ! foll[+ curi\s problem1 ,i am told3 ,place fifte5 %eep 9 f\r p5s s t !re %all be ! same numb] ( %eep 9 ea* p54 ,no answ] :atev] is v\*saf$1 s ,i ?\ght ,i w\ld inve/igate ! matt]4 ,i saw t 9 deal+ ) apples or bricks ! ?+ w\ld appe> to be q impossible1 s9ce f\r times any numb] mu/ be an ev5 numb]1 :ile fifte5 is an odd numb]4 ,i ?\ght1 !re=e1 t !re mu/ be "s quality peculi> to ! %eep t 0 n g5]ally kn[n4 ,s ,i decid$ to int]view "s f>m]s on ! subject4 ,! fir/ "o po9t$ out t if we put "o p5 inside ano!r1 l ! r+s ( a t>get1 & plac$ all %eep 9 ! smalle/ p51 x w\ld be all "r4 ,b ,i object$ to ?is1 because y admitt$ly place all ! %eep 9 "o p51 n 9 f\r p5s4 ,! second man said t if ,i plac$ f\r %eep 9 ea* ( ?ree p5s & ?ree %eep 9 ! la/ p5 7t is fifte5 %eep 9 all71 & "o ( ! ewes 9 ! la/ p5 _h a lamb dur+ ! night1 !re w\ld be ! same numb] 9 ea* p5 9 ! morn+4 ,?is also fail$ to satisfy me4 ,illu/ration ,! ?ird f>m] said1 ,i've got f\r hurdle p5s d[n 9 "o ( my fields1 & a small flock ( we!rs1 s if y w j /ep d[n ) me ,i w %[ y h[ x is done4 ,! illu/ration depicts my fri5d z he is ab\t to demon/rate ! matt] to me4 ,8 lucid explanation 0 evid5tly t :i* 0 9 ! m9d ( ! writ] ( ! article 9 ! cyclopdia4 ,:at 0 x8 ,c y place ^? fifte5 %eep8 #bfc4--,,k+ ,,ar?ur',s ,,knights4 ,k+ ,ar?ur sat at ! ,r\nd ,table on ?ree successive ev5+s ) 8 knights--,beleobus1 ,c>adoc1 ,driam1 ,eric1 ,floll1 & ,galahad--b on no occasion did any p]son h z 8 neighb\r "o :o _h be=e sat next to him4 ,on ! fir/ ev5+ !y sat 9 alphabetical ord] r\nd ! table4 ,b aft]w>ds ,k+ ,ar?ur arrang$ ! two next sitt+s s t he might h ,beleobus z ne> to him z possible & ,galahad z f> away f him z c\ld be manag$4 ,h[ did he seat ! knights to ! be/ advantage1 rememb]+ t rule t no knight may h ! same neighb\r twice8 #bfd4--,,! ,,city ,,lun*eons4 ,twelve m5 connect$ ) a l>ge firm 9 ! ,city ( ,london sit d[n to lun*eon toge!r e day 9 ! same room4 ,! tables are small ones t only accommodate two p]sons at ! same "t4 ,c y %[ h[ ^! twelve m5 may lun* toge!r on elev5 days 9 pairs1 s t no two ( !m %all ev] sit twice toge!r8 ,we w repres5t ! m5 by ! fir/ twelve lett]s ( ! alphabet1 & suppose ! fir/ day's pair+ to be z foll[s-- 7,a ,b7 7,c ,d7 7,e ,f7 7,g ,h7 7,i ,j7 7,k ,l74 ,!n give any pair+ y l = ! next day1 say-- 7,a ,c7 7,b ,d7 7,e ,g7 7,f ,h7 7,i ,k7 7,j ,l71 & s on1 until y h complet$ y\r elev5 l9es1 ) no pair ev] occurr+ twice4 ,!re are a good _m diff]5t arrangem5ts possible4 ,try to f9d "o ( !m4 #bfe4--;,a ,,puzzle ,,= ,,c>d-,,play]s4 ,twelve memb]s ( a club arrang$ to play bridge toge!r on elev5 ev5+s1 b no play] 0 ev] to h ! same p>tn] m ?an once1 or ! same oppon5t m ?an twice4 ,c y draw up a s*eme %[+ h[ !y may all sit d[n at ?ree tables e ev5+8 ,call ! twelve play]s by ! fir/ twelve lett]s ( ! alphabet & try to gr\p !m4 #bff4--;,a ,,t5nis ,,t\rnam5t4 ,f\r m>ri$ c\ples play$ a mix$ d\ble t5nis t\rnam5t1 a man & a lady always play+ aga9/ a man & a lady4 ,b no p]son ev] play$ ) or aga9/ any o!r p]son m ?an once4 ,c y %[ h[ !y all c\ld h play$ toge!r 9 ! two c\rts on ?ree successive days8 ,?is is a little puzzle ( a q practical k9d1 & x is j p]plex+ en\gh to be int]e/+4 #bfg4--,,! ,,wrong ,,hats4 ,"o ( ! mo/ p]plex+ ?+s ,i h come across lately1 said ,mr4 ,wilson1 is ?is4 ,eight m5 _h be5 d9+ n wisely b too well at a c]ta9 ,london re/aurant4 ,!y 7 ! la/ to leave1 b n "o man 0 9 a condition to id5tify 8 own hat4 ,n[1 consid]+ t !y took _! hats at r&om1 :at are ! *ances t e man took a hat t did n belong to him8 ,! fir/ ?+1 said ,mr4 ,wat]son1 is to see 9 h[ _m diff]5t ways ! eight hats c\ld be tak54 ,t is q easy1 ,mr4 ,/ubbs expla9$4 ,multiply toge!r ! numb]s1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 & #h4 ,let me see--half a m9ute--yes2 !re are #dj1cbj diff]5t ways4 ,n[ all y've got to d is to see 9 h[ _m ( ^! cases no man has 8 own hat1 said ,mr4 ,wat]son4 ,?ank y1 ,i'm n tak+ any1 said ,mr4 ,packhur/4 ,i don't envy ! man :o attempts ! task ( writ+ out all ^? =ty-?\s&-odd cases & !n pick+ out ! ones he wants4 ,!y all agre$ t life is n long en\gh = t sort ( amusem5t2 & z nobody saw any o!r way ( gett+ at ! answ]1 ! matt] 0 po/pon$ indef9itely4 ,c y solve ! puzzle8 #bfh4--,,! ,,peal ,,( ,,bells4 ,a correspond5t1 :o is app>5tly mu* int]e/$ 9 campanology1 asks me h[ he is to con/ruct :at he calls a true & correct peal = f\r bells4 ,he says t e possible p]mutation ( ! f\r bells mu/ be rung once1 & once only4 ,he adds t no bell mu/ move m ?an "o place at a "t1 t no bell mu/ make m ?an two successive /rokes 9 ei!r ! fir/ or ! la/ place1 & t ! la/ *ange mu/ be able to pass into ! fir/4 ,^! fanta/ic conditions w be f\nd to be obs]v$ 9 ! little peal = ?ree bells1 z foll[s3-- #a #b #c #b #a #c #b #c #a #c #b #a #c #a #b #a #c #b ,h[ are we to give him a correct solution = 8 f\r bells8 #bfi4--,,?ree ,,m5 ,,9 ,a ,,boat4 ,a c]ta9 g5]\s ,london manufactur] gives 8 workm5 e ye> a week's holiday at ! seaside at 8 own exp5se4 ,"o ye> fifte5 ( 8 m5 paid a visit to ,h]ne ,bay4 ,on ! morn+ ( _! dep>ture f ,london !y 7 address$ by _! employ]1 :o express$ ! hope t !y w\ld h a v pleasant "t4 ,i h be5 giv5 to und]/&1 he add$1 t "s ( y fell[s are v fond ( r[+1 s ,i propose on ?is occasion to provide y ) ?is recreation1 & at ! same "t give y an amus+ little puzzle to solve4 ,dur+ ! sev5 days t y are at ,h]ne ,bay e "o ( y w g out e day at ! same "t = a r[1 b !re mu/ always be ?ree m5 9 a boat & no m4 ,no two m5 may ev] g out 9 a boat toge!r m ?an once1 & no man is all[$ to g out twice 9 ! same boat4 ,if y c manage to d ?is1 & use z few diff]5t boats z possible1 y may *>ge ! firm ) ! exp5se4 ,"o ( ! m5 tells me t ! exp]i5ce he has ga9$ 9 su* matt]s soon enabl$ him to "w out ! answ] to ! entire satisfaction ( !mselves & _! employ]4 ,b ! amus+ "p ( ! ?+ is t !y nev] really solv$ ! little my/]y4 ,i f9d _! me?od to h be5 q incorrect1 & ,i ?9k x w amuse my read]s to discov] h[ ! m5 %\ld h be5 plac$ 9 ! boats4 ,z _! names happ5 to h be5 ,&rews1 ,bak]1 ,c>t]1 ,danby1 ,edw>ds1 ,fri?1 ,gay1 ,h>t1 ,isaacs1 ,jackson1 ,k5t1 ,lang1 ,mason1 ,napp]1 & ,onsl[1 we c call !m by _! initials & write out ! five gr\ps = ea* ( ! sev5 days 9 ! foll[+ simple way3 #a #b #c #d #e ,fir/ ,day3 7,,abc7 7,,def7 7,,ghi7 7,,jkl7 7,,mno74 ,! m5 )9 ea* pair ( brackets are "h se5 to be 9 ! same boat1 & !re=e ,a c nev] g out ) ,b or ) ,c aga91 & ,c c nev] g out aga9 ) ,b4 ,! same applies to ! o!r f\r boats4 ,! figures %[ ! numb] on ! boat1 s t ,a1 ,b1 or ,c1 = example1 c nev] g out 9 boat ,no4 #a aga94 #bgj4--,,! ,,glass ,,balls4 ,a numb] ( clev] m>ksm5 7 /ay+ at a c\ntry h\se1 & ! ho/1 to provide a little amusem5t1 susp5d$ /r+s ( glass balls1 z %[n 9 ! illu/ration1 to be fir$ at4 ,aft] !y _h all put _! skill to a suffici5t te/1 somebody ask$ ! foll[+ "q3 ,:at is ! total numb] ( diff]5t ways 9 :i* ^! sixte5 balls may be brok51 if we mu/ always break ! l[e/ ball t rema9s on any /r+8 ,?us1 "o way w\ld be to break all ! f\r balls on ea* /r+ 9 succession1 tak+ ! /r+s f left to "r4 ,ano!r w\ld be to break all ! f\r? balls on ! f\r /r+s fir/1 !n break ! ?ree rema9+ on ! fir/ /r+1 !n take ! balls on ! ?ree o!r /r+s alt]nately f "r to left1 & s on4 ,!re is su* a va/ numb] ( diff]5t ways 7s9ce e little v>iation ( ord] makes a diff]5t way7 t "o is apt to be at fir/ impress$ by ! great difficulty ( ! problem4 ,yet x is really q simple :5 once y h hit on ! prop] me?od ( attack+ x4 ,h[ _m diff]5t ways are !re8 ,illu/ration #bga4--;,,fifte5 ,,lett] ,,puzzle4 ,,ale ,,foe ,,hod ,,bgn ,,cab ,,h5 ,,jog ,,kfm ,,hag ,,gem ,,mob ,,bfh ,,fan ,,k9 ,,jek ,,dfl ,,jam ,,him ,,gcl ,,ljh ,,aid ,,jib ,,fcj ,,njd ,,oak ,,fig ,,hck ,,mln ,,b$ ,,oil ,,mcd ,,blk ,,ice ,,con ,,dgk ,! above is ! solution ( a puzzle ,i gave 9 ,tit-bits 9 ! summ] ( #ahif4 ,x 0 requir$ to take ! lett]s1 ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 ,g1 ,h1 ,i1 ,j1 ,k1 ,l1 ,m1 ,n1 & ,o1 & ) !m =m ?irty-five gr\ps ( ?ree lett]s s t ! comb9ations %\ld include ! greate/ numb] possible ( common ,engli% words4 ,no two lett]s may appe> toge!r 9 a gr\p m ?an once4 ,?us1 ,a & ,l hav+ be5 toge!r 9 ,,ale1 mu/ nev] be f\nd toge!r aga92 nor may ,a appe> aga9 9 a gr\p ) ,e1 nor ,l ) ,e4 ,^! conditions w be f\nd compli$ ) 9 ! above solution1 & ! numb] ( words =m$ is tw5ty-"o4 ,_m p]sons h s9ce tri$ h>d to beat ?is numb]1 b s f> h n succe$$4 ,m ?an ?irty-five comb9ations ( ! fifte5 lett]s _c be =m$ )9 ! conditions4 ,!oretically1 !re _c possibly be m ?an tw5ty-?ree words =m$1 because only ?is numb] ( comb9ations is possible ) a v[el or v[els 9 ea*4 ,& z no ,engli% ^w c be =m$ f ?ree ( ! giv5 v[els 7,a1 ,e1 ,i1 & ,o71 we mu/ r$uce ! numb] ( possible words to tw5ty-two4 ,?is is correct !oretically1 b practically t tw5ty-second ^w _c be got 94 ,if ,,jek1 %[n above1 7 a ^w x w\ld be all "r2 b x is n1 & no am\nt ( juggl+ ) ! o!r lett]s has result$ 9 a bett] answ] ?an ! "o %[n4 ,i %\ld1 say t prop] n\ns & abbreviations1 su* z ,joe1 ,jim1 ,alf1 ,hal1 ,flo1 ,ike1 etc41 are disall[$4 ,n[1 ! pres5t puzzle is a v>iation ( ! above4 ,x is simply ?is3 ,in/ead ( us+ ! fifte5 lett]s giv51 ! read] is all[$ to select any fifte5 diff]5t lett]s ( ! alphabet t he may pref]4 ,!n con/ruct ?irty-five gr\ps 9 accordance ) ! conditions1 & %[ z _m good ,engli% words z possible4 #bgb4--,,! ,,n9e ,,s*oolboys4 ,?is is a new & int]e/+ companion puzzle to ! ,fifte5 ,s*oolgirls 7see solution ( ,no4 #bfi71 & ev5 9 ! simple/ possible =m 9 :i* ,i pres5t x !re are unque/ionable difficulties4 ,n9e s*oolboys walk out 9 triplets on ! six week days s t no boy ev] walks side by side ) any o!r boy m ?an once4 ,h[ w\ld y arrange !m8 ,if we repres5t !m by ! fir/ n9e lett]s ( ! alphabet1 !y might be gr\p$ on ! fir/ day z foll[s3-- ,a ,b ,c ,d ,e ,f ,g ,h ,i ,!n ,a c nev] walk aga9 side by side ) ,b1 or ,b ) ,c1 or ,d ) ,e1 & s on4 ,b ,a c1 ( c\rse1 walk side by side ) ,c4 ,x is "h n a "q ( be+ toge!r 9 ! same triplet1 b ( walk+ side by side 9 a triplet4 ,"u ^! conditions !y c walk out on six days2 "u ! ,s*oolgirls conditions !y c only walk on f\r days4 #bgc4--,,! ,,r\nd ,,table4 ,seat ! same n p]sons at a r\nd table on 7n - #a77n - #b7 -------------- #b occasions s t no p]son %all ev] h ! same two neighb\rs twice4 ,?is is1 ( c\rse1 equival5t to say+ t e p]son mu/ sit once1 & once only1 betwe5 e possible pair4 #bgd4--,,! ,,m\se-,,trap ,,puzzle4 ,illu/ration #f #bj #b #ai #ac #ba #g #e #c #ah #ag #h #ae #aa #ad #af #a #i #aj #d #ab ,?is is a mod]n v]sion1 ) a diff]5ce1 ( an old puzzle ( ! same "n4 ,numb] tw5ty-"o c>ds1 #a1 #b1 #c1 etc41 up to #ba1 & place !m 9 a circle 9 ! p>ticul> ord] %[n 9 ! illu/ration4 ,^! c>ds repres5t mice4 ,y />t f any c>d1 call+ t c>d "o1 & c\nt1 "o1 two1 ?ree1 etc41 9 a clockwise direction1 & :5 y\r c\nt agrees ) ! numb] on ! c>d1 y h made a cat*1 & y remove ! c>d4 ,!n />t at ! next c>d1 call+ t "o1 & try aga9 to make ano!r cat*4 ,& s on4 ,suppos+ y />t at #ah1 call+ t c>d "o1 y\r fir/ cat* w be #ai4 ,remove #ai & y\r next cat* is #aj4 ,remove #aj & y\r next cat* is #a4 ,remove ! #a1 & if y c\nt up to #ba 7y mu/ nev] g beyond71 y _c make ano!r cat*4 ,n[1 ! ideal is to cat* all ! tw5ty-"o mice1 b ?is is n "h possible1 & if x 7 x w\ld m]ely require tw5ty-"o diff]5t trials1 at ! mo/1 to succe$4 ,b ! read] may make any two c>ds *ange places be=e he beg9s4 ,?us1 y c *ange ! #f ) ! #b1 or ! #g ) ! #aa1 or any o!r pair4 ,?is c be done 9 sev]al ways s z to enable y to cat* all ! tw5ty-"o mice1 if y !n />t at ! "r place4 ,y may nev] pass ov] a cat*2 y mu/ always remove ! c>d & />t afre%4 #bge4--,,! ,,sixte5 ,,%eep4 ,illu/ration3 #j #j #j #j --------------------- #j #j #j #j #j #j #j #j ----------- #j #j #j #j ,"h is a new puzzle ) mat*es & c\nt]s or co9s4 ,9 ! illu/ration ! mat*es repres5t hurdles & ! c\nt]s %eep4 ,! sixte5 hurdles on ! outside1 & ! %eep1 mu/ be reg>d$ z immovable2 ! puzzle has to d entirely ) ! n9e hurdles on ! inside4 ,x w be se5 t at pres5t ^! n9e hurdles enclose f\r gr\ps ( #h1 #c1 #c1 & #b %eep4 ,! f>m] requires to readju/ "s ( ! hurdles s z to enclose #f1 #f1 & #d %eep4 ,c y d x by only replac+ two hurdles8 ,:5 y h succe$$1 !n try to d x by replac+ ?ree hurdles2 !n f\r1 five1 six1 & sev5 9 succession4 ,( c\rse1 ! hurdles mu/ be legitimately laid on ! dott$ l9es1 & no su* tricks are all[$ z leav+ unconnect$ ends ( hurdles1 or two hurdles plac$ side by side1 or m]ely mak+ hurdles *ange places4 ,9 fact1 ! conditions are s simple t any f>m lab\r] w und]/& x directly4 #bgf4--,,! ,,eight ,,villas4 ,9 "o ( ! outly+ suburbs ( ,london a man _h a squ>e plot ( gr\nd on :i* he decid$ to build eight villas1 z %[n 9 ! illu/ration1 ) a common recreation gr\nd 9 ! middle4 ,aft] ! h\ses 7 complet$1 & all or "s ( !m let1 he discov]$ t ! numb] ( occupants 9 ! ?ree h\ses =m+ a side ( ! squ>e 0 9 e case n9e4 ,he did n /ate h[ ! occupants 7 di/ribut$1 b ,i h %[n by ! numb]s on ! sides ( ! h\ses "o way 9 :i* x might h happ5$4 ,! puzzle is to discov] ! total numb] ( ways 9 :i* all or any ( ! h\ses might be occupi$1 s t !re %\ld be n9e p]sons on ea* side4 ,9 ord] t !re may be no misund]/&+1 ,i w expla9 t al?\gh ,b is :at we call a reflection ( ,a1 ^! w\ld c\nt z two diff]5t arrangem5ts1 :ile ,c1 if x is turn$ r\nd1 w give f\r arrangem5ts2 & if turn$ r\nd 9 front ( a mirror1 f\r o!r arrangem5ts4 ,all eight mu/ be c\nt$4 ,illu/ration3 _/ _/ _/ #b #e #b _/ _/ #e #e _/ _/ _/ #b #e #b --------- --------- --------- #a #f #b #b #f #a #a #f #b --------- --------- --------- #f #f #f #f #d #d --------- --------- --------- #b #f #a #a #f #b #d #b #c --------- --------- --------- ,a ,b ,c #bgg4--;,,c\nt] ,,crosses4 ,all t we ne$ = ?is puzzle is n9e c\nt]s1 numb]$ #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 & #i4 ,x w be se5 t 9 ! illu/ration ,a ^! are arrang$ s z to =m a ,greek cross1 :ile 9 ! case ( ,b !y =m a ,lat9 cross4 ,9 bo? cases ! read] w f9d t ! sum ( ! numb]s 9 ! upright ( ! cross is ! same z ! sum ( ! numb]s 9 ! horizontal arm4 ,x is q easy to hit on su* an arrangem5t by trial1 b ! problem is to discov] 9 exactly h[ _m diff]5t ways x may be done 9 ea* case4 ,rememb] t rev]sals & reflections d n c\nt z diff]5t4 ,t is to say1 if y turn ?is page r\nd y get f\r arrangem5ts ( ! ,greek cross1 & if y turn x r\nd aga9 9 front ( a mirror y w get f\r m4 ,b ^! eight are all reg>d$ z "o & ! same4 ,n[1 h[ _m diff]5t ways are !re 9 ea* case8 ,illu/ration3 7#a7 7#b7 7#b7 7#d7 7#e7 7#a7 7#f7 7#g7 7#c7 7#d7 7#i7 7#e7 7#f7 7#c7 7#g7 7#h7 ,a 7#h7 ,b 7#i7 #bgh4--;,a ,,dormitory ,,puzzle4 ,9 a c]ta9 conv5t !re 7 eight l>ge dormitories on "o floor1 approa*$ by a spiral /aircase 9 ! c5tre1 z %[n 9 our plan4 ,on an inspection "o ,monday by ! abbess x 0 f\nd t ! s\? aspect 0 s mu* pref]r$ t six times z _m nuns slept on ! s\? side z on ea* ( ! o!r ?ree sides4 ,%e object$ to ?is ov]cr[d+1 & ord]$ t x %\ld be r$uc$4 ,on ,tuesday %e f\nd t five times z _m slept on ! s\? side z on ea* ( ! o!r sides4 ,aga9 %e compla9$4 ,on ,w$nesday %e f\nd f\r times z _m on ! s\? side1 on ,?ursday ?ree times z _m1 & on ,friday twice z _m4 ,urg+ ! nuns to fur!r ef=ts1 %e 0 pleas$ to f9d on ,saturday t an equal numb] slept on ea* ( ! f\r sides ( ! h\se4 ,:at is ! smalle/ numb] ( nuns !re c\ld h be51 & h[ might !y h arrang$ !mselves on ea* ( ! six nights8 ,no room may ev] be unoccupi$4 ,illu/ration --------- --------- _/ -99- _/ --------- --------- #bgi4--,,! ,,b>rels ,,( ,,balsam4 ,a m]*ant ( ,bagdad _h t5 b>rels ( preci\s balsam = sale4 ,!y 7 numb]$1 & 7 arrang$ 9 two r[s1 "o on top ( ! o!r1 z %[n 9 ! picture4 ,! small] ! numb] on ! b>rel1 ! great] 0 its value4 ,s t ! be/ quality 0 numb]$ #a & ! wor/ numb]$ #aj1 & all ! o!r numb]s ( graduat+ values4 ,n[1 ! rule ( ,ahm$ ,assan1 ! m]*ant1 0 t he nev] put a b>rel ei!r b5ea? or to ! "r ( "o ( less value4 ,! arrangem5t %[n is1 ( c\rse1 ! simple/ way ( comply+ ) ?is condition4 ,b !re are _m o!r ways--su*1 = example1 z ?is3-- #a #b #e #g #h #c #d #f #i #aj ,"h1 aga91 no b>rel has a small] numb] ?an itself on its "r or b5ea? x4 ,! puzzle is to discov] 9 h[ _m diff]5t ways ! m]*ant ( ,bagdad might h arrang$ 8 b>rels 9 ! two r[s )\t break+ 8 rule4 ,c y c\nt ! numb] ( ways8 #bhj4--;,,build+ ,,! ,,tetrah$ron4 ,i possess a tetrah$ron1 or triangul> pyramid1 =m$ ( six /icks glu$ toge!r1 z %[n 9 ! illu/ration4 ,c y c\nt correctly ! numb] ( diff]5t ways 9 :i* ^! six /icks might h be5 /uck toge!r s z to =m ! pyramid8 ,"s fri5ds work$ at x toge!r "o ev5+1 ea* p]son provid+ himself ) six lucif] mat*es to aid 8 ?\ghts2 b x 0 f\nd t no two results 7 ! same4 ,y see1 if we remove "o ( ! /icks & turn x r\nd ! o!r way1 t w be a diff]5t pyramid4 ,if we make two ( ! /icks *ange places ! result w aga9 be diff]5t4 ,b rememb] t e pyramid may be made to /& on ei!r ( its f\r sides )\t be+ a diff]5t "o4 ,h[ _m ways are !re altoge!r8 ,illu/ration #bha4--,,pa9t+ ,a ,,pyramid4 ,?is puzzle conc]ns ! pa9t+ ( ! f\r sides ( a tetrah$ron1 or triangul> pyramid4 ,if y cut out a piece ( c>dbo>d ( ! triangul> %ape %[n 9 ,fig4 #a1 & !n cut half ?r\gh along ! dott$ l9es1 x w fold up & =m a p]fect triangul> pyramid4 ,& ,i w\ld fir/ rem9d my read]s t ! prim>y col\rs ( ! sol> spectrum are sev5--violet1 indigo1 blue1 gre51 yell[1 orange1 & r$4 ,:5 ,i 0 a *ild ,i 0 taught to rememb] ^! by ! unga9ly ^w =m$ by ! initials ( ! col\rs1 ,vibgyor4 ,illu/ration ,9 h[ _m diff]5t ways may ! triangul> pyramid be col\r$1 us+ 9 e case "o1 two1 ?ree1 or f\r col\rs ( ! sol> spectrum8 ,( c\rse a side c only receive a s+le col\r1 & no side c be left uncol\r$4 ,b !re is "o po9t t ,i mu/ make q cle>4 ,! f\r sides are n to be reg>d$ z individually di/9ct4 ,t is to say1 if y pa9t y\r pyramid z %[n 9 ,fig4 #b 7:]e ! bottom side is gre5 & ! o!r side t is out ( view is yell[71 & !n pa9t ano!r 9 ! ord] %[n 9 ,fig4 #c1 ^! are really bo? ! same & c\nt z "o way4 ,= if y tilt ov] ,no4 #b to ! "r x w s fall z to repres5t ,no4 #c4 ,! avoidance ( repetitions ( ?is k9d is ! real puzzle ( ! ?+4 ,if a col\r$ pyramid _c be plac$ s t x exactly resembles 9 its col\rs & _! relative ord] ano!r pyramid1 !n !y are diff]5t4 ,rememb] t "o way w\ld be to col\r all ! f\r sides r$1 ano!r to col\r two sides gre51 & ! rema9+ sides yell[ & blue2 & s on4 #bhb4--,,! ,,antiqu>y',s ,,*a94 ,an antiqu>y possess$ a numb] ( curi\s old l9ks1 :i* he took to a blacksmi?1 & told him to jo9 toge!r to =m "o /raight piece ( *a91 ) ! sole condition t ! two circul> l9ks 7 n to be toge!r4 ,! foll[+ illu/ration %[s ! appe>ance ( ! *a9 & ! =m ( ea* l9k4 ,n[1 suppos+ ! own] %\ld sep>ate ! l9ks aga91 & !n take !m to ano!r smi? & repeat 8 =m] in/ructions exactly1 :at are ! *ances aga9/ ! l9ks be+ put toge!r exactly z !y 7 by ! fir/ man8 ,rememb] t e successive l9k c be jo9$ on to ano!r 9 "o ( two ways1 j z y c put a r+ on y\r f+] 9 two ways1 or l9k y\r =ef+]s & ?umbs 9 two ways4 ,illu/ration #bhc4--,,! ,,fifte5 ,,dom9oes4 ,9 ?is case we d n use ! complete set ( tw5ty-eight dom9oes to be f\nd 9 ! ord9>y box4 ,we disp5se ) all ^? dom9oes t h a five or a six on !m & limit ourselves to ! fifte5 t rema91 :]e ! d\ble-f\r is ! highe/4 ,9 h[ _m diff]5t ways may ! fifte5 dom9oes be arrang$ 9 a /raight l9e 9 accordance ) ! simple rule ( ! game t a numb] mu/ always be plac$ aga9/ a simil> numb]--t is1 a f\r aga9/ a f\r1 a blank aga9/ a blank1 & s on8 ,left to "r & "r to left ( ! same arrangem5t are to be c\nt$ z two diff]5t ways4 #chd4--,,! ,,cross ,,t>get4 -- 9999 -- 9999 ------ 99 ------ 99 99 ------ 99 -- -- ,9 ! illu/ration we h a some:at curi\s t>get design$ by an ecc5tric %>p%oot]4 ,8 idea 0 t 9 ord] to score y mu/ hit f\r circles 9 z _m %ots s t ^? f\r %ots %all =m a squ>e4 ,x w be se5 by ! results record$ on ! t>get t two attempts h be5 successful4 ,! fir/ man hit ! f\r circles at ! top ( ! cross1 & ?us =m$ 8 squ>e4 ,! second man int5d$ to hit ! f\r 9 ! bottom arm1 b 8 second %ot1 on ! left1 w5t too high4 ,?is compell$ him to complete 8 f\r 9 a diff]5t way ?an he int5d$4 ,x w ?us be se5 t ?\gh x is immat]ial :i* circle y hit at ! fir/ %ot1 ! second %ot may commit y to a def9ite proc$ure if y are to get y\r squ>e4 ,n[1 ! puzzle is to say 9 j h[ _m diff]5t ways x is possible to =m a squ>e on ! t>get ) f\r %ots4 #bhe4--,,! ,,f\r ,,po/age ,,/amps4 --------------- #a #b #c #d --------------- #e #f #g #h --------------- #i #aj #aa #ab --------------- ,x is z easy z c\nt+1 is an expression "o sometimes he>s4 ,b m]e c\nt+ may be puzzl+ at times4 ,take ! foll[+ simple example4 ,suppose y h j b\ght twelve po/age /amps1 9 ?is =m--?ree by f\r--& a fri5d asks y to oblige him ) f\r /amps1 all jo9$ toge!r--no /amp hang+ on by a m]e corn]4 ,9 h[ _m diff]5t ways is x possible = y to te> (f ^? f\r /amps8 ,y see1 y c give him #a1 #b1 #c1 #d1 or #b1 #c1 #f1 #g1 or #a1 #b1 #c1 #f1 or #a1 #b1 #c1 #g1 or #b1 #c1 #d1 #h1 & s on4 ,c y c\nt ! numb] ( diff]5t ways 9 :i* ^? f\r /amps might be deliv]$8 ,!re are n _m m ?an fifty ways1 s x is n a big c\nt4 ,c y get ! exact numb]8 #bhf4--,,pa9t+ ,,! ,,die4 ,9 h[ _m diff]5t ways may ! numb]s on a s+le die be m>k$1 ) ! only condition t ! #a & #f1 ! #b & #e1 & ! #c & #d mu/ be on opposite sides8 ,x is a simple en\gh "q1 & yet x w puzzle a good _m p4 #bhg4--;,,an ,,acro/ic ,,puzzle4 ,9 ! mak+ or solv+ ( d\ble acro/ics1 has x ev] occurr$ to y to consid] ! v>iety & limitation ( ! pair ( initial & f9al lett]s available = cross words8 ,y may h to f9d a ^w beg9n+ ) ,a & end+ ) ,b1 or ,a & ,c1 or ,a & ,d1 & s on4 ,"s comb9ations are obvi\sly impossible--su*1 = example1 z ^? ) ,q at ! end4 ,b let u assume t a good ,engli% ^w c be f\nd = e case4 ,!n h[ _m possible pairs ( lett]s are available8 ,,*essbo>d ,,problems4 ,y & ,i w goe to ! *esse4 ,,gre5e',s ,groatswor? ( ,wit4 ,dur+ a heavy gale a *imney-pot 0 hurl$ ?r\gh ! air1 & cra%$ ^u ! pavem5t j 9 front ( a p$e/rian4 ,he q calmly said1 ,i h no use = x3 ,i d n smoke4 ,"s read]s1 :5 !y happ5 to see a puzzle repres5t$ on a *essbo>d ) *ess pieces1 are apt to make ! equally inconsequ5t rem>k1 ,i h no use = x3 ,i d n play *ess4 ,?is is l>gely a result ( ! common1 b errone\s1 notion t ! ord9>y *ess puzzle ) :i* we are famili> 9 ! press 7dignifi$1 = "s reason1 ) ! "n problem7 has a vital connection ) ! game ( *ess itself4 ,b !re is no condition 9 ! game t y %all *eckmate y\r oppon5t 9 two moves1 9 ?ree moves1 or 9 f\r moves1 :ile ! majority ( ! positions giv5 9 ^! puzzles are su* t "o play] w\ld h s great a sup]iority 9 pieces t ! o!r w\ld h resign$ be=e ! situations 7 rea*$4 ,& ! solv+ ( !m helps y b little1 & t q indirectly1 9 play+ ! game1 x be+ well kn[n t1 z a rule1 ! be/ *ess problemi/s are indiff]5t play]s1 & vice v]sa4 ,occasionally a man w be f\nd /rong on bo? subjects1 b he is ! exception to ! rule4 ,yet ! simple *equ]$ bo>d & ! *>act]i/ic moves ( ! pieces l5d !mselves 9 a v rem>kable mann] to ! devis+ ( ! mo/ ent]ta9+ puzzles4 ,!re is room = su* inf9ite v>iety t ! true puzzle lov] _c af=d to neglect !m4 ,x 0 ) a view to secur+ ! int]e/ ( read]s :o are fright5$ (f by ! m]e pres5tation ( a *essbo>d t s _m puzzles ( ?is class 7 orig9ally publi%$ by me 9 v>i\s fanciful dresses4 ,"s ( ^! pos]s ,i /ill reta9 9 _! disguis$ =m2 o!rs ,i h translat$ into t]ms ( ! *essbo>d4 ,9 ! majority ( cases ! read] w n ne$ any k :atev] ( *ess1 b ,i h ?\ght x be/ to assume ?r\gh\t t he is acqua9t$ ) ! t]m9ology1 ! moves1 & ! notation ( ! game4 ,i fir/ deal ) a few que/ions affect+ ! *essbo>d itself2 !n ) c]ta9 /atical puzzles relat+ to ! ,rook1 ! ,bi%op1 ! ,que51 & ! ,knight 9 turn2 !n dynamical puzzles ) ! pieces 9 ! same ord]2 &1 f9ally1 ) "s miscellane\s puzzles on ! *essbo>d4 ,x is hop$ t ! =mul & tables giv5 at ! end ( ! /atical puzzles w be ( int]e/1 z !y are1 = ! mo/ "p1 publi%$ = ! fir/ "t4 ,,! ,,*essbo>d4 ,good company's a *essbo>d4 ,,byron',s ,don ,juan1 xiii4 #hi4 ,a *essbo>d is ess5tially a squ>e plane divid$ into sixty-f\r small] squ>es by /raight l9es at "r angles4 ,orig9ally x 0 n *equ]$ 7t is1 made ) its r[s & columns alt]nately black & :ite1 or ( any o!r two col\rs71 & ?is improvem5t 0 introduc$ m]ely to help ! eye 9 actual play4 ,! utility ( ! *equ]s is unque/ionable4 ,= example1 x facilitates ! op]ation ( ! bi%ops1 enabl+ u to see at ! m]e/ glance t our k+ or pawns on black squ>es are n op5 to attack f an oppon5t's bi%op runn+ on ! :ite diagonals4 ,yet ! *equ]+ ( ! bo>d is n ess5tial to ! game ( *ess4 ,also1 :5 we are prop\nd+ puzzles on ! *essbo>d1 x is (t5 well to rememb] t additional int]e/ may result f g5]aliz+ = bo>ds conta9+ any numb] ( squ>es1 or f limit+ ourselves to "s p>ticul> *equ]$ arrangem5t1 n necess>ily a squ>e4 ,we w give a few puzzles deal+ ) *equ]$ bo>ds 9 ?is g5]al way4 #bhh4--;,,*equ]$ ,,bo>d ,,divisions4 ,i rec5tly ask$ myself ! "q3 ,9 h[ _m diff]5t ways may a *essbo>d be divid$ into two p>ts ( ! same size & %ape by cuts along ! l9es divid+ ! squ>es8 ,! problem soon prov$ to be bo? fasc9at+ & bri/l+ ) difficulties4 ,i pres5t x 9 a simplifi$ =m1 tak+ a bo>d ( small] dim5sions4 ,illu/ration3 ------99------ ---------99--- ---------99--- ,h ,h ,h ------99------ ------9999--- ---9999---99--- ,h ,h ,h ,h ,h ------99------ ------99------ ---99---99---99--- ,h ,h ,h ,h ,h ------99------ ---9999------ ---99---9999--- ,h ,h ,h ------99------ ---99--------- ---99--------- ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ ------99------ ---------99--- ---------99--- ,h ,h ,h ---9999------ ---999999--- ------9999--- ,h ,h ,h ---999999--- ---999999--- ------99------ ,h ,h ,h ------9999--- ---999999--- ---9999------ ,h ,h ,h ------99------ ---99--------- ---99--------- ,x is obvi\s t a bo>d ( f\r squ>es c only be s divid$ 9 "o way--by a /raight cut d[n ! c5tre--because we %all n c\nt rev]sals & reflections z diff]5t4 ,9 ! case ( a bo>d ( sixte5 squ>es--f\r by f\r--!re are j six diff]5t ways4 ,i h giv5 all ^! 9 ! diagram1 & ! read] w n f9d any o!rs4 ,n[1 take ! l>g] bo>d ( ?irty-six squ>es1 & try to discov] 9 h[ _m ways x may be cut into two p>ts ( ! same size & %ape4 #bhi4--,,lions ,,& ,,cr[ns4 ,! "y lady 9 ! illu/ration is confront$ ) a little cutt+-out difficulty 9 :i* ! read] may be glad to assi/ h]4 ,%e wi%es1 = "s reason t %e has n communicat$ to me1 to cut t squ>e piece ( valuable mat]ial into f\r p>ts1 all ( exactly ! same size & %ape1 b x is important t e piece %all conta9 a lion & a cr[n4 ,z %e insi/s t ! cuts c only be made along ! l9es divid+ ! squ>es1 %e is consid]ably p]plex$ to f9d out h[ x is to be done4 ,c y %[ h] ! way8 ,!re is only "o possible me?od ( cutt+ ! /uff4 ,illu/ration3 ------ ------ ,l,l,l ------ ,c,c ------ ,c,c ------ ,l ------ ------ #bij4--;,,bo>ds ,,) ,,an ,,odd ,,numb] ,,( ,,squ>es4 ,we w "h consid] ! "q ( ^? bo>ds t conta9 an odd numb] ( squ>es4 ,we w suppose t ! c5tral squ>e is fir/ cut out1 s z to leave an ev5 numb] ( squ>es = division4 ,n[1 x is obvi\s t a squ>e ?ree by ?ree c only be divid$ 9 "o way1 z %[n 9 ,fig4 #a4 ,x w be se5 t ! pieces ,a & ,b are ( ! same size & %ape1 & t any o!r way ( cutt+ w\ld only produce ! same %ap$ pieces1 s rememb] t ^! v>iations are n c\nt$ z diff]5t ways4 ,! puzzle ,i propose is to cut ! bo>d five by five 7,fig4 #b7 into two pieces ( ! same size & %ape 9 z _m diff]5t ways z possible4 ,i h %[n 9 ! illu/ration "o way ( do+ x4 ,h[ _m diff]5t ways are !re altoge!r8 ,a piece :i* :5 turn$ ov] resembles ano!r piece is n consid]$ to be ( a diff]5t %ape4 ,illu/ration3 ---99------ ,h ---9999--- ,,hhhhh ---9999--- ,h ------99--- ,fig #a ,illu/ration3 --------------- 99999999------ ,h ------9999------ ,,hhhhh ------9999------ ,h ------99999999 ,h ---99------------ ,fig #b #bia4--,,! ,,gr& ,,lama',s ,,problem4 ,once ^u a "t !re 0 a ,gr& ,lama :o _h a *essbo>d made ( pure gold1 magnific5tly engrav$1 &1 ( c\rse1 ( great value4 ,e ye> a t\rnam5t 0 held at ,lhassa among ! prie/s1 & :5ev] any "o beat ! ,gr& ,lama x 0 consid]$ a great hon\r1 & 8 "n 0 inscrib$ on ! back ( ! bo>d1 & a co/ly jewel set 9 ! p>ticul> squ>e on :i* ! *eckmate _h be5 giv54 ,aft] ?is sov]eign pontiff _h be5 defeat$ on f\r occasions he di$--possibly ( *agr94 ,illu/ration3 ------------------------ 99 ------------------------ 99 ------------------------ 99 ------------------------ 99 ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ,n[ ! new ,gr& ,lama 0 an inf]ior *ess-play]1 & pref]r$ o!r =ms ( innoc5t amusem5t1 su* z cutt+ (f p's heads4 ,s he disc\rag$ *ess z a degrad+ game1 t did n improve ei!r ! m9d or ! morals1 & aboli%$ ! t\rnam5t summ>ily4 ,!n he s5t = ! f\r prie/s :o _h _h ! effront]y to play bett] ?an a ,gr& ,lama1 & address$ !m z foll[s3 ,mis]able & hea!ni% m51 call+ y\rselves prie/s6 ,"k ye n t to lay claim to a capacity to d any?+ bett] ?an my pr$ecessor is a capital (f5ce8 ,take t *essbo>d &1 be=e day dawns ^u ! torture *amb]1 cut x into f\r equal p>ts ( ! same %ape1 ea* conta9+ sixte5 p]fect squ>es1 ) "o ( ! gems 9 ea* "p6 ,if 9 ?is y fail1 !n %all o!r sports be devis$ = y\r special delectation4 ,g6 ,! f\r prie/s succe$$ 9 _! app>5tly hopeless task4 ,c y %[ h[ ! bo>d may be divid$ into f\r equal p>ts1 ea* ( exactly ! same %ape1 by cuts along ! l9es divid+ ! squ>es1 ea* "p to conta9 "o ( ! gems8 #bib4--,,! ,,abbot',s ,,w9d[4 ,illu/ration ,once ^u a "t ! ,"l ,abbot ( ,/4 ,edmondsbury1 9 consequ5ce ( devotions too /rong = 8 head1 fell sick & 0 unable to leave 8 b$4 ,z he lay awake1 toss+ 8 head re/lessly f side to side1 ! att5tive monks notic$ t some?+ 0 di/urb+ 8 m9d2 b nobody d>$ ask :at x might be1 = ! abbot 0 ( a /]n disposition1 & nev] w\ld brook inquisitiv5ess4 ,sudd5ly he call$ = ,fa!r ,john1 & t v5]able monk 0 soon at ! b$side4 ,fa!r ,john1 said ! ,abbot1 do/ ?\ "k t ,i came into ?is wick$ _w on a ,*ri/mas ,ev58 ,! monk nodd$ ass5t4 ,& h ,i n (t5 told !e t1 hav+ be5 born on ,*ri/mas ,ev51 ,i h no love = ! ?+s t are odd8 ,look !re6 ,! ,abbot po9t$ to ! l>ge dormitory w9d[1 ( :i* ,i give a sket*4 ,! monk look$1 & 0 p]plex$4 ,do/ ?\ n see t ! sixty-f\r lights add up an ev5 numb] v]tically & horizontally1 b t all ! diagonal l9es1 except f\rte5 are ( a numb] t is odd8 ,:y is ?is8 ,( a tru?1 my ,"l ,abbot1 x is ( ! v nature ( ?+s1 & _c be *ang$4 ,nay1 b x %all be *ang$4 ,i comm& !e t c]ta9 ( ! lights be clos$ ?is day1 s t e l9e %all h an ev5 numb] ( lights4 ,see ?\ t ?is be done )\t delay1 le/ ! cell>s be lock$ up = a mon? & o!r griev\s tr\bles befall !e4 ,fa!r ,john 0 at 8 wits' end1 b aft] consultation ) "o :o 0 le>n$ 9 /range my/]ies1 a way 0 f\nd to satisfy ! :im ( ! ,"l ,abbot4 ,:i* lights 7 block$ up1 s t ^? :i* rema9$ add$ up an ev5 numb] 9 e l9e horizontally1 v]tically1 & diagonally1 :ile ! lea/ possible ob/ruction ( light 0 caus$8 #bic4--,,! ,,*9ese ,,*essbo>d4 ,into h[ l>ge a numb] ( diff]5t pieces may ! *essbo>d be cut 7by cuts along ! l9es only71 no two pieces be+ exactly alike8 ,rememb] t ! arrangem5t ( black & :ite con/itutes a diff]5ce4 ,?us1 a s+le black squ>e w be diff]5t f a s+le :ite squ>e1 a r[ ( ?ree conta9+ two :ite squ>es w diff] f a r[ ( ?ree conta9+ two black1 & s on4 ,if two pieces _c be plac$ on ! table s z to be exactly alike1 !y c\nt z diff]5t4 ,& z ! back ( ! bo>d is pla91 ! pieces _c be turn$ ov]4 #bid4--,,! ,,*essbo>d ,,s5t5ce4 ,illu/ration ,i once set myself ! amus+ task ( s dissect+ an ord9>y *essbo>d into lett]s ( ! alphabet t !y w\ld =m a complete s5t5ce4 ,x w be se5 f ! illu/ration t ! pieces assembl$ give ! s5t5ce1 ,,cut ,,?y ,,life1 ) ! /ops betwe54 ,! ideal s5t5ce w\ld1 ( c\rse1 h only "o full /op1 b t ,i did n succe$ 9 obta9+4 ,! s5t5ce is an appeal to ! transgressor to cut himself adrift f ! evil life he is liv+4 ,c y fit ^! pieces toge!r to =m a p]fect *essbo>d8 ,,/atical ,,*ess ,,puzzles4 ,!y also s]ve :o only /& & wait4 ,,milton4 #bie4--,,! ,,eight ,,rooks4 ,illu/ration3 ------------------------ ------------------------ ------------------------ ------------------------ ,r ,r ,r ,r ,r ,r ,r ,r ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ,,fig4 #a4 ,illu/ration3 ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,r ------------------------ ,,fig4 #b4 ,x w be se5 9 ! fir/ diagram t e squ>e on ! bo>d is ei!r occupi$ or attack$ by a rook1 & t e rook is gu>d$ 7if !y 7 alt]nately black & :ite rooks we %\ld say attack$7 by ano!r rook4 ,plac+ ! eight rooks on any r[ or file obvi\sly w h ! same effect4 ,9 diagram #b e squ>e is aga9 ei!r occupi$ or attack$1 b 9 ?is case e rook is ungu>d$4 ,n[1 9 h[ _m diff]5t ways c y s place ! eight rooks on ! bo>d t e squ>e %all be occupi$ or attack$ & no rook ev] gu>d$ by ano!r8 ,i d n wi% to g into ! "q ( rev]sals & reflections on ?is occasion1 s t plac+ ! rooks on ! o!r diagonal w c\nt z diff]5t1 & simil>ly ) o!r repetitions obta9$ by turn+ ! bo>d r\nd4 #bif4--,,! ,,f\r ,,lions4 ,! puzzle is to f9d 9 h[ _m diff]5t ways ! f\r lions may be plac$ s t !re %all nev] be m ?an "o lion 9 any r[ or column4 ,m]e rev]sals & reflections w n c\nt z diff]5t4 ,?us1 reg>d+ ! example giv51 if we place ! lions 9 ! o!r diagonal1 x w be consid]$ ! same arrangem5t4 ,= if y hold ! second arrangem5t 9 front ( a mirror or give x a qu>t] turn1 y m]ely get ! fir/ arrangem5t4 ,x is a simple little puzzle1 b requires a c]ta9 am\nt ( c>eful consid]ation4 ,illu/ration ------------ ,l ------------ ,l ------------ ,l ------------ ,l ------------ #big4--;,,bi%ops--,,ungu>d$4 ,place z few bi%ops z possible on an ord9>y *essbo>d s t e squ>e ( ! bo>d %all be ei!r occupi$ or attack$4 ,x w be se5 t ! rook has m scope ?an ! bi%op3 = :]ev] y place ! =m]1 x w always attack f\rte5 o!r squ>es2 :]eas ! latt] w attack sev51 n9e1 elev51 or ?irte5 squ>es1 accord+ to ! position ( ! diagonal on :i* x is plac$4 ,& x is well "h to /ate t :5 we speak ( diagonals 9 connection ) ! *essbo>d1 we d n limit ourselves to ! two long diagonals f corn] to corn]1 b include all ! %ort] l9es t are p>allel to ^!4 ,to prev5t misund]/&+ on future occasions1 x w be well = ! read] to note c>efully ?is fact4 #bih4--;,,bi%ops--,,gu>d$4 ,n[1 h[ _m bi%ops are necess>y 9 ord] t e squ>e %all be ei!r occupi$ or attack$1 & e bi%op gu>d$ by ano!r bi%op8 ,& h[ may !y be plac$8 #bii4--;,,bi%ops ,,9 ,,convocation4 ,illu/ration3 ------------------------ ,b ,b ,b ,b ,b ,b ,b ,b ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ,b ,b ,b ,b ,b ,b ------------------------ ,! greate/ numb] ( bi%ops t c be plac$ at ! same "t on ! *essbo>d1 )\t any bi%op attack+ ano!r1 is f\rte54 ,i %[1 9 diagram1 ! simple/ way ( do+ ?is4 ,9 fact1 on a squ>e *equ]$ bo>d ( any numb] ( squ>es ! greate/ numb] ( bi%ops t c be plac$ )\t attack is always two less ?an twice ! numb] ( squ>es on ! side4 ,x is an int]e/+ puzzle to discov] 9 j h[ _m diff]5t ways ! f\rte5 bi%ops may be s plac$ )\t mutual attack4 ,i %all give an exce$+ly simple rule = det]m9+ ! numb] ( ways = a squ>e *equ]$ bo>d ( any numb] ( squ>es4 #cjj4--,,! ,,eight ,,que5s4 ,illu/ration3 ------------------------ 44,q ---------444------------ 44,q44 ---444------------------ ,q44 ------------------------ ,q ------------------------ ,q ------------------------ 44,q ------------------444--- 44,q44 ------------444--------- ,q44 ------------------------ ,! que5 is by f> ! /ronge/ piece on ! *essbo>d4 ,if y place h] on "o ( ! f\r squ>es 9 ! c5tre ( ! bo>d1 %e attacks no few] ?an tw5ty-sev5 o!r squ>es2 & if y try to hide h] 9 a corn]1 %e /ill attacks tw5ty-"o squ>es4 ,eight que5s may be plac$ on ! bo>d s t no que5 attacks ano!r1 & x is an old puzzle 7fir/ propos$ by ,nauck 9 #ahej1 & x has q a little lit]ature ( its own7 to discov] 9 j h[ _m diff]5t ways ?is may be done4 ,i %[ "o way 9 ! diagram1 & !re are 9 all twelve ( ^! fundam5tally diff]5t ways4 ,^! twelve produce n9ety-two ways if we reg>d rev]sals & reflections z diff]5t4 ,! diagram is 9 a way a symmetrical arrangem5t4 ,if y turn ! page upside d[n1 x w reproduce itself exactly2 b if y look at x ) "o ( ! o!r sides at ! bottom1 y get ano!r way t is n id5tical4 ,!n if y reflect ^! two ways 9 a mirror y get two m ways4 ,n[1 all ! o!r elev5 solutions are non-symmetrical1 & !re=e ea* ( !m may be pres5t$ 9 eight ways by ^! rev]sals & reflections4 ,x w ?us be se5 :y ! twelve fundam5tally diff]5t solutions produce only n9ety-two arrangem5ts1 z ,i h said1 & n n9ety-six1 z w\ld happ5 if all twelve 7 non-symmetrical4 ,x is well to h a cle> und]/&+ on ! matt] ( rev]sals & reflections :5 deal+ ) puzzles on ! *essbo>d4 ,c ! read] place ! eight que5s on ! bo>d s t no que5 %all attack ano!r & s t no ?ree que5s %all be 9 a /raight l9e 9 any oblique direction8 ,ano!r glance at ! diagram w %[ t ?is arrangem5t w n answ] ! conditions1 = 9 ! two directions indicat$ by ! dott$ l9es !re are ?ree que5s 9 a /raight l9e4 ,!re is only "o ( ! twelve fundam5tal ways t w solve ! puzzle4 ,c y f9d x8 #cja4--,,! ,,eight ,,/>s4 ,illu/ration3 ------------------------ _/_/_/ _/_/_/ ------------------------ _/_/_/ _/_/_/ 99 ------------------------ _/_/_/ _/_/_/ ------------------------ _/_/_/_/_/_/ ------------------------ _/_/_/_/_/_/ ------------------------ _/_/_/ _/_/_/ ------------------------ _/_/_/ _/_/_/ ------------------------ _/_/_/ _/_/_/ ------------------------ ,! puzzle 9 ?is case is to place eight />s 9 ! diagram s t no /> %all be 9 l9e ) ano!r /> horizontally1 v]tically1 or diagonally4 ,"o /> is already plac$1 & t mu/ n be mov$1 s !re are only sev5 = ! read] n[ to place4 ,b y mu/ n place a /> on any "o ( ! %ad$ squ>es4 ,!re is only "o way ( solv+ ?is little puzzle4 #cjb4--;,a ,,problem ,,9 ,,mosaics4 ,! art ( produc+ pictures or designs by means ( jo9+ toge!r pieces ( h>d sub/ances1 ei!r naturally or artificially col\r$1 is ( v great antiquity4 ,x 0 c]ta9ly kn[n 9 ! "t ( ! ,ph>aohs1 & we f9d a ref]5ce 9 ! ,book ( ,e/h] to a pavem5t ( r$1 & blue1 & :ite1 & black m>ble4 ,"s ( ?is anci5t "w t has come d[n to u1 especially "s ( ! ,roman mosaics1 w\ld seem to %[ cle>ly1 ev5 :]e design is n at fir/ evid5t1 t mu* ?\ght 0 be/[$ ^u app>5tly disord]ly arrangem5ts4 ,:]e1 = example1 ! "w has be5 produc$ ) a v limit$ numb] ( col\rs1 !re are evid5ces ( great +5uity 9 prev5t+ ! same t9ts com+ 9 close proximity4 ,lady read]s :o are famili> ) ! con/ruction ( pat*work quilts w "k h[ desirable x is sometimes1 :5 !y are limit$ 9 ! *oice ( mat]ial1 to prev5t pieces ( ! same /uff com+ too ne> toge!r4 ,n[1 ?is puzzle w apply equally to pat*work quilts or tesselat$ pavem5ts4 ,x w be se5 f ! diagram h[ a squ>e piece ( floor+ may be pav$ ) sixty-two squ>e tiles ( ! eight col\rs violet1 r$1 yell[1 gre51 orange1 purple1 :ite1 & blue 7indicat$ by ! initial lett]s71 s t no tile is 9 l9e ) a simil>ly col\r$ tile1 v]tically1 horizontally1 or diagonally4 ,sixty-f\r su* tiles c\ld n possibly be plac$ "u ^! conditions1 b ! two %ad$ squ>es happ5 to be occupi$ by iron v5tilators4 ,illu/ration3 ------------------------ ,v ,r ,y ,g ,o ,p ,w ,b ------------------------ ,w ,b ,o ,p ,y ,g ,v ,r ------9999------9999------ ,g ,p ,h ,w ,h ,v ,b ,h ,r ,h ,y ,o ------9999------9999------ ,r ,y ,b ,o ,g ,v ,p ,w ------------------------ ,b ,g ,r ,y ,p ,w ,o ,v ------------------------ ,o ,v ,p ,w ,r ,y ,b ,g ------------------------ ,p ,w ,g ,b ,v ,o ,r ,y ------------------------ _/_/_/ ,o ,v ,r ,w ,b ,g _/_/_/ ------------------------ ,! puzzle is ?is4 ,^! two v5tilators h to be remov$ to ! positions indicat$ by ! d>kly bord]$ tiles1 & two tiles plac$ 9 ^? bottom corn] squ>es4 ,c y readju/ ! ?irty-two tiles s t no two ( ! same col\r %all /ill be 9 l9e8 #cjc4--,,"u ,,! ,,veil4 ,illu/ration3 ------------------------ ,v ,e ,i ,l ------------------------ ,i ,l ,v ,e ------------------------ ,i ,v ,l ,e ------------------------ ,l ,e ,i ,v ------------------------ ,v ,i ,e ,l ------------------------ ,e ,l ,v ,i ------------------------ ,e ,v ,l ,i ------------------------ ,l ,i ,e ,v ------------------------ ,if ! read] w exam9e ! above diagram1 he w see t ,i h s plac$ eight ,v's1 eight ,e's1 eight ,i's1 & eight ,l's 9 ! diagram t no lett] is 9 l9e ) a simil> "o horizontally1 v]tically1 or diagonally4 ,?us1 no ,v is 9 l9e ) ano!r ,v1 no ,e ) ano!r ,e1 & s on4 ,!re are a great _m diff]5t ways ( arrang+ ! lett]s "u ?is condition4 ,! puzzle is to f9d an arrangem5t t produces ! greate/ possible numb] ( f\r-lett] words1 read+ upw>ds & d[nw>ds1 backw>ds & =w>ds1 or diagonally4 ,all repetitions c\nt z diff]5t words1 & ! five v>iations t may be us$ are3 ,,veil1 ,,vile1 ,,levi1 ,,live1 & ,,evil4 ,?is w be made p]fectly cle> :5 ,i say t ! above arrangem5t scores eight1 because ! top & bottom r[ bo? give ,,veil2 ! second & sev5? columns bo? give ,,veil2 & ! two diagonals1 />t+ f ! ,l 9 ! #e? r[ & ,e 9 ! #h? r[1 bo? give ,,live & ,,evil4 ,!re are !re=e eight diff]5t read+s ( ! words 9 all4 ,?is difficult ^w puzzle is giv5 z an example ( ! use ( *essbo>d analysis 9 solv+ su* ?+s4 ,only a p]son :o is famili> ) ! ,eight ,que5s problem c\ld hope to solve x4 #cjd4--;,,ba*et',s ,,squ>e4 ,"o ( ! olde/ c>d puzzles is by ,claude ,casp> ,ba*et de ,mziriac1 fir/ publi%$1 ,i believe1 9 ! #afbd edition ( 8 "w4 ,re>range ! sixte5 c\rt c>ds 7includ+ ! aces7 9 a squ>e s t 9 no r[ ( f\r c>ds1 horizontal1 v]tical1 or diagonal1 %all be f\nd two c>ds ( ! same suit or ! same value4 ,?is 9 itself is easy en\gh1 b a po9t ( ! puzzle is to f9d 9 h[ _m diff]5t ways ?is may be done4 ,! em95t ,fr5* ma!matician ,a4 ,labosne1 9 8 mod]n edition ( ,ba*et1 gives ! answ] incorrectly4 ,& yet ! puzzle is really q easy4 ,any arrangem5t produces sev5 m by turn+ ! squ>e r\nd & reflect+ x 9 a mirror4 ,^! are c\nt$ z diff]5t by ,ba*et4 ,note r[ ( f\r c>ds1 s t ! only diagonals we h "h to consid] are ! two long ones4 #cje4--,,! ,,?irty-,,six ,,lett]-,,blocks4 ,illu/ration ,! illu/ration repres5ts a box conta9+ ?irty-six lett]-blocks4 ,! puzzle is to re>range ^! blocks s t no ,a %all be 9 a l9e v]tically1 horizontally1 or diagonally ) ano!r ,a1 no ,b ) ano!r ,b1 no ,c ) ano!r ,c1 & s on4 ,y w f9d x impossible to get all ! lett]s into ! box "u ^! conditions1 b ! po9t is to place z _m z possible4 ,( c\rse no lett]s o!r ?an ^? %[n may be us$4 #cjf4--,,! ,,cr[d$ ,,*essbo>d4 ,illu/ration ,! puzzle is to re>range ! fifty-"o pieces on ! *essbo>d s t no que5 %all attack ano!r que51 no rook attack ano!r rook1 no bi%op attack ano!r bi%op1 & no knight attack ano!r knight4 ,no notice is to be tak5 ( ! int]v5tion ( pieces ( ano!r type f t "u consid]ation--t is1 two que5s w be consid]$ to attack "o ano!r al?\gh !re may be1 say1 a rook1 a bi%op1 & a knight betwe5 !m4 ,& s ) ! rooks & bi%ops4 ,x is n difficult to dispose ( ea* type ( piece sep>ately2 ! difficulty comes 9 :5 y h to f9d room = all ! arrangem5ts on ! bo>d simultane\sly4 #cjg4--,,! ,,col\r$ ,,c\nt]s4 ,illu/ration ,! diagram repres5ts tw5ty-five col\r$ c\nt]s1 ,r$1 ,blue1 ,yell[1 ,orange1 & ,gre5 7indicat$ by _! initials71 & !re are five ( ea* col\r1 numb]$ #a1 #b1 #c1 #d1 & #e4 ,! problem is s to place !m 9 a squ>e t nei!r col\r nor numb] %all be f\nd repeat$ 9 any "o ( ! five r[s1 five columns1 & two diagonals4 ,c y s re>range !m8 #cjh4--,,! ,,g5tle ,,art ,,( ,,/amp-,,lick+4 ,! ,insurance ,act is a mo/ prolific s\rce ( ent]ta9+ puzzles1 p>ticul>ly ent]ta9+ if y happ5 to be among ! exempt4 ,"o's initiation into ! g5tle art ( /amp-lick+ sugge/s ! foll[+ little pos]3 ,if y h a c>d divid$ into sixte5 spaces 7#d #d71 & are provid$ ) pl5ty ( /amps ( ! values #a;d41 #b;d41 #c;d41 #d;d41 & #e;d41 :at is ! greate/ value t y c /ick on ! c>d if ! ,*ancellor ( ! ,ex*equ] =bids y to place any /amp 9 a /raight l9e 7t is1 horizontally1 v]tically1 or diagonally7 ) ano!r /amp ( simil> value8 ,( c\rse1 only "o /amp c be affix$ 9 a space4 ,! read] w probably f9d1 :5 he sees ! solution1 t1 l ! /amps !mselves1 he is lick$ ,he w mo/ likely be twop5ce %ort ( ! maximum4 ,a fri5d ask$ ! ,po/ ,(fice h[ x 0 to be done2 b !y s5t him to ! ,cu/oms & ,excise (fic]1 :o s5t him to ! ,insurance ,commission]s1 :o s5t him to an approv$ society1 :o pr(anely s5t him--b no matt]4 #cji4--,,! ,,=ty-,,n9e ,,c\nt]s4 ,illu/ration ,c y re>range ! above =ty-n9e c\nt]s 9 a squ>e s t no lett]1 & also no numb]1 %all be 9 l9e ) a simil> "o1 v]tically1 horizontally1 or diagonally8 ,"h ,i1 ( c\rse1 mean 9 ! l9es p>allel ) ! diagonals1 9 ! *essbo>d s5se4 #caj4--,,! ,,?ree ,,%eep4 ,illu/ration ,a f>m] _h ?ree %eep & an arrangem5t ( sixte5 p5s1 divid$ (f by hurdles 9 ! mann] indicat$ 9 ! illu/ration4 ,9 h[ _m diff]5t ways c\ld he place ^? %eep1 ea* 9 a sep>ate p51 s t e p5 %\ld be ei!r occupi$ or 9 l9e 7horizontally1 v]tically1 or diagonally7 ) at lea/ "o %eep8 ,i h giv5 "o arrangem5t t fulfils ! conditions4 ,h[ _m o!rs c y f9d8 ,m]e rev]sals & reflections mu/ n be c\nt$ z diff]5t4 ,! read] may reg>d ! %eep z que5s4 ,! problem is !n to place ! ?ree que5s s t e squ>e %all be ei!r occupi$ or attack$ by at lea/ "o que5--9 ! maximum numb] ( diff]5t ways4 #caa4--,,! ,,five ,,dogs ,,puzzle4 ,9 #ahfc1 ,c4,f4 de ,ja5is* fir/ discuss$ ! ,five ,que5s ,puzzle--to place five que5s on ! *essbo>d s t e squ>e %all be attack$ or occupi$--:i* 0 prop\nd$ by 8 fri5d1 a ,mr4 de ,r4 ,ja5is* %[$ t if no que5 may attack ano!r !re are n9ety-"o diff]5t ways ( plac+ ! five que5s1 rev]sals & reflections n c\nt+ z diff]5t4 ,if ! que5s may attack "o ano!r1 ,i h record$ hundr$s ( ways1 b x is n practicable to enum]ate !m exactly4 ,illu/ration ,! illu/ration is suppos$ to repres5t an arrangem5t ( sixty-f\r k5nels4 ,x w be se5 t five k5nels ea* conta9 a dog1 & on fur!r exam9ation x w be se5 t e "o ( ! sixty-f\r k5nels is 9 a /raight l9e ) at lea/ "o dog--ei!r horizontally1 v]tically1 or diagonally4 ,take any k5nel y l1 & y w f9d t y c draw a /raight l9e to a dog 9 "o or o!r ( ! ?ree ways m5tion$4 ,! puzzle is to replace ! five dogs & discov] 9 j h[ _m diff]5t ways !y may be plac$ 9 five k5nels 9 a /raight r[1 s t e k5nel %all always be 9 l9e ) at lea/ "o dog4 ,rev]sals & reflections are "h c\nt$ z diff]5t4 #cab4--,,! ,,five ,,cresc5ts ,,( ,,byzantium4 ,:5 ,philip ( ,mac$on1 ! fa!r ( ,alex&] ! ,great1 f\nd himself confront$ ) great difficulties 9 ! siege ( ,byzantium1 he set 8 m5 to und]m9e ! walls4 ,8 desires1 h[ev]1 misc>ri$1 = no soon] _h ! op]ations be5 begun ?an a cresc5t moon sudd5ly appe>$ 9 ! heav5s & discov]$ 8 plans to 8 adv]s>ies4 ,! ,byzant9es 7 naturally elat$1 & 9 ord] to %[ _! gratitude !y erect$ a /atue to ,diana1 & ! cresc5t became !nce=w>d a symbol ( ! /ate4 ,9 ! temple t conta9$ ! /atue 0 a squ>e pavem5t compos$ ( sixty-f\r l>ge & co/ly tiles4 ,^! 7 all pla91 ) ! exception ( five1 :i* bore ! symbol ( ! cresc5t4 ,^! five 7 = occult reasons s plac$ t e tile %\ld be wat*$ ov] by 7t is1 9 a /raight l9e1 v]tically1 horizontally1 or diagonally )7 at lea/ "o ( ! cresc5ts4 ,! arrangem5t adopt$ by ! ,byzant9e ar*itect 0 z foll[s3-- ,illu/ration ,n[1 to cov] up "o ( ^! five cresc5ts 0 a capital (f5ce1 ! dea? be+ some?+ v pa9ful & l+]+4 ,b on a c]ta9 occasion ( fe/ivity x 0 necess>y to lay d[n on ?is pavem5t a squ>e c>pet ( ! l>ge/ dim5sions possible1 & ,i h %[n 9 ! illu/ration by d>k %ad+ ! l>ge/ dim5sions t w\ld be available4 ,! puzzle is to %[ h[ ! ar*itect1 if he _h =ese5 ?is "q ( ! c>pet1 might h s arrang$ 8 five cresc5t tiles 9 accordance ) ! requir$ conditions1 & yet h all[$ = ! l>ge/ possible squ>e c>pet to be laid d[n )\t any "o ( ! five cresc5t tiles be+ cov]$1 or any portion ( !m4 #cac4--,,que5s ,,& ,,bi%op ,,puzzle4 ,x w be se5 t e squ>e ( ! bo>d is ei!r occupi$ or attack$4 ,! puzzle is to sub/itute a bi%op = ! rook on ! same squ>e1 & !n place ! f\r que5s on o!r squ>es s t e squ>e %all aga9 be ei!r occupi$ or attack$4 ,illu/ration #cad4--,,! ,,s\!rn ,,cross4 ,illu/ration ,9 ! above illu/ration we h five ,planets & eighty-"o ,fix$ ,/>s1 five ( ! latt] be+ hidd5 by ! ,planets4 ,x w be f\nd t e ,/>1 ) ! exception ( ! t5 t h a black spot 9 _! c5tres1 is 9 a /raight l9e1 v]tically1 horizontally1 or diagonally1 ) at lea/ "o ( ! ,planets4 ,! puzzle is s to re>range ! ,planets t all ! ,/>s %all be 9 l9e ) "o or m ( !m4 ,9 re>rang+ ! ,planets1 ea* ( ! five may be mov$ once 9 a /raight l9e1 9 ei!r ( ! ?ree directions m5tion$4 ,!y w1 ( c\rse1 obscure five o!r ,/>s 9 place ( ^? at pres5t cov]$4 #cae4--,,! ,,hat-,,peg ,,puzzle4 ,"h is a five-que5 puzzle t ,i gave 9 a fanciful dress 9 #ahig4 ,z ! que5s 7 !re repres5t$ z hats on sixty-f\r pegs1 ,i w keep to ! title1 ,! ,hat-,peg ,puzzle4 ,x w be se5 t e squ>e is occupi$ or attack$4 ,! puzzle is to remove "o que5 to a diff]5t squ>e s t /ill e squ>e is occupi$ or attack$1 !n move a second que5 "u a simil> condition1 !n a ?ird que51 & f9ally a f\r? que54 ,aft] ! f\r? move e squ>e mu/ be attack$ or occupi$1 b no que5 mu/ !n attack ano!r4 ,( c\rse1 ! moves ne$ n be que5 moves2 y c move a que5 to any "p ( ! bo>d4 ,illu/ration #caf4--,,! ,,amazons4 ,illu/ration ,?is puzzle is bas$ on "o by ,capta9 ,turton4 ,remove ?ree ( ! que5s to o!r squ>es s t !re %all be elev5 squ>es on ! bo>d t are n attack$4 ,! removal ( ! ?ree que5s ne$ n be by que5 moves4 ,y may take !m up & place !m any:]e4 ,!re is only "o solution4 #cag4--;,a ,,puzzle ,,) ,,pawns4 ,place two pawns 9 ! middle ( ! *essbo>d1 "o at ,q #d & ! o!r at ,k #e4 ,n[1 place ! rema9+ f\rte5 pawns 7sixte5 9 all7 s t no ?ree %all be 9 a /raight l9e 9 any possible direction4 ,note t ,i purposely d n say que5s1 because by ! words any possible direction ,i g beyond attacks on diagonals4 ,! pawns mu/ be reg>d$ z m]e po9ts 9 space--at ! c5tres ( ! squ>es4 ,see dott$ l9es 9 ! case ( ,no4 #cjj1 ,! ,eight ,que5s4 #cah4--,,lion-,,hunt+4 ,illu/ration ,my fri5d ,capta9 ,po?am ,hall1 ! r5[n$ hunt] ( big game1 says !re is no?+ m exhil>at+ ?an a bru% ) a h]d--a pack--a team--a flock--a sw>m 7x has tak5 me a full qu>t] ( an h\r to recall ! "r ^w1 b ,i h x at la/7--a pride ( lions4 ,:y a numb] ( lions are call$ a pride1 a numb] ( :ales a s*ool1 & a numb] ( foxes a skulk are my/]ies ( philology into :i* ,i w n ent]4 ,well1 ! capta9 says t if a spirit$ lion crosses y\r pa? 9 ! des]t x becomes lively1 = ! lion has g5]ally be5 look+ = ! man j z mu* z ! man has s\ght ! k+ ( ! =e/4 ,& yet :5 !y meet !y always qu>rel & fight x out4 ,a little contemplation ( ?is un=tunate & long-/&+ feud betwe5 two e/imable families has l$ me to figure out a few calculations z to ! probability ( ! man & ! lion cross+ "o ano!r's pa? 9 ! jungle4 ,9 all ^! cases "o has to />t on c]ta9 m or less arbitr>y assumptions4 ,t is :y 9 ! above illu/ration ,i h ?\ght x necess>y to repres5t ! pa?s 9 ! des]t ) su* rigid regul>ity4 ,?\gh ! capta9 assures me t ! tracks ( ! lions usually run mu* 9 ?is way1 ,i h d\bts4 ,! puzzle is simply to f9d out 9 h[ _m diff]5t ways ! man & ! lion may be plac$ on two diff]5t spots t are n on ! same pa?4 ,by pa?s x mu/ be und]/ood t ,i only ref] to ! rul$ l9es4 ,?us1 ) ! exception ( ! f\r corn] spots1 ea* combatant is always on two pa?s & no m4 ,x w be se5 t !re is a lot ( scope = evad+ "o ano!r 9 ! des]t1 :i* is j :at "o has always und]/ood4 #cai4--,,! ,,knight-,,gu>ds4 ,illu/ration ,! knight is ! irresponsible l[ com$ian ( ! *essbo>d4 ,he is a v unc]ta91 sneak+1 & demoraliz+ rascal1 says an ,am]ican writ]4 ,he c only move two squ>es1 b makes up 9 ! quality ( 8 locomotion = its quantity1 = he c spr+ "o squ>e sideways & "o =w>d simultane\sly1 l a cat2 c /& on "o leg 9 ! middle ( ! bo>d & jump to any "o ( eight squ>es he *ooses2 c get on "o side ( a f5ce & blackgu>d ?ree or f\r m5 on ! o!r2 has an objectionable way ( ins]t+ himself 9 safe places :]e he c sc>e ! k+ & compel him to move1 & !n gobble a que54 ,= pure cuss$ness ! knight has no equal1 & :5 y *ase him out ( "o hole he skips into ano!r4 ,attempts h be5 made ov] & ov] aga9 to obta9 a %ort1 simple1 & exact def9ition ( ! move ( ! knight--)\t success4 ,x really consi/s 9 mov+ "o squ>e l a rook1 & !n ano!r squ>e l a bi%op--! two op]ations be+ done 9 "o leap1 s t x does n matt] :e!r ! fir/ squ>e pass$ ov] is occupi$ by ano!r piece or n4 ,x is1 9 fact1 ! only leap+ move 9 *ess4 ,b difficult z x is to def9e1 a *ild c le>n x by inspection 9 a few m9utes4 ,i h %[n 9 ! diagram h[ twelve knights 7! fewe/ possible t w p]=m ! feat7 may be plac$ on ! *essbo>d s t e squ>e is ei!r occupi$ or attack$ by a knight4 ,exam9e e squ>e 9 turn1 & y w f9d t ?is is s4 ,n[1 ! puzzle 9 ?is case is to discov] :at is ! smalle/ possible numb] ( knights t is requir$ 9 ord] t e squ>e %all be ei!r occupi$ or attack$1 & e knight protect$ by ano!r knight4 ,& h[ w\ld y arrange !m8 ,x w be f\nd t ( ! twelve %[n 9 ! diagram only f\r are ?us protect$ by be+ a knight's move f ano!r knight4 ,,! ,,gu>d$ ,,*essbo>d4 ,on an ord9>y *essbo>d1 #h by #h1 e squ>e c be gu>d$--t is1 ei!r occupi$ or attack$--by #e que5s1 ! fewe/ possible4 ,!re are exactly #ia fundam5tally diff]5t arrangem5ts 9 :i* no que5 attacks ano!r que54 ,if e que5 mu/ attack 7or be protect$ by7 ano!r que51 !re are at fewe/ #da arrangem5ts1 & ,i h record$ "s #aej ways 9 :i* "s ( ! que5s are attack$ & "s n1 b ?is la/ case is v difficult to enum]ate exactly4 ,on an ord9>y *essbo>d e squ>e c be gu>d$ by #h rooks 7! fewe/ possible7 9 #dj1cbj ways1 if no rook may attack ano!r rook1 b x is n kn[n h[ _m ( ^! are fundam5tally diff]5t4 7,see solution to ,no4 #bie1 ,! ,eight ,rooks47 ,i h n enum]at$ ! ways 9 :i* e rook %all be protect$ by ano!r rook4 ,on an ord9>y *essbo>d e squ>e c be gu>d$ by #h bi%ops 7! fewe/ possible71 if no bi%op may attack ano!r bi%op4 ,t5 bi%ops are necess>y if e bi%op is to be protect$4 7,see ,nos4 #big & #bih1 ,bi%ops ungu>d$ & ,bi%ops gu>d$47 ,on an ord9>y *essbo>d e squ>e c be gu>d$ by #ab knights if all b #d are unprotect$4 ,b if e knight mu/ be protect$1 #ad are necess>y4 7,see ,no4 #cai1 ,! ,knight-,gu>ds47 ,deal+ ) ! que5 on n bo>ds g5]ally1 :]e n is less ?an #h1 ! foll[+ results w be ( int]e/3-- #a que5 gu>ds #b bo>d 9 #a fundam5tal way4 #a que5 gu>ds #c bo>d 9 #a fundam5tal way4 #b que5s gu>d #d bo>d 9 #c fundam5tal ways 7protect$74 #c que5s gu>d #d bo>d 9 #b fundam5tal ways 7n protect$74 #c que5s gu>d #e bo>d 9 #cg fundam5tal ways 7protect$74 #c que5s gu>d #e bo>d 9 #b fundam5tal ways 7n protect$74 #c que5s gu>d #f bo>d 9 #a fundam5tal way 7protect$74 #d que5s gu>d #f bo>d 9 #ag fundam5tal ways 7n protect$74 #d que5s gu>d #g bo>d 9 #e fundam5tal ways 7protect$74 #d que5s gu>d #g bo>d 9 #a fundam5tal way 7n protect$74 ,,non-,,attack+ ,,*essbo>d ,,arrangem5ts4 ,we "k t n que5s may always be plac$ on a squ>e bo>d ( n squ>es 7if n be great] ?an #c7 )\t any que5 attack+ ano!r que54 ,b no g5]al =mula = enum]at+ ! numb] ( diff]5t ways 9 :i* x may be done has yet be5 discov]$2 probably x is undiscov]able4 ,! kn[n results are z foll[s3-- ,:]e n #d !re is #a fundam5tal solution & #b 9 all4 ,:]e n #e !re are #b fundam5tal solutions & #aj 9 all4 ,:]e n #f !re is #a fundam5tal solution & #d 9 all4 ,:]e n #g !re are #f fundam5tal solutions & #dj 9 all4 ,:]e n #h !re are #ab fundam5tal solutions & #ib 9 all4 ,:]e n #i !re are #df fundam5tal solutions4 ,:]e n #aj !re are #ib fundam5tal solutions4 ,:]e n #aa !re are #cda fundam5tal solutions4 ,obvi\sly n rooks may be plac$ )\t attack on an n bo>d 9 n6 ways1 b h[ _m ( ^! are fundam5tally diff]5t ,i h only work$ out 9 ! f\r cases :]e n equals #b1 #c1 #d1 & #e4 ,! answ]s "h are respectively #a1 #b1 #g1 & #bc4 7,see ,no4 #bif1 ,! ,f\r ,lions47 ,we c place #bn-#b bi%ops on an n bo>d 9 #bn ways4 7,see ,no4 #bii1 ,bi%ops 9 ,convocation47 ,= bo>ds conta9+ #b1 #c1 #d1 #e1 #f1 #g1 #h squ>es1 on a side !re are respectively #a1 #b1 #c1 #f1 #aj1 #bj1 #cf fundam5tally diff]5t arrangem5ts4 ,:]e n is odd !re are #b7n-#a7 su* arrangem5ts1 ea* giv+ #d by rev]sals & reflections1 & #bn-#c - #b7n-#c7 giv+ #h4 ,:]e n is ev5 !re are #b7n-#b71 ea* giv+ #d by rev]sals & reflections1 & #bn-#c - #b7n-#d71 ea* giv+ #h4 ,we c place 7n#a7 knights on an n bo>d )\t attack1 :5 n is odd1 9 #a fundam5tal way2 & n knights on an n bo>d1 :5 n is ev51 9 #a fundam5tal way4 ,9 ! fir/ case we place all ! knights on ! same col\r z ! c5tral squ>e2 9 ! second case we place !m all on black1 or all on :ite1 squ>es4 ,,! ,,two ,,pieces ,,problem4 ,on a bo>d ( n squ>es1 two que5s1 two rooks1 two bi%ops1 or two knights c always be plac$1 irrespective ( attack or n1 9 7n#d - n7 ways4 ,! foll[+ =mul w %[ 9 h[ _m ( ^! ways ! two pieces may be plac$ ) attack & )\t3-- ,) ,attack4 ,)\t ,attack4 #b ,que5s #en - #fn n #cn#d - #ajn #in - #bn ------------------- ------------------------------ #c #f #b ,rooks n - n n#d - #bn n ---------------------- #b #b ,bi%ops #dn - #fn #bn #cn#d - #dn #cn - #bn -------------------- ----------------------------- #f #f #b ,knights #dn - #abn #h n#d - #in #bdn -------------------- #b 7,see ,no4 #cah1 ,lion ,hunt+47 ,,dynamical ,,*ess ,,puzzles4 ,pu% on--keep mov+4 ,,?os4 ,,morton3 ,cure = ! ,he>ta*e4 #cbj4--,,! ,,rook',s ,,t\r4 ,illu/ration3 ------------------------ ------------------------ ------------------------ ------------------------ ,r ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ,! puzzle is to move ! s+le rook ov] ! :ole bo>d1 s t x %all visit e squ>e ( ! bo>d once1 & only once1 & end its t\r on ! squ>e f :i* x />ts4 ,y h to d ?is 9 z few moves z possible1 & unless y are v c>eful y w take j "o move too _m4 ,( c\rse1 a squ>e is reg>d$ equally z visit$ :e!r y m]ely pass ov] x or make x a /opp+-place1 & we w n quibble ov] ! po9t :e!r ! orig9al squ>e is actually visit$ twice4 ,we w assume t x is n4 #cba4--,,! ,,rook',s ,,j\rney4 ,?is puzzle ,i call ,! ,rook's ,j\rney1 because ! ^w t\r 7d]iv$ f a turn]'s :eel7 implies t we return to ! po9t f :i* we set out1 & we d n d ?is 9 ! pres5t case4 ,we %\ld n be satisfi$ ) a p]sonally conduct$ holiday t\r t end$ by leav+ u1 say1 9 ! middle ( ! ,sah>a4 ,! rook "h makes tw5ty-"o moves1 9 ! c\rse ( :i* j\rney x visits e squ>e ( ! bo>d once & only once1 /opp+ at ! squ>e m>k$ #aj at ! end ( its t5? move1 & end+ at ! squ>e m>k$ #ba4 ,two consecutive moves _c be made 9 ! same direction--t is to say1 y mu/ make a turn aft] e move4 ,illu/ration3 ------------------------ ,r ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ #ba #aj ------------------------ #cbb4--,,! ,,langui%+ ,,maid54 ,illu/ration3 ------------------------------------- ,kt -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- ,m -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- ,a wick$ b>on 9 ! good old days imprison$ an innoc5t maid5 9 "o ( ! deepe/ dungeons b5ea? ! ca/le moat4 ,x w be se5 f our illu/ration t !re 7 sixty-?ree cells 9 ! dungeon1 all connect$ by op5 doors1 & ! maid5 0 *a9$ 9 ! cell 9 :i* %e is %[n4 ,n[1 a valiant knight1 :o lov$ ! damsel1 succe$$ 9 rescu+ h] f ! enemy4 ,hav+ ga9$ an entrance to ! dungeon at ! po9t :]e he is se51 he succe$$ 9 rea*+ ! maid5 aft] ent]+ e cell once & only once4 ,take y\r p5cil & try to trace out su* a r\te4 ,:5 y h succe$$1 !n try to discov] a r\te 9 tw5ty-two /raight pa?s ?r\gh ! cells4 ,x c be done 9 ?is numb] )\t ent]+ any cell a second "t4 #cbc4--;,a ,,dungeon ,,puzzle4 ,illu/ration3 ---------------------------------------- 4444444444444 4444444 4444444444444 4 4 4 4 4 4 --4---- ----4----4----4----4---- ----4-- 4 4 4 4 4 4 4444444 4444444 4444444 4444444 4 4 -- ----4---- ---- ---- ---- ----4---- -- 4 4 4444444 4444444 4444444 4444444 4 4 4 4 4 4 --4---- ----4----4----4----4---- ----4-- 4 4 4 4 4 4 4444444444444 4444444 4 4444444 4 4 -- ---- ---- ---- ---- ----4----4---- -- 4 4 4444444444444 4444444 4 4444444 4 4 4 4 4 4 --4---- ----4----4----4----4---- ----4-- 4 4 4 4 4 4 4444444 4444444 4444444 4444444 4 4 -- ----4---- ---- ---- ---- ----4---- -- 4 4 4444444 4444444 4444444 4444444 4 4 4 4 4 4 --4---- ----4----4----4----4---- ----4-- 4 4 4 4 4 4 4444444444444 4 ,p 4444444444444 -- ---- ---- ---- ---- ---- ---- ---- -- ,a ,fr5* prison]1 = 8 s9s 7or o!r p's71 0 conf9$ 9 an und]gr\nd dungeon conta9+ sixty-f\r cells1 all communicat+ ) op5 doorways1 z %[n 9 our illu/ration4 ,9 ord] to r$uce ! t$ium ( 8 re/rict$ life1 he set himself v>i\s puzzles1 & ?is is "o ( !m4 ,/>t+ f ! cell 9 :i* he is %[n1 h[ c\ld he visit e cell once1 & only once1 & make z _m turn+s z possible8 ,8 fir/ attempt is %[n by ! dott$ track4 ,x w be f\nd t !re are z _m z fifty-five /raight l9es 9 8 pa?1 b aft] _m attempts he improv$ ^u ?is4 ,c y get m ?an fifty-five8 ,y may end y\r pa? 9 any cell y l4 ,try ! puzzle ) a p5cil on *essbo>d diagrams1 or y may reg>d !m z rooks' moves on a bo>d4 #cbd4--,,! ,,lion ,,& ,,! ,,man4 ,9 a public place 9 ,rome !re once /ood a prison divid$ into sixty-f\r cells1 all op5 to ! sky & all communicat+ ) "o ano!r1 z %[n 9 ! illu/ration4 ,! sports t "h took place 7 wat*$ f a high t[]4 ,! fav\rite game 0 to place a ,*ri/ian 9 "o corn] cell & a lion 9 ! diagonally opposite corn] & !n leave !m ) all ! inn] doors op54 ,! consequ5t effect 0 sometimes mo/ laughable4 ,on "o occasion ! man 0 giv5 a sword4 ,he 0 no c[>d1 & 0 z anxi\s to f9d ! lion z ! lion und\bt$ly 0 to f9d him4 ,illu/ration3 ---------------------------------------- ,l -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- -- ---- ---- ---- ---- ---- ---- ---- -- ,c -- ---- ---- ---- ---- ---- ---- ---- -- ,! man visit$ e cell once & only once 9 ! fewe/ possible /raight l9es until he rea*$ ! lion's cell4 ,! lion1 curi\sly en\gh1 also visit$ e cell once & only once 9 ! fewe/ possible /raight l9es until he f9ally rea*$ ! man's cell4 ,!y />t$ toge!r & w5t at ! same spe$2 yet1 al?\gh !y occasionally got glimpses ( "o ano!r1 !y nev] once met4 ,! puzzle is to %[ ! r\te t ea* happ5$ to take4 #cbe4--;,,an ,,episcopal ,,visitation4 ,! :ite squ>es on ! *essbo>d repres5t ! p>i%es ( a diocese4 ,place ! bi%op on any squ>e y l1 & s contrive t 7us+ ! ord9>y bi%op's move ( *ess7 he %all visit e "o ( 8 p>i%es 9 ! fewe/ possible moves4 ,( c\rse1 all ! p>i%es pass$ ?r\gh on any move are reg>d$ z visit$4 ,y c visit any squ>es m ?an once1 b y are n all[$ to move twice betwe5 ! same two adjo9+ squ>es4 ,:at are ! fewe/ possible moves8 ,! bi%op ne$ n end 8 visitation at ! p>i% f :i* he fir/ set out4 #cbf4--;,a ,,new ,,c\nt] ,,puzzle4 ,"h is a new puzzle ) mov+ c\nt]s1 or co9s1 t at fir/ glance looks z if x mu/ be absurdly simple4 ,b x w be f\nd q a little p]plexity4 ,i give x 9 ?is place = a reason t ,i w expla9 :5 we come to ! next puzzle4 ,copy ! simple diagram1 enl>g$1 on a %eet ( pap]2 !n place two :ite c\nt]s on ! po9ts #a & #b1 & two r$ c\nt]s on #i & #aj1 ,! puzzle is to make ! r$ & :ite *ange places4 ,y may move ! c\nt]s "o at a "t 9 any ord] y l1 along ! l9es f po9t to po9t1 ) ! only re/riction t a r$ & a :ite c\nt] may nev] /& at once on ! same /raight l9e4 ,?us ! fir/ move c only be f #a or #b to #c1 or f #i or #aj to #g4 ,illu/ration3 #d #h _/ _/ #b #f #aj _/ _/ #c #g _/ _/ #a #e #i #cbg4--;,a ,,new ,,bi%op',s ,,puzzle4 ,illu/ration3 ------------ b b b b ------------ ------------ ------------ ,b ,b ,b ,b ------------ ,?is is q a fasc9at+ little puzzle4 ,place eight bi%ops 7f\r black & f\r :ite7 on ! r$uc$ *essbo>d1 z %[n 9 ! illu/ration4 ,! problem is to make ! black bi%ops *ange places ) ! :ite ones1 no bi%op ev] attack+ ano!r ( ! opposite col\r4 ,!y mu/ move alt]nately--fir/ a :ite1 !n a black1 !n a :ite1 & s on4 ,:5 y h succe$$ 9 do+ x at all1 try to f9d ! fewe/ possible moves4 ,if y leave out ! bi%ops /&+ on black squ>es1 & only play on ! :ite squ>es1 y w discov] my la/ puzzle turn$ on its side4 #cbh4--,,! ,,que5',s ,,t\r4 ,! puzzle ( mak+ a complete t\r ( ! *essbo>d ) ! que5 9 ! fewe/ possible moves 79 :i* squ>es may be visit$ m ?an once7 0 fir/ giv5 by ! late ,sam ,loyd 9 8 ,*ess ,/rategy4 ,b ! solution %[n bel[ is ! "o he gave 9 ,am]ican ,*ess-,nuts 9 #ahfh4 ,i h record$ at lea/ six diff]5t solutions 9 ! m9imum numb] ( moves--f\rte5--b ?is "o is ! be/ ( all1 = reasons ,i w expla94 ,illu/ration3 ------------------------ 44444444444444444444444444444 4 4 -4--------------------4- 4 4 4 44444444444444444444444444 4 4 4 -4----4---------------44- 4 4 4 4 4 44444444444444444 4 4 4 4 4 4 4 4 -4----4---4------4---4---4- 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 44 4 44 4 -4--4-4---4---4---4--4-4--4- 4 4 4 4 44 4 4 4 4 4 4 4 4 44 4 4 4 4 4 44 4 444 4 4 -4--4----4---44--4----4--4- 4 4 44 44 44 4 4 4 4 4 44 4 4 4 4 4 4 4 444 4 4 4 4 -4--4-4---4--4---4---4-4--4- 4 44 4 44 4 44 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 44 4 -4-4-4-4--4------4---4-4-4-4- 44 4 4 4 4 44 44 4 44 4444444444444 4 4 4 ------------------------ ,if y w look at ! lett]$ squ>e y w und]/& t !re are only t5 really diff]5tly plac$ squ>es on a *essbo>d--^? enclos$ by a d>k l9e--all ! o!rs are m]e rev]sals or reflections4 ,= example1 e ,a is a corn] squ>e1 & e ,j a c5tral squ>e4 ,consequ5tly1 z ! solution %[n has a turn+-po9t at ! enclos$ ,d squ>e1 we c obta9 a solution />t+ f & end+ at any squ>e m>k$ ,d--by j turn+ ! bo>d ab\t4 ,n[1 ?is s*eme w give y a t\r />t+ f any ,a1 ,b1 ,c1 ,d1 ,e1 ,f1 or ,h1 :ile no o!r r\te t ,i "k c be adapt$ to m ?an five diff]5t />t+-po9ts4 ,!re is no ,que5's ,t\r 9 f\rte5 moves 7rememb] a t\r mu/ be re-entrant7 t may />t f a ,g1 ,i1 or ,j4 ,b we c h a non-re-entrant pa? ov] ! :ole bo>d 9 f\rte5 moves1 />t+ f any giv5 squ>e4 ,h5ce ! foll[+ puzzle3-- ,illu/ration3 ------------99------------ ,a ,b ,c ,g ,g ,c ,b ,a 9999---------99------------ ,b ,d ,e ,h ,h ,e ,d ,b ---9999------99------------ ,c ,e ,f ,i ,i ,f ,e ,c ------9999---99------------ ,g ,h ,i ,j ,j ,i ,h ,g ---------9999------------ ,g ,h ,i ,j ,j ,i ,h ,g ------------------------ ,c ,e ,f ,i ,i ,f ,e ,c ------------------------ ,b ,d ,e ,h ,h ,e ,d ,b ------------------------ ,a ,b ,c ,g ,g ,c ,b ,a ------------------------ ,/>t f ! ,j 9 ! enclos$ "p ( ! lett]$ diagram & visit e squ>e ( ! bo>d 9 f\rte5 moves1 end+ :]ev] y l4 #cbi4--,,! ,,/> ,,puzzle4 ,illu/ration3 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ 99 99 99 99 99 99 99 99 ------------------------ ,put ! po9t ( y\r p5cil on "o ( ! :ite />s & 7)\t ev] lift+ y\r p5cil f ! pap]7 /rike out all ! />s 9 f\rte5 cont9u\s /raight /rokes1 end+ at ! second :ite />4 ,y\r /raight /rokes may be 9 any direction y l1 only e turn+ mu/ be made on a />4 ,!re is no objection to /rik+ out any /> m ?an once4 ,9 ?is case1 :]e bo? y\r />t+ & end+ squ>es are fix$ inconv5i5tly1 y _c obta9 a solution by break+ a ,que5's ,t\r1 or 9 any o!r way by que5 moves alone4 ,b y are all[$ to use oblique /raight l9es--su* z f ! upp] :ite /> direct to a corn] />4 #ccj4--,,! ,,ya*t ,,race4 ,n[ !n1 ye l&-lubb]s1 hoi/ y\r baby-jib-topsails1 break out y\r sp9nak]s1 ease (f y\r balloon %eets1 & get y\r head-sails set6 ,our race consi/s 9 />t+ f ! po9t at :i* ! ya*t is ly+ 9 ! illu/ration & t\*+ e "o ( ! sixty-f\r buoys 9 f\rte5 /raight c\rses1 return+ 9 ! f9al tack to ! buoy f :i* we />t4 ,! sev5? c\rse mu/ f9i% at ! buoy f :i* a flag is fly+4 ,?is puzzle w call = a lot ( skilful seaman%ip on acc\nt ( ! %>p angles at :i* x w occasionally be necess>y to tack4 ,! po9t ( a lead p5cil & a good nautical eye are all ! outfit t we require4 ,illu/ration ,?is is difficult1 because ( ! condition z to ! flag-buoy1 & because x is a re-entrant t\r4 ,b aga9 we are all[$ ^? oblique l9es4 #cca4--,,! ,,sci5tific ,,skat]4 ,illu/ration ,x w be se5 t ?is skat] has m>k$ on ! ice sixty-f\r po9ts or />s1 & he proposes to />t f 8 pres5t position ne> ! corn] & ent] e "o ( ! po9ts 9 f\rte5 /raight l9es4 ,h[ w he d x8 ,( c\rse !re is no objection to 8 pass+ ov] any po9t m ?an once1 b 8 la/ /raight /roke mu/ br+ him back to ! position f :i* he />t$4 ,x is m]ely a matt] ( tak+ y\r p5cil & />t+ f ! spot on :i* ! skat]'s foot is at pres5t re/+1 & /rik+ out all ! />s 9 f\rte5 cont9u\s /raight l9es1 return+ to ! po9t f :i* y set out4 #ccb4--,,! ,,=ty-,,n9e ,,/>s4 ,illu/ration ,! puzzle 9 ?is case is simply to take y\r p5cil &1 />t+ f "o black />1 /rike out all ! />s 9 twelve /raight /rokes1 end+ at ! o!r black />4 ,x w be se5 t ! attempt %[n 9 ! illu/ration requires fifte5 /rokes4 ,c y d x 9 twelve8 ,e turn+ mu/ be made on a />1 & ! l9es mu/ be p>allel to ! sides & diagonals ( ! squ>e1 z %[n4 ,9 ?is case we are deal+ ) a *essbo>d ( r$uc$ dim5sions1 b only que5 moves 7)\t go+ outside ! b\nd>y z 9 ! la/ case7 are requir$4 #ccc4--,,! ,,que5',s ,,j\rney4 ,illu/ration ,place ! que5 on h] own squ>e1 z %[n 9 ! illu/ration1 & !n try to discov] ! greate/ di/ance t %e c travel ov] ! bo>d 9 five que5's moves )\t pass+ ov] any squ>e a second "t4 ,m>k ! que5's pa? on ! bo>d1 & note c>efully also t %e mu/ nev] cross h] own track4 ,x seems simple en\gh1 b ! read] may f9d t he has tripp$4 #ccd4--,,/4 ,,george ,,& ,,! ,,dragon4 ,illu/ration ,"h is a little puzzle on a r$uc$ *essbo>d ( =ty-n9e squ>es4 ,/4 ,george wi%es to kill ! dragon4 ,kill+ dragons 0 a well-kn[n pa/ime ( 81 &1 be+ a knight1 x 0 only natural t he %\ld desire to p]=m ! feat 9 a s]ies ( knight's moves4 ,c y %[ h[1 />t+ f t c5tral squ>e1 he may visit once1 & only once1 e squ>e ( ! bo>d 9 a *a9 ( *ess knight's moves1 & end by captur+ ! dragon on 8 la/ move8 ,( c\rse a v>iety ( diff]5t ways are op5 to him1 s try to discov] a r\te t =ms "s pretty design :5 y h m>k$ ea* successive leap by a /raight l9e f squ>e to squ>e4 #cce4--;,,f>m] ,,lawr5ce',s ,,cornfields4 ,"o ( ! mo/ beautiful di/ricts )9 easy di/ance ( ,london = a summ] ramble is t "p ( ,buck+ham%ire kn[n z ! ,valley ( ! ,*ess--at lea/1 x 0 a few ye>s ago1 be=e x 0 discov]$ by ! speculative build]4 ,at ! beg9n+ ( ! pres5t c5tury !re liv$1 n f> f ,latim]s1 a wor?y b ecc5tric f>m] nam$ ,lawr5ce4 ,"o ( 8 que] notions 0 t e p]son :o liv$ ne> ! banks ( ! riv] ,*ess ought to be 9 "s way acqua9t$ ) ! noble game ( ! same "n1 & 9 ord] to impress ?is fact on 8 m5 & 8 neighb\rs he adopt$ at times /range t]m9ology4 ,= example1 :5 "o ( 8 ewes pres5t$ him ) a lamb1 he w\ld say t x _h que5$ a pawn2 :5 he put up a new b>n aga9/ ! highway1 he call$ x ca/l+ on ! k+'s side2 & :5 he s5t a man ) a gun to keep 8 neighb\r's birds (f 8 fields1 he spoke ( x z attack+ 8 oppon5t's rooks4 ,ev]ybody 9 ! neighb\rhood us$ to be amus$ at ,f>m] ,lawr5ce's little jokes1 & "o boy 7! wag ( ! village7 :o got 8 e>s pull$ by ! old g5tleman = /eal+ 8 *e/nuts w5t s f> z to call him a silly old *ess-protector6 ,"o ye> he _h a l>ge squ>e field divid$ into =ty-n9e squ>e plots1 z %[n 9 ! illu/ration4 ,! :ite squ>es 7 s[n ) :eat & ! black squ>es ) b>ley4 ,:5 ! h>ve/ "t came r\nd he gave ord]s t 8 m5 7 fir/ to cut ! corn 9 ! pat* m>k$ #a1 & t ea* successive cutt+ %\ld be exactly a knight's move f ! la/ "o1 ! ?irte5? cutt+ be+ 9 ! pat* m>k$ #ac1 ! tw5ty-fif? 9 ! pat* m>k$ #be1 ! ?irty-sev5? 9 ! "o m>k$ #cg1 & ! la/1 or =ty-n9? cutt+1 9 ! pat* m>k$ #di4 ,?is 0 too mu* = poor ,hodge1 & ea* day ,f>m] ,lawr5ce _h to g d[n to ! field & %[ :i* piece _h to be op]at$ ^u4 ,b ! problem w p]haps pres5t no difficulty to my read]s4 ,illu/ration #ccf4--,,! ,,greyh\nd ,,puzzle4 ,9 ?is puzzle ! tw5ty k5nels d n communicate ) "o ano!r by doors1 b are divid$ (f by a l[ wall4 ,! solit>y occupant is ! greyh\nd :i* lives 9 ! k5nel 9 ! top left-h& corn]4 ,:5 he is all[$ 8 lib]ty he has to obta9 x by visit+ e k5nel once & only once 9 a s]ies ( knight's moves1 end+ at ! bottom "r-h& corn]1 :i* is op5 to ! _w4 ,! l9es 9 ! above diagram %[ "o solution4 ,! puzzle is to discov] 9 h[ _m diff]5t ways ! greyh\nd may ?us make 8 exit f 8 corn] k5nel4 ,illu/ration #ccg4--,,! ,,f\r ,,kang>oos4 ,illu/ration ,9 introduc+ a little ,commonweal? problem1 ,i mu/ fir/ expla9 t ! diagram repres5ts ! sixty-f\r fields1 all prop]ly f5c$ (f f "o ano!r1 ( an ,au/ralian settlem5t1 ?\gh ,i ne$ h>dly say t our ki? & k9 d[n "u always d set out _! l& 9 ?is me?odical & exact mann]4 ,x w be se5 t 9 e "o ( ! f\r corn]s is a kang>oo4 ,:y kang>oos h a m>k$ pref]5ce = corn] plots has nev] be5 satisfactorily expla9$1 & x w\ld be out ( place to discuss ! po9t "h4 ,i %\ld also add t kang>oos1 z is well kn[n1 always leap 9 :at we call knight's moves4 ,9 fact1 *ess play]s w\ld probably h adopt$ ! bett] t]m kang>oo's move _h n *ess be5 inv5t$ be=e kang>oos4 ,! puzzle is simply ?is4 ,"o morn+ ea* kang>oo w5t = 8 morn+ hop1 & 9 sixte5 consecutive knight's leaps visit$ j fifte5 diff]5t fields & jump$ back to 8 corn]4 ,no field 0 visit$ by m ?an "o ( ! kang>oos4 ,! diagram %[s h[ !y arrang$ matt]s4 ,:at y are ask$ to d is to %[ h[ !y might h p]=m$ ! feat )\t any kang>oo ev] cross+ ! horizontal l9e 9 ! middle ( ! squ>e t divides ! bo>d into two equal p>ts4 #cch4--,,! ,,bo>d ,,9 ,,comp>tm5ts4 ,illu/ration ,we _c divide ! ord9>y *essbo>d into f\r equal squ>e comp>tm5ts1 & describe a complete t\r1 or ev5 pa?1 9 ea* comp>tm5t4 ,b we may divide x into f\r comp>tm5ts1 z 9 ! illu/ration1 two conta9+ ea* tw5ty squ>es1 & ! o!r two ea* twelve squ>es1 & s obta9 an int]e/+ puzzle4 ,y are ask$ to describe a complete re-entrant t\r on ?is bo>d1 />t+ :]e y l1 b visit+ e squ>e 9 ea* successive comp>tm5t be=e pass+ into ano!r "o1 & mak+ ! f9al leap back to ! squ>e f :i* ! knight set out4 ,x is n difficult1 b w be f\nd v ent]ta9+ & n un9/ructive4 ,:e!r a re-entrant t\r or a complete knight's pa? is possible or n on a rectangul> bo>d ( giv5 dim5sions dep5ds n only on its dim5sions1 b also on its %ape4 ,a t\r is obvi\sly n possible on a bo>d conta9+ an odd numb] ( cells1 su* z #e by #e or #g by #g1 = ?is reason3 ,e successive leap ( ! knight mu/ be f a :ite squ>e to a black & a black to a :ite alt]nately4 ,b if !re be an odd numb] ( cells or squ>es !re mu/ be "o m squ>e ( "o col\r ?an ( ! o!r1 !re=e ! pa? mu/ beg9 f a squ>e ( ! col\r t is 9 excess1 & end on a simil> col\r1 & z a knight's move f "o col\r to a simil> col\r is impossible ! pa? _c be re-entrant4 ,b a p]fect t\r may be made on a rectangul> bo>d ( any dim5sions provid$ ! numb] ( squ>es be ev51 & t ! numb] ( squ>es on "o side be n less ?an #f & on ! o!r n less ?an #e4 ,9 o!r words1 ! smalle/ rectangul> bo>d on :i* a re-entrant t\r is possible is "o t is #f by #e4 ,a complete knight's pa? 7n re-entrant7 ov] all ! squ>es ( a bo>d is nev] possible if !re be only two squ>es on "o side2 nor is x possible on a squ>e bo>d ( small] dim5sions ?an #e by #e4 ,s t on a bo>d #d by #d we c nei!r describe a knight's t\r nor a complete knight's pa?2 we mu/ leave "o squ>e unvisit$4 ,yet on a bo>d #d by #c 7conta9+ f\r squ>es few]7 a complete pa? may be describ$ 9 sixte5 diff]5t ways4 ,x may int]e/ ! read] to discov] all ^!4 ,e pa? t />ts f & ends at diff]5t squ>es is "h c\nt$ z a diff]5t solution1 & ev5 rev]se r\tes are call$ diff]5t4 #cci4--,,! ,,f\r ,,knights' ,,t\rs4 ,illu/ration ,i w repeat t if a *essbo>d be cut into f\r equal p>ts1 z indicat$ by ! d>k l9es 9 ! illu/ration1 x is n possible to p]=m a knight's t\r1 ei!r re-entrant or n1 on "o ( ! p>ts4 ,! be/ re-entrant attempt is %[n1 9 :i* ea* knight has to trespass twice on o!r p>ts4 ,! puzzle is to cut ! bo>d diff]5tly into f\r p>ts1 ea* ( ! same size & %ape1 s t a re-entrant knight's t\r may be made on ea* "p4 ,cuts along ! dott$ l9es w n d1 z ! f\r c5tral squ>es ( ! bo>d w\ld be ei!r deta*$ or hang+ on by a m]e ?read4 #cdj4--,,! ,,cubic ,,knight',s ,,t\r4 ,"s few ye>s ago ,i happ5$ to read some:]e t ,abnit ,v&]monde1 a clev] ma!matician1 :o 0 born 9 #agcf & di$ 9 #agic1 _h devot$ a good deal ( /udy to ! "q ( knight's t\rs4 ,beyond :at may be ga!r$ f a few fragm5t>y ref]5ces1 ,i am n aw>e ( ! exact nature or results ( 8 inve/igations1 b "o ?+ attract$ my att5tion1 & t 0 ! /atem5t t he _h propos$ ! "q ( a t\r ( ! knight ov] ! six surfaces ( a cube1 ea* surface be+ a *essbo>d4 ,:e!r he obta9$ a solution or n ,i d n "k1 b ,i h nev] se5 "o publi%$4 ,s ,i at once set to "w to ma/] ?is int]e/+ problem4 ,p]haps ! read] may l to attempt x4 #cda4--,,! ,,f\r ,,frogs4 ,illu/ration ,9 ! illu/ration we h eight toad/ools1 ) :ite frogs on #a & #c & black frogs on #f & #h4 ,! puzzle is to move "o frog at a "t1 9 any ord]1 along "o ( ! /raight l9es f toad/ool to toad/ool1 until !y h ex*ang$ places1 ! :ite frogs be+ left on #f & #h & ! black ones on #a & #c4 ,if y use f\r c\nt]s on a simple diagram1 y w f9d ?is q easy1 b x is a little m puzzl+ to d x 9 only sev5 plays1 any numb] ( successive moves by "o frog c\nt+ z "o play4 ,( c\rse1 m ?an "o frog _c be on a toad/ool at ! same "t4 #cdb4--,,! ,,m&>9',s ,,puzzle4 ,! foll[+ puzzle has an add$ int]e/ f ! circum/ance t a correct solution ( x secur$ = a c]ta9 "y ,*9aman ! h& ( 8 *>m+ bride4 ,! weal?ie/ m&>9 )9 a radius ( a hundr$ miles ( ,pek+ 0 ,hi-,*um-,*op1 & 8 beautiful daught]1 ,peeky-,bo1 _h innum]able admir]s4 ,"o ( h] mo/ ard5t lov]s 0 ,w9ky-,hi1 & :5 he ask$ ! old m&>9 = 8 cons5t to _! m>riage1 ,hi-,*um-,*op pres5t$ him ) ! foll[+ puzzle & promis$ 8 cons5t if ! y\? br\ght him ! correct answ] )9 a week4 ,w9ky-,hi1 foll[+ a habit :i* obta9s among c]ta9 solv]s to ?is day1 gave x to all 8 fri5ds1 & :5 he _h comp>$ _! solutions he h&$ 9 ! be/ "o z 8 own4 ,luckily x 0 q "r4 ,! m&>9 !reupon fulfill$ 8 promise4 ,! fatt$ pup 0 kill$ = ! w$d+ fea/1 & :5 ,hi-,*um-,*op pass$ ,w9ky-,hi ! liv] w+ all pres5t knew t x 0 a tok5 ( et]nal goodwill1 9 accordance ) ,*9ese cu/om f "t immemorial4 ,! m&>9 _h a table divid$ into tw5ty-five squ>es1 z %[n 9 ! diagram4 ,on ea* ( tw5ty-f\r ( ^! squ>es 0 plac$ a numb]$ c\nt]1 j z ,i h indicat$4 ,! puzzle is to get ! c\nt]s 9 num]ical ord] by mov+ !m "o at a "t 9 :at we call knight's moves4 ,c\nt] #a %\ld be :]e #af is1 #b :]e #aa is1 #d :]e #ac n[ is1 & s on4 ,x w be se5 t all ! c\nt]s on %ad$ squ>es are 9 _! prop] positions4 ,( c\rse1 two c\nt]s may nev] be on a squ>e at ! same "t4 ,c y p]=m ! feat 9 ! fewe/ possible moves8 ,illu/ration ,9 ord] to make ! mann] ( mov+ p]fectly cle> ,i w po9t out t ! fir/ knight's move c only be made by #a or by #b or by #aj4 ,suppos+ #a moves1 !n ! next move mu/ be by #bc1 #d1 #h1 or #ba4 ,z !re is nev] m ?an "o squ>e vacant1 ! ord] 9 :i* ! c\nt]s move may be writt5 out z foll[s3 #a--ba--ad--ah--bb1 etc4 ,a r\gh diagram %\ld be made on a l>g] scale = practice1 & numb]$ c\nt]s or pieces ( c>dbo>d us$4 #cdc4--;,,ex]cise ,,= ,,prison]s4 ,! foll[+ is ! plan ( ! nor? w+ ( a c]ta9 gaol1 %[+ ! sixte5 cells all communicat+ by op5 doorways4 ,fifte5 prison]s 7 numb]$ & arrang$ 9 ! cells z %[n4 ,!y 7 all[$ to *ange _! cells z mu* z !y lik$1 b if two prison]s 7 ev] 9 ! same cell toge!r !re 0 a sev]e puni%m5t promis$ !m4 ,illu/ration ,n[1 9 ord] to r$uce _! gr[+ obesity1 & to comb9e physical ex]cise ) m5tal recreation1 ! prison]s decid$1 on ! sugge/ion ( "o ( _! numb] :o 0 int]e/$ 9 knight's t\rs1 to try to =m !mselves into a p]fect knight's pa? )\t break+ ! prison regulations1 & leav+ ! bottom "r-h& corn] cell vacant1 z orig9ally4 ,! joke ( ! matt] is t ! arrangem5t at :i* !y arriv$ 0 z foll[s3-- #h #c #ab #a #aa #ad #i #f #d #g #b #ac #ae #aj #e ,! w>d]s fail$ to detect ! important fact t ! m5 c\ld n possibly get into ?is position )\t two ( !m hav+ be5 at "s "t 9 ! same cell toge!r4 ,make ! attempt ) c\nt]s on a rul$ diagram1 & y w f9d t ?is is s4 ,o!rwise ! solution is correct en\gh1 ea* memb] be+1 z requir$1 a knight's move f ! prec$+ numb]1 & ! orig9al corn] cell vacant4 ,! puzzle is to />t ) ! m5 plac$ z 9 ! illu/ration & %[ h[ x might h be5 done 9 ! fewe/ moves1 :ile giv+ a complete re/ to z _m prison]s z possible4 ,z !re is nev] m ?an "o vacant cell = a man to ent]1 x is only necess>y to write d[n ! numb]s ( ! m5 9 ! ord] 9 :i* !y move4 ,x is cle> t v few m5 c be left ?r\gh\t 9 _! cells undi/urb$1 b ,i w leave ! solv] to discov] j h[ _m1 z ?is is a v ess5tial "p ( ! puzzle4 #cdd4--,,! ,,k5nel ,,puzzle4 ,illu/ration ,a man has tw5ty-five dog k5nels all communicat+ ) ea* o!r by doorways1 z %[n 9 ! illu/ration4 ,he wi%es to arrange 8 tw5ty dogs s t !y %all =m a knight's /r+ f dog ,no4 #a to dog ,no4 #bj1 ! bottom r[ ( five k5nels to be left empty1 z at pres5t4 ,?is is to be done by mov+ "o dog at a "t into a vacant k5nel4 ,! dogs are well tra9$ to ob$i5ce1 & may be tru/$ to rema9 9 ! k5nels 9 :i* !y are plac$1 except t if two are plac$ 9 ! same k5nel toge!r !y w fight x out to ! dea?4 ,h[ is ! puzzle to be solv$ 9 ! fewe/ possible moves )\t two dogs ev] be+ toge!r8 #cde4--,,! ,,two ,,pawns4 ,illu/ration ,"h is a neat little puzzle 9 c\nt+4 ,9 h[ _m diff]5t ways may ! two pawns advance to ! eigh? squ>e8 ,y may move !m 9 any ord] y l to =m a diff]5t sequ5ce4 ,= example1 y may move ! ,q ,r ,p 7"o or two squ>es7 fir/1 or ! ,k ,r ,p fir/1 or "o pawn z f> z y l be=e t\*+ ! o!r4 ,any sequ5ce is p]missible1 only 9 ?is puzzle z soon z a pawn rea*es ! eigh? squ>e x is dead1 & rema9s !re unconv]t$4 ,c y c\nt ! numb] ( diff]5t sequ5ces8 ,at fir/ x w /rike y z be+ v difficult1 b ,i w %[ t x is really q simple :5 prop]ly attack$4 ,,v>i\s ,,*ess ,,puzzles4 ,*esse-play is a good & wittie ex]cise ( ! m9de = "s k9de ( m54 ,burton's ,anatomy ( ,melan*oly4 #cdf4--,,sett+ ,,! ,,bo>d4 ,i h a s+le *essbo>d & a s+le set ( *essm54 ,9 h[ _m diff]5t ways may ! m5 be correctly set up = ! beg9n+ ( a game8 ,i f9d t mo/ p slip at a p>ticul> po9t 9 mak+ ! calculation4 #cdg4--;,,c\nt+ ,,! ,,rectangles4 ,c y say correctly j h[ _m squ>es & o!r rectangles ! *essbo>d conta9s8 ,9 o!r words1 9 h[ great a numb] ( diff]5t ways is x possible to indicate a squ>e or o!r rectangle enclos$ by l9es t sep>ate ! squ>es ( ! bo>d8 #cdh4--,,! ,,rook]y4 ,illu/ration ,! ,:ite rooks _c move outside ! little squ>e 9 :i* !y are enclos$ except on ! f9al move1 9 giv+ *eckmate4 ,! puzzle is h[ to *eckmate ,black 9 ! fewe/ possible moves ) ,no4 #h rook1 ! o!r rooks be+ left 9 num]ical ord] r\nd ! sides ( _! squ>e ) ! break betwe5 #a & #g4 #cdi4--,,/alemate4 ,"s ye>s ago ! puzzle 0 propos$ to con/ruct an imag9>y game ( *ess1 9 :i* ,:ite %all be /alemat$ 9 ! fewe/ possible moves ) all ! ?irty-two pieces on ! bo>d4 ,c y build up su* a position 9 few] ?an tw5ty moves8 #cej4--,,! ,,=sak5 ,,k+4 ,illu/ration ,set up ! position %[n 9 ! diagram4 ,!n ! condition ( ! puzzle is--,:ite to play & *eckmate 9 six moves4 ,not)/&+ ! complexities1 ,i w %[ h[ ! mann] ( play may be cond5s$ into q a few l9es1 m]ely /at+ "h t ! fir/ two moves ( ,:ite _c be v>i$4 #cea4--,,! ,,crusad]4 ,! foll[+ is a prize puzzle prop\nd$ by me "s ye>s ago4 ,produce a game ( *ess :i*1 aft] sixte5 moves1 %all leave ,:ite ) all 8 sixte5 m5 on _! orig9al squ>es & ,black 9 possession ( 8 k+ alone 7n necess>ily on 8 own squ>e74 ,:ite is !n to =ce mate 9 ?ree moves4 #ceb4--;,,immovable ,,pawns4 ,/>t+ f ! ord9>y arrangem5t ( ! pieces z = a game1 :at is ! smalle/ possible numb] ( moves necess>y 9 ord] to arrive at ! foll[+ position8 ,! moves = bo? sides mu/1 ( c\rse1 be play$ /rictly 9 accordance ) ! rules ( ! game1 ?\gh ! result w necess>ily be a v weird k9d ( *ess4 ,illu/ration #cec4--,,?irty-,,six ,,mates4 ,illu/ration ,place ! rema9+ eight ,:ite pieces 9 su* a position t ,:ite %all h ! *oice ( ?irty-six diff]5t mates on ! move4 ,e move t *eckmates & leaves a diff]5t position is a diff]5t mate4 ,! pieces already plac$ mu/ n be mov$4 #ced4--;,,an ,,amaz+ ,,dilemma4 ,9 a game ( *ess betwe5 ,mr4 ,black & ,mr4 ,:ite1 ,black 0 9 difficulties1 & z usual 0 oblig$ to cat* a tra94 ,s he propos$ t ,:ite %\ld complete ! game 9 8 abs5ce on condition t no moves :atev] %\ld be made = ,black1 b only ) ! ,:ite pieces4 ,mr4 ,:ite accept$1 b to 8 dismay f\nd x utt]ly impossible to w9 ! game "u su* conditions4 ,try z he w\ld1 he c\ld n *eckmate 8 oppon5t4 ,on :i* squ>e did ,mr4 ,black leave 8 k+8 ,! o!r pieces are 9 _! prop] positions 9 ! diagram4 ,:ite may leave ,black 9 *eck z (t5 z he likes1 = x makes no diff]5ce1 z he c nev] arrive at a *eckmate position4 ,illu/ration #cee4--;,,*eckmate6 ,illu/ration ,/roll+ into "o ( ! rooms ( a ,london club1 ,i notic$ a position left by two play]s :o _h gone4 ,?is position is %[n 9 ! diagram4 ,x is evid5t t ,:ite has *eckmat$ ,black4 ,b h[ did he d x8 ,t is ! puzzle4 #cef4--,,que] ,,*ess4 ,c y place two ,:ite rooks & a ,:ite knight on ! bo>d s t ! ,black k+ 7:o mu/ be on "o ( ! f\r squ>es 9 ! middle ( ! bo>d7 %all be 9 *eck ) no possible move op5 to him8 ,9 o!r words1 ! read] w say1 ! k+ is to be %[n *eckmat$4 ,well1 y c use ! t]m if y wi%1 ?\gh ,i int5tionally d n employ x myself4 ,! m]e fact t !re is no ,:ite k+ on ! bo>d w\ld be a suffici5t reason = my n do+ s4 #ceg4--;,,anci5t ,,*9ese ,,puzzle4 ,illu/ration ,my next puzzle is suppos$ to be ,*9ese1 _m hundr$s ( ye>s old1 & nev] fails to int]e/4 ,:ite to play & mate1 mov+ ea* ( ! ?ree pieces once1 & once only4 #ceh4--,,! ,,six ,,pawns4 ,9 h[ _m diff]5t ways may ,i place six pawns on ! *essbo>d s t !re %all be an ev5 numb] ( unoccupi$ squ>es 9 e r[ & e column8 ,we are n "h consid]+ ! diagonals at all1 & e diff]5t six squ>es occupi$ makes a diff]5t solution1 s we h n to exclude rev]sals or reflections4 #cei4--;,,c\nt] ,,solitaire4 ,"h is a little game ( solitaire t is q easy1 b n s easy z to be un9t]e/+4 ,y c ei!r rule out ! squ>es on a %eet ( c>dbo>d or pap]1 or y c use a portion ( y\r *essbo>d4 ,i h %[n numb]$ c\nt]s 9 ! illu/ration s z to make ! solution easy & intelligible to all1 b *ess pawns or draughts w s]ve j z well 9 practice4 ,illu/ration ,! puzzle is to remove all ! c\nt]s except "o1 & ?is "o t is left mu/ be ,no4 #a4 ,y remove a c\nt] by jump+ ov] ano!r c\nt] to ! next space beyond1 if t squ>e is vacant1 b y _c make a leap 9 a diagonal direction4 ,! foll[+ moves w make ! play q cle>3 #a-i1 #b-aj1 #a-b1 & s on4 ,"h #a jumps ov] #i1 & y remove #i f ! bo>d2 !n #b jumps ov] #aj1 & y remove #aj2 !n #a jumps ov] #b1 & y remove #b4 ,e move is ?us a capture1 until ! la/ capture ( all is made by ,no4 #a4 #cfj4--;,,*essbo>d ,,solitaire4 ,illu/ration ,"h is an ext5sion ( ! la/ game ( solitaire4 ,all y ne$ is a *essbo>d & ! ?irty-two pieces1 or ! same numb] ( draughts or c\nt]s4 ,9 ! illu/ration numb]$ c\nt]s are us$4 ,! puzzle is to remove all ! c\nt]s except two1 & ^! two mu/ h orig9ally be5 on ! same side ( ! bo>d2 t is1 ! two left mu/ ei!r belong to ! gr\p #a to #af or to ! o!r gr\p1 #ag to #cb4 ,y remove a c\nt] by jump+ ov] x ) ano!r c\nt] to ! next squ>e beyond1 if t squ>e is vacant1 b y _c make a leap 9 a diagonal direction4 ,! foll[+ moves w make ! play q cle>3 #c-aa1 #d-ab1 #c-d1 #ac-c4 ,"h #c jumps ov] #aa1 & y remove #aa2 #d jumps ov] #ab1 & y remove #ab2 & s on4 ,x w be f\nd a fasc9at+ little game ( pati5ce1 & ! solution requires ! ex]cise ( "s +5uity4 #cfa4--,,! ,,mon/rosity4 ,"o ,*ri/mas ,eve ,i 0 travell+ by rail to a little place 9 "o ( ! s\!rn c\nties4 ,! comp>tm5t 0 v full1 & ! pass5g]s 7 w$g$ 9 v tightly4 ,my neighb\r 9 "o ( ! corn] seats 0 closely /udy+ a position set up on "o ( ^? little fold+ *essbo>ds t c be c>ri$ conv5i5tly 9 ! pocket1 & ,i c\ld sc>cely avoid look+ at x myself4 ,"h is ! position3-- ,illu/ration ,my fell[-pass5g] sudd5ly turn$ 8 head & caught ! look ( bewild]m5t on my face4 ,d y play *ess8 he ask$4 ,yes1 a little4 ,:at is t8 ,a problem8 ,problem8 ,no2 a game4 ,impossible6 ,i exclaim$ r rudely4 ,! position is a p]fect mon/rosity6 ,he took f 8 pocket a po/c>d & h&$ x to me4 ,x bore an address at "o side & on ! o!r ! words #dc4 ,k to ,kt #h4 ,x is a correspond5ce game4 he exclaim$4 ,t is my fri5d's la/ move1 & ,i am consid]+ my reply4 ,b y really mu/ excuse me2 ! position seems utt]ly impossible4 ,h[ on e>?1 = example-- ,ah6 he broke 9 smil+ly4 ,i see2 y are a beg9n]2 y play to w94 ,( c\rse y w\ldn't play to lose or draw6 ,he laugh$ al\d4 ,y h mu* to le>n4 ,my fri5d & myself d n play = results ( t antiquat$ k9d4 ,we seek 9 *ess ! wond]ful1 ! :imsical1 ! weird4 ,did y ev] see a position l t8 ,i inw>dly congratulat$ myself t ,i nev] _h4 ,t position1 sir1 mat]ializes ! s9u\s evolvem5ts & syncretic1 syn!tic1 & syn*ron\s concat5ations ( two c]ebral individualities4 ,x is ! product ( an amphot]ic & int]calatory int]*ange (-- ,h y se5 ! ev5+ pap]1 sir8 int]rupt$ ! man opposite1 hold+ out a newspap]4 ,i notic$ on ! m>g9 beside 8 ?umb "s p5cill$ writ+4 ,?ank+ him1 ,i took ! pap] & read--,insane1 b q h>mless4 ,he is 9 my *>ge4 ,aft] t ,i let ! poor fell[ run on 9 8 wild way until bo? got out at ! next /ation4 ,b t que] position became fix$ indelibly 9 my m9d1 ) ,black's la/ move #dc4 ,k to ,kt #h2 & a %ort "t aft]w>ds ,i f\nd x actually possible to arrive at su* a position 9 =ty-?ree moves4 ,c ! read] con/ruct su* a sequ5ce8 ,h[ did ,:ite get 8 rooks & k+'s bi%op into _! pres5t positions1 consid]+ ,black c nev] h mov$ 8 k+'s bi%op8 ,no odds 7 giv51 & e move 0 p]fectly legitimate4 ,,measur+1 ,,weigh+1 ,,& ,,pack+ ,,puzzles4 ,measure /ill = measure4 ,measure = ,measure1 v4 #a4 ,app>5tly ! fir/ pr9t$ puzzle involv+ ! measur+ ( a giv5 quantity ( liquid by p\r+ f "o vessel to o!rs ( kn[n capacity 0 t prop\nd$ by ,niccola ,fontana1 bett] kn[n z ,t>taglia 7! /amm]]71 #aejj-aeei4 ,x consi/s 9 divid+ #bd oz4 ( valuable balsam into ?ree equal p>ts1 ! only measures available be+ vessels hold+ #e1 #aa1 & #ac ounces respectively4 ,!re are _m diff]5t solutions to ?is puzzle 9 six manipulations1 or p\r+s f "o vessel to ano!r4 ,ba*et de ,mziriac repr9t$ ?is & o!r ( ,t>taglia's puzzles 9 8 ,problmes plaisans et dlectables 7#afab74 ,x is ! g5]al op9ion t puzzles ( ?is class c only be solv$ by trial1 b ,i ?9k =mul c be con/ruct$ = ! solution g5]ally ( c]ta9 relat$ cases4 ,x is a practically unexplor$ field = inve/igation4 ,! classic weigh+ problem is1 ( c\rse1 t propos$ by ,ba*et4 ,x entails ! det]m9ation ( ! lea/ numb] ( weights t w\ld s]ve to weigh any integral numb] ( p\nds f #a lb4 to #dj lbs4 inclusive1 :5 we are all[$ to put a weight 9 ei!r ( ! two pans4 ,! answ] is #a1 #c1 #i1 & #bg lbs4 ,t>taglia _h previ\sly prop\nd$ ! same puzzle ) ! condition t ! weights may only be plac$ 9 "o pan4 ,! answ] 9 t case is #a1 #b1 #d1 #h1 #af1 #cb lbs4 ,major ,mac,mahon has solv$ ! problem q g5]ally4 ,a full acc\nt w be f\nd 9 ,ball's ,ma!matical ,recreations 7#e? edition74 ,pack+ puzzles1 9 :i* we are requir$ to pack a maximum numb] ( articles ( giv5 dim5sions into a box ( kn[n dim5sions1 are1 ,i believe1 ( q rec5t introduction4 ,at lea/ ,i _c recall any example 9 ! books ( ! old writ]s4 ,"o w\ld r expect to f9d 9 ! toy %ops ! idea pres5t$ z a me*anical puzzle1 b ,i d n ?9k ,i h ev] se5 su* a ?+4 ,! ne>e/ approa* to x w\ld appe> to be ! puzzles ( ! jig-saw "*1 :]e !re is only "o dep? ( ! pieces to be adju/$4 #cfb4--,,! ,,wassail ,,b[l4 ,"o ,*ri/mas ,eve ?ree ,we>y ,willies came into possession ( :at 0 to !m a v]itable wassail b[l1 9 ! =m ( a small b>rel1 conta9+ exactly six qu>ts ( f9e ale4 ,"o ( ! m5 possess$ a five-p9t jug & ano!r a ?ree-p9t jug1 & ! problem = !m 0 to divide ! liquor equally among/ !m )\t wa/e4 ,( c\rse1 !y are n to use any o!r vessels or measures4 ,if y c %[ h[ x 0 to be done at all1 !n try to f9d ! way t requires ! fewe/ possible manipulations1 e sep>ate p\r+ f "o vessel to ano!r1 or d[n a man's ?roat1 c\nt+ z a manipulation4 #cfc4--,,! ,,doctor',s ,,qu]y4 ,a curi\s little po9t occurr$ to me 9 my disp5s>y ?is morn+1 said a doctor4 ,i _h a bottle conta9+ t5 ounces ( spirits ( w9e1 & ano!r bottle conta9+ t5 ounces ( wat]4 ,i p\r$ a qu>t] ( an ounce ( spirits into ! wat] & %ook !m up toge!r4 ,! mixture 0 !n cle>ly =ty to "o4 ,!n ,i p\r$ back a qu>t]-ounce ( ! mixture1 s t ! two bottles %\ld aga9 ea* conta9 ! same quantity ( fluid4 ,:at proportion ( spirits to wat] did ! spirits ( w9e bottle !n conta98 #cfd4--,,! ,,b>rel ,,puzzle4 ,! m5 9 ! illu/ration are disput+ ov] ! liquid cont5ts ( a b>rel4 ,:at ! p>ticul> liquid is x is impossible to say1 = we are unable to look into ! b>rel2 s we w call x wat]4 ,"o man says t ! b>rel is m ?an half full1 :ile ! o!r insi/s t x is n half full4 ,:at is _! easie/ way ( settl+ ! po9t8 ,x is n necess>y to use /ick1 /r+1 or implem5t ( any k9d = measur+4 ,i give ?is m]ely z "o ( ! simple/ possible examples ( ! value ( ord9>y sagacity 9 ! solv+ ( puzzles4 ,:at are app>5tly v difficult problems may frequ5tly be solv$ 9 a simil>ly easy mann] if we only use a little common s5se4 ,illu/ration #cfe4--,,new ,,measur+ ,,puzzle4 ,"h is a new pos] 9 measur+ liquids t w be f\nd int]e/+4 ,a man has two t5-qu>t vessels full ( w9e1 & a five-qu>t & a f\r-qu>t measure4 ,he wants to put exactly ?ree qu>ts into ea* ( ! two measures4 ,h[ is he to d x8 ,& h[ _m manipulations 7p\r+s f "o vessel to ano!r7 d y require8 ,( c\rse1 wa/e ( w9e1 tilt+1 & o!r tricks are n all[$4 #cff4--,,! ,,hone/ ,,dairyman4 ,an hone/ dairyman 9 prep>+ 8 milk = public consumption employ$ a c m>k$ ,b1 conta9+ milk1 & a c m>k$ ,a1 conta9+ wat]4 ,f c ,a he p\r$ en\gh to d\ble ! cont5ts ( c ,b4 ,!n he p\r$ f c ,b into c ,a en\gh to d\ble its cont5ts4 ,!n he f9ally p\r$ f c ,a into c ,b until _! cont5ts 7 exactly equal4 ,aft] ^! op]ations he w\ld s5d ! c ,a to ,london1 & ! puzzle is to discov] :at are ! relative proportions ( milk & wat] t he provides = ! ,london]s' breakfa/-tables4 ,d !y get equal proportions ( milk & wat]--or two p>ts ( milk & "o ( wat]--or :at8 ,x is an int]e/+ "q1 ?\gh1 curi\sly en\gh1 we are n told h[ mu* milk or wat] he puts into ! cans at ! />t ( 8 op]ations4 #cfg4--,,w9e ,,& ,,wat]4 ,mr4 ,goodfell[ has adopt$ a capital idea ( late4 ,:5 he gives a little d9n] p>ty & ! "t arrives to smoke1 aft] ! dep>ture ( ! ladies1 he sometimes f9ds t ! conv]sation is apt to become too political1 too p]sonal1 too sl[1 or too sc&al\s4 ,!n he always manages to introduce to ! company "s new pos] t he has secret$ up 8 sleeve = ! occasion4 ,?is inv>iably results 9 no end ( int]e/+ discussion & debate1 & puts ev]ybody 9 a good hum\r4 ,"h is a little puzzle t he prop\nd$ ! o!r night1 & x is extraord9>y h[ ! company diff]$ 9 _! answ]s4 ,he fill$ a w9e-glass half full ( w9e1 & ano!r glass twice ! size "o-?ird full ( w9e4 ,!n he fill$ up ea* glass ) wat] & empti$ ! cont5ts ( bo? into a tumbl]4 ,n[1 he said1 :at "p ( ! mixture is w9e & :at "p wat]8 ,c y give ! correct answ]8 #cfh4--,,! ,,keg ,,( ,,w9e4 ,"h is a curi\s little problem4 ,a man _h a t5-gallon keg full ( w9e & a jug4 ,"o day he drew (f a jugful ( w9e & fill$ up ! keg ) wat]4 ,lat] on1 :5 ! w9e & wat] _h got ?or\ghly mix$1 he drew (f ano!r jugful & aga9 fill$ up ! keg ) wat]4 ,x 0 !n f\nd t ! keg conta9$ equal proportions ( w9e & wat]4 ,c y f9d f ^! facts ! capacity ( ! jug8 #cfi4--,,mix+ ,,! ,,tea4 ,mrs4 ,spoon] call$ ?is morn+1 said ! hone/ groc] to 8 assi/ant4 ,%e wants tw5ty p\nds ( tea at #bs4 #d;d4 p] lb4 ,( c\rse we h a good #bs4 #f;d4 tea1 a slightly inf]ior at #bs4 #c;d41 & a *eap ,indian at #as4 #i;d41 b %e is v p>ticul> always ab\t h] prices4 ,:at d y propose to d8 ask$ ! innoc5t assi/ant4 ,d8 exclaim$ ! groc]4 ,:y1 j mix up ! ?ree teas 9 diff]5t proportions s t ! tw5ty p\nds w "w out fairly at ! lady's price4 ,only don't put 9 m ( ! be/ tea ?an y c help1 z we make less pr(it on t1 & ( c\rse y w use only our complete p\nd packets4 ,don't d any weigh+4 ,h[ 0 ! poor fell[ to mix ! ?ree teas8 ,c\ld y h %[n him h[ to d x8 #cgj4--;,a ,,pack+ ,,puzzle4 ,z we all "k by exp]i5ce1 consid]able +5uity is (t5 requir$ 9 pack+ articles into a box if space is n to be unduly wa/$4 ,a man once told me t he _h a l>ge numb] ( iron balls1 all exactly two in*es 9 diamet]1 & he wi%$ to pack z _m ( ^! z possible into a rectangul> box #bdi_/aj in*es long1 #bbd_/e in*es wide1 & #ad in*es deep4 ,n[1 :at is ! greate/ numb] ( ! balls t he c\ld pack into t box8 #cga4--;,,gold ,,pack+ ,,9 ,,russia4 ,! editor ( ! ,times newspap] 0 invit$ by a high ,russian (ficial to inspect ! gold /or$ 9 res]ve at ,/4 ,pet]sburg1 9 ord] t he might satisfy himself t x 0 n ano!r ,humb]t safe4 ,he repli$ t x w\ld be ( no use :atev]1 = al?\gh ! gold might appe> to be !re1 he w\ld be q unable f a m]e inspection to decl>e t :at he saw 0 really gold4 ,a correspond5t ( ! ,daily ,mail !reupon took up ! *all5ge1 b1 al?\gh he 0 greatly impress$ by :at he saw1 he 0 compell$ to confess 8 incompet5ce 7)\t empty+ & c\nt+ ! cont5ts ( e box & sack1 & assay+ e piece ( gold7 to give any assurance on ! subject4 ,9 pres5t+ ! foll[+ little puzzle1 ,i wi% x to be also und]/ood t ,i d n gu>antee ! real exi/5ce ( ! gold1 & ! po9t is n at all mat]ial to our purpose4 ,moreov]1 if ! read] says t gold is n usually put up 9 slabs ( ! dim5sions t ,i give1 ,i c only claim problematic lic5ce4 ,russian (ficials 7 engag$ 9 pack+ #hjj gold slabs1 ea* measur+ #ab in*es long1 #aa in*es wide1 & #a in* deep4 ,:at are ! int]ior dim5sions ( a box ( equal l5g? & wid?1 & necess>y dep?1 t w exactly conta9 !m )\t any space be+ left ov]8 ,n m ?an twelve slabs may be laid on edge1 accord+ to ! rules ( ! gov]nm5t4 ,x is an int]e/+ little problem 9 pack+1 & n at all difficult4 #cgb4--,,! ,,b>rels ,,( ,,honey4 ,illu/ration ,once ^u a "t !re 0 an ag$ m]*ant ( ,bagdad :o 0 mu* respect$ by all :o knew him4 ,he _h ?ree sons1 & x 0 a rule ( 8 life to treat !m all exactly alike4 ,:5ev] "o receiv$ a pres5t1 ! o!r two 7 ea* giv5 "o ( equal value4 ,"o day ?is wor?y man fell sick & di$1 bequea?+ all 8 possessions to 8 ?ree sons 9 equal %>es4 ,! only difficulty t arose 0 ov] ! /ock ( honey4 ,!re 7 exactly tw5ty-"o b>rels4 ,! old man _h left in/ructions t n only %\ld e son receive an equal quantity ( honey1 b %\ld receive exactly ! same numb] ( b>rels1 & t no honey %\ld be transf]r$ f b>rel to b>rel on acc\nt ( ! wa/e involv$4 ,n[1 z sev5 ( ^! b>rels 7 full ( honey1 sev5 7 half-full1 & sev5 7 empty1 ?is 0 f\nd to be q a puzzle1 especially z ea* bro!r object$ to tak+ m ?an f\r b>rels (1 ! same description--full1 half-full1 or empty4 ,c y %[ h[ !y succe$$ 9 mak+ a correct division ( ! prop]ty8 ,,cross+ ,,riv] ,,problems ,my boat is on ! %ore4 ,,byron4 ,?is is ano!r m$ival class ( puzzles4 ,probably ! e>lie/ example 0 by ,abbot ,alcu91 :o 0 born 9 ,york%ire 9 #gce & di$ at ,t\rs 9 #hjd4 ,& ev]ybody kn[s ! /ory ( ! man ) ! wolf1 goat1 & basket ( cabbages ^: boat w\ld only take "o ( ! ?ree at a "t ) ! man himself4 ,8 difficulties arose f 8 be+ unable to leave ! wolf alone ) ! goat1 or ! goat alone ) ! cabbages4 ,^! puzzles 7 consid]$ by ,t>taglia & ,ba*et1 & h be5 lat] inve/igat$ by ,lucas1 ,de ,font5ey1 ,delannoy1 ,t>ry1 & o!rs4 ,9 ! puzzles ,i give !re w be f\nd "o or two new conditions :i* add to ! complexity some:at4 ,i also include a pulley problem t practically involves ! same pr9ciples4 ,illu/ration #cgc4--;,,cross+ ,,! ,,/ream4 ,dur+ a c\ntry ramble ,mr4 & ,mrs4 ,s(tleigh f\nd !mselves 9 a pretty little dilemma4 ,!y _h to cross a /ream 9 a small boat :i* 0 capable ( c>ry+ only #aej lbs4 weight4 ,b ,mr4 ,s(tleigh & 8 wife ea* weigh$ exactly #aej lbs41 & ea* ( _! sons weigh$ #ge lbs4 ,& !n !re 0 ! dog1 :o c\ld n be induc$ on any t]ms to swim4 ,on ! pr9ciple ( ladies fir/1 !y at once s5t ,mrs4 ,s(tleigh ov]2 b ?is 0 a /upid ov]sight1 because %e _h to come back aga9 ) ! boat1 s no?+ 0 ga9$ by t op]ation4 ,h[ did !y all succe$ 9 gett+ across8 ,! read] w f9d x mu* easi] ?an ! ,s(tleigh family did1 = _! greate/ enemy c\ld n h tru?fully call$ !m a brilliant qu>tette--:ile ! dog 0 a p]fect fool4 #cgd--;,,cross+ ,,! ,,riv] ,,axe4 ,_m ye>s ago1 9 ! days ( ! smuggl] kn[n z ,rob ,roy ( ! ,we/1 a piratical b& buri$ on ! coa/ ( ,s\? ,devon a quantity ( treasure :i* 01 ( c\rse1 ab&on$ by !m 9 ! usual inexplicable way4 ,"s "t aft]w>ds its :]eab\ts 0 discov]$ by ?ree c\ntrym51 :o visit$ ! spot "o night & divid$ ! spoil betwe5 !m1 ,giles tak+ treasure to ! value ( #hjj1 ,jasp] #ejj wor?1 & ,timo?y #cjj wor?4 ,9 return+ !y _h to cross ! riv] ,axe at a po9t :]e !y _h left a small boat 9 read9ess4 ,"h1 h[ev]1 0 a difficulty !y _h n anticipat$4 ,! boat w\ld only c>ry two m51 or "o man & a sack1 & !y _h s little confid5ce 9 "o ano!r t no p]son c\ld be left alone on ! l& or 9 ! boat ) m ?an 8 %>e ( ! spoil1 ?\gh two p]sons 7be+ a *eck on ea* o!r7 might be left ) m ?an _! %>es4 ,! puzzle is to %[ h[ !y got ov] ! riv] 9 ! fewe/ possible cross+s1 tak+ _! treasure ) !m4 ,no tricks1 su* z ropes1 fly+ bridges1 curr5ts1 swimm+1 or simil> dodges1 may be employ$4 #cge4--;,,five ,,jeal\s ,,husb&s4 ,dur+ c]ta9 local floods five m>ri$ c\ples f\nd !mselves surr\nd$ by wat]1 & _h to escape f _! unpleasant position 9 a boat t w\ld only hold ?ree p]sons at a "t4 ,e husb& 0 s jeal\s t he w\ld n all[ 8 wife to be 9 ! boat or on ei!r bank ) ano!r man 7or ) o!r m57 unless he 0 himself pres5t4 ,%[ ! quicke/ way ( gett+ ^! five m5 & _! wives across into safety4 ,call ! m5 ,a1 ,b1 ,c1 ,d1 ,e1 & _! respective wives a1 b1 c1 d1 e4 ,to g ov] & return c\nts z two cross+s4 ,no tricks su* z ropes1 swimm+1 curr5ts1 etc41 are p]mitt$4 #cgf4--,,! ,,f\r ,,elopem5ts4 ,colonel ,b---- 0 a wid[] ( a v taciturn disposition4 ,8 treatm5t ( 8 f\r daught]s 0 unusually sev]e1 almo/ cruel1 & !y n unnaturally felt dispos$ to res5t x4 ,be+ *>m+ girls ) e virtue & _m accompli%m5ts1 x is n surpris+ t ea* _h a fond admir]4 ,b ! fa!r =bade ! "y m5 to call at 8 h\se1 int]cept$ all lett]s1 & plac$ 8 daught]s "u /rict] sup]vision ?an ev]4 ,b love1 :i* scorns locks & keys & g>d5 walls1 0 equal to ! occasion1 & ! f\r y\?s conspir$ toge!r & plann$ a g5]al elopem5t4 ,at ! foot ( ! t5nis lawn at ! bottom ( ! g>d5 ran ! silv] ,?ames1 & "o night1 aft] ! f\r girls _h be5 safely conduct$ f a dormitory w9d[ to t]ra firma1 !y all crept s(tly d[n to ! bank ( ! riv]1 :]e a small boat belong+ to ! ,colonel 0 moor$4 ,) ?is !y propos$ to cross to ! opposite side & make _! way to a lane :]e conveyances 7 wait+ to c>ry !m 9 _! flight4 ,alas6 "h at ! wat]'s br9k _! difficulties already began4 ,! "y m5 7 s extremely jeal\s t n "o ( !m w\ld all[ 8 prospective bride to rema9 at any "t 9 ! company ( ano!r man1 or m51 unless he himself 7 pres5t also4 ,n[1 ! boat w\ld only hold two p]sons1 ?\gh x c\ld1 ( c\rse1 be r[$ by "o1 & x seem$ impossible t ! f\r c\ples w\ld ev] get across4 ,b midway 9 ! /ream 0 a small isl&1 & ?is seem$ to pres5t a way out ( ! difficulty1 because a p]son or p]sons c\ld be left !re :ile ! boat 0 r[$ back or to ! opposite %ore4 ,if !y _h be5 prep>$ = _! difficulty !y c\ld h easily work$ out a solution to ! little pos] at any o!r "t4 ,b !y 7 n[ s hurri$ & excit$ 9 _! flight t ! confusion !y soon got into 0 exce$+ly amus+--or w\ld h be5 to any "o except !mselves4 ,z a consequ5ce !y took twice z long & cross$ ! riv] twice z (t5 z 0 really necess>y4 ,mean:ile1 ! ,colonel1 :o 0 a v light sleep]1 ?\ght he he>d a spla% ( o>s4 ,he quickly rais$ ! al>m among 8 h\sehold1 & ! "y ladies 7 f\nd to be miss+4 ,somebody 0 s5t to ! police-/ation1 & a numb] ( (fic]s soon aid$ 9 ! pursuit ( ! fugitives1 :o1 9 consequ5ce ( t delay 9 cross+ ! riv]1 7 quickly ov]tak54 ,! f\r girls return$ sadly to _! homes1 & aft]w>ds broke (f _! engagem5ts 9 disgu/4 ,= a consid]able "t x 0 a my/]y h[ ! p>ty ( eight manag$ to cross ! riv] 9 t little boat )\t any girl be+ ev] left ) a man1 unless h] betro!d 0 also pres5t4 ,! fav\rite me?od is to take eight c\nt]s or pieces ( c>dbo>d & m>k !m ,a1 ,b1 ,c1 ,d1 a1 b1 c1 d1 to repres5t ! f\r m5 & _! prospective brides1 & c>ry !m f "o side ( a table to ! o!r 9 a mat*box 7to repres5t ! boat71 a p5ny be+ plac$ 9 ! middle ( ! table z ! isl&4 ,read]s are n[ ask$ to f9d ! quicke/ me?od ( gett+ ! p>ty across ! riv]4 ,h[ _m passages are necess>y f l& to l&8 ,by l& is und]/ood ei!r %ore or isl&4 ,?\gh ! boat w\ld n necess>ily call at ! isl& e "t ( cross+1 ! possibility ( its do+ s mu/ be provid$ =4 ,= example1 x w\ld n d = a man to be alone 9 ! boat 7?\gh x 7 und]/ood t he int5d$ m]ely to cross f "o bank to ! opposite "o7 if !re happ5$ to be a girl alone on ! isl& o!r ?an ! "o to :om he 0 engag$4 #cgg4--,,/eal+ ,,! ,,ca/le ,,treasure4 ,! +5i\s mann] 9 :i* a box ( treasure1 consi/+ pr9cipally ( jewels & preci\s /ones1 0 /ol5 f ,gloomhur/ ,ca/le has be5 h&$ d[n z a tradition 9 ! ,de ,g\rney family4 ,! ?ieves consi/$ ( a man1 a y\?1 & a small boy1 ^: only mode ( escape ) ! box ( treasure 0 by means ( a high w9d[4 ,outside ! w9d[ 0 fix$ a pulley1 ov] :i* ran a rope ) a basket at ea* end4 ,:5 "o basket 0 on ! gr\nd ! o!r 0 at ! w9d[4 ,! rope 0 s dispos$ t ! p]sons 9 ! basket c\ld nei!r help !mselves by means ( x nor receive help f o!rs4 ,9 %ort1 ! only way ! baskets c\ld be us$ 0 by plac+ a heavi] weight 9 "o ?an 9 ! o!r4 ,n[1 ! man weigh$ #aie lbs41 ! y\? #aje lbs41 ! boy #ij lbs41 & ! box ( treasure #ge lbs4 ,! weight 9 ! desc5d+ basket c\ld n exce$ t 9 ! o!r by m ?an #ae lbs4 )\t caus+ a desc5t s rapid z to be mo/ dang]\s to a human be+1 ?\gh x w\ld n injure ! /ol5 prop]ty4 ,only two p]sons1 or "o p]son & ! treasure1 c\ld be plac$ 9 ! same basket at "o "t4 ,h[ did !y all manage to escape & take ! box ( treasure ) !m8 ,! puzzle is to f9d ! %orte/ way ( p]=m+ ! feat1 :i* 9 itself is n difficult4 ,rememb]1 a p]son _c help himself by hang+ on to ! rope1 ! only way be+ to g d[n ) a bump1 ) ! weight 9 ! o!r basket z a c\nt]poise4 ,,problems ,,conc]n+ ,,games4 ,! little pleasure ( ! game4 ,,mat!w ,,prior4 ,e game l5ds itself to ! prop\nd+ ( a v>iety ( puzzles4 ,!y c be made1 z we h se51 out ( ! *essbo>d & ! peculi> moves ( ! *ess pieces4 ,i w n[ give j a few examples ( puzzles ) play+ c>ds & dom9oes1 & also g out ( doors & consid] "o or two little pos]s 9 ! cricket field1 at ! football mat*1 & ! horse race & motor-c> race4 #cgh4--;,,dom9oes ,,9 ,,progression4 ,illu/ration ,x w be se5 t ,i h play$ six dom9oes1 9 ! illu/ration1 9 accordance ) ! ord9>y rules ( ! game1 #d aga9/ #d1 #a aga9/ #a1 & s on1 & yet ! sum ( ! spots on ! successive dom9oes1 #d1 #e1 #f1 #g1 #h1 #i1 are 9 ari?metical progression2 t is1 ! numb]s tak5 9 ord] h a common diff]5ce ( #a4 ,9 h[ _m diff]5t ways may we play six dom9oes1 f an ord9>y box ( tw5ty-eight1 s t ! numb]s on !m may lie 9 ari?metical progression8 ,we mu/ always play f left to "r1 & numb]s 9 decreas+ ari?metical progression 7su* z #i1 #h1 #g1 #f1 #e1 #d7 are n admissible4 #cgi4--,,! ,,five ,,dom9oes4 ,illu/ration ,"h is a new little puzzle t is n difficult1 b w probably be f\nd ent]ta9+ by my read]s4 ,x w be se5 t ! five dom9oes are s arrang$ 9 prop] sequ5ce 7t is1 ) #a aga9/ #a1 #b aga9/ #b1 & s on71 t ! total numb] ( pips on ! two end dom9oes is five1 & ! sum ( ! pips on ! ?ree dom9oes 9 ! middle is also five4 ,!re are j ?ree o!r arrangem5ts giv+ five = ! additions4 ,!y are3 -- 7#a--j7 7#j--j7 7#j--b7 7#b--a7 7#a--c7 7#d--j7 7#j--j7 7#j--b7 7#b--a7 7#a--j7 7#b--j7 7#j--j7 7#j--a7 7#a--c7 7#c--j7 ,n[1 h[ _m simil> arrangem5ts are !re ( five dom9oes t %all give six in/ead ( five 9 ! two additions8 #chj4--,,! ,,dom9o ,,frame ,,puzzle4 ,illu/ration ,x w be se5 9 ! illu/ration t ! full set ( tw5ty-eight dom9oes is arrang$ 9 ! =m ( a squ>e frame1 ) #f aga9/ #f1 #b aga9/ #b1 blank aga9/ blank1 & s on1 z 9 ! game4 ,x w be f\nd t ! pips 9 ! top r[ & left-h& column bo? add up #dd4 ,! pips 9 ! o!r two sides sum to #ei & #cb respectively4 ,! puzzle is to re>range ! dom9oes 9 ! same =m s t all ( ! f\r sides %all sum to #dd4 ,rememb] t ! dom9oes mu/ be correctly plac$ "o aga9/ ano!r z 9 ! game4 #cha4--,,! ,,c>d ,,frame ,,puzzle4 ,9 ! illu/ration we h a frame con/ruct$ f ! t5 play+ c>ds1 ace to t5 ( diamonds4 ,! *ildr5 :o made x want$ ! pips on all f\r sides to add up alike1 b !y fail$ 9 _! attempt & gave x up z impossible4 ,x w be se5 t ! pips 9 ! top r[1 ! bottom r[1 & ! left-h& side all add up #ad1 b ! "r-h& side sums to #bc4 ,n[1 :at !y 7 try+ to d is q possible4 ,c y re>range ! t5 c>ds 9 ! same =mation s t all f\r sides %all add up alike8 ,( c\rse !y ne$ n add up #ad1 b any numb] y *oose to select4 ,illu/ration #chb4--,,! ,,cross ,,( ,,c>ds4 ,illu/ration ,9 ?is case we use only n9e c>ds--! ace to n9e ( diamonds4 ,! puzzle is to arrange !m 9 ! =m ( a cross1 exactly 9 ! way %[n 9 ! illu/ration1 s t ! pips 9 ! v]tical b> & 9 ! horizontal b> add up alike4 ,9 ! example giv5 x w be f\nd t bo? directions add up #bc4 ,:at ,i want to "k is1 h[ _m diff]5t ways are !re ( re>rang+ ! c>ds 9 ord] to br+ ab\t ?is result8 ,x w be se5 t1 )\t affect+ ! solution1 we may ex*ange ! #e ) ! #f1 ! #e ) ! #g1 ! #h ) ! #c1 & s on4 ,also we may make ! horizontal & ! v]tical b>s *ange places4 ,b su* obvi\s manipulations z ^! are n to be reg>d$ z diff]5t solutions4 ,!y are all m]e v>iations ( "o fundam5tal solution4 ,n[1 h[ _m ( ^! fundam5tally diff]5t solutions are !re8 ,! pips ne$ n1 ( c\rse1 always add up #bc4 #chc4--,,! ,t ,,c>d ,,puzzle4 ,illu/ration ,an ent]ta9+ little puzzle ) c>ds is to take ! n9e c>ds ( a suit1 f ace to n9e inclusive1 & arrange !m 9 ! =m ( ! lett] ,t1 z %[n 9 ! illu/ration1 s t ! pips 9 ! horizontal l9e %all c\nt ! same z ^? 9 ! column4 ,9 ! example giv5 !y add up tw5ty-?ree bo? ways4 ,n[1 x is q easy to get a s+le correct arrangem5t4 ,! puzzle is to discov] 9 j h[ _m diff]5t ways x may be done4 ,?\gh ! numb] is high1 ! solution is n really difficult if we attack ! puzzle 9 ! "r mann]4 ,! rev]se way obta9$ by reflect+ ! illu/ration 9 a mirror we w n c\nt z diff]5t1 b all o!r *anges 9 ! relative positions ( ! c>ds w "h c\nt4 ,h[ _m diff]5t ways are !re8 #chd4--;,,c>d ,,triangles4 ,"h y pick out ! n9e c>ds1 ace to n9e ( diamonds1 & arrange !m 9 ! =m ( a triangle1 exactly z %[n 9 ! illu/ration1 s t ! pips add up ! same on ! ?ree sides4 ,9 ! example giv5 x w be se5 t !y sum to #bj on ea* side1 b ! p>ticul> numb] is ( no importance s long z x is ! same on all ?ree sides4 ,! puzzle is to f9d out 9 j h[ _m diff]5t ways ?is c be done4 ,if y simply turn ! c>ds r\nd s t "o ( ! o!r two sides is ne>e/ to y ?is w n c\nt z diff]5t1 = ! ord] w be ! same4 ,also1 if y make ! #d1 #i1 #e *ange places ) ! #g1 #c1 #h1 & at ! same "t ex*ange ! #a & ! #f1 x w n be diff]5t4 ,b if y only *ange ! #a & ! #f x w be diff]5t1 because ! ord] r\nd ! triangle is n ! same4 ,?is explanation w prev5t any d\bt aris+ z to ! conditions4 ,illu/ration #che4--,,/r& ,,pati5ce4 ,! idea = ?is came to me :5 consid]+ ! game ( ,pati5ce t ,i gave 9 ! ,/r& ,magaz9e = ,decemb]1 #aiaj1 :i* has be5 repr9t$ 9 ,erne/ ,b]gholt's ,second ,book ( ,pati5ce ,games1 "u ! new "n ( ,k+ ,alb]t4 ,make two piles ( c>ds z foll[s3 #i ,d1 #h ,s1 #g ,d1 #f ,s1 #e ,d1 #d ,s1 #c ,d1 #b ,s1 #a ,d1 & #i ,h1 #h ,c1 #g ,h1 #f ,c1 #e ,h1 #d ,c1 #c ,h1 #b ,c1 #a ,h1 ) ! #i ( diamonds at ! bottom ( "o pile & ! #i ( he>ts at ! bottom ( ! o!r4 ,! po9t is to ex*ange ! spades ) ! clubs1 s t ! diamonds & clubs are /ill 9 num]ical ord] 9 "o pile & ! he>ts & spades 9 ! o!r4 ,!re are f\r vacant spaces 9 addition to ! two spaces occupi$ by ! piles1 & any c>d may be laid on a space1 b a c>d c only be laid on ano!r ( ! next high] value--an ace on a two1 a two on a ?ree1 & s on4 ,pati5ce is requir$ to discov] ! %orte/ way ( do+ ?is4 ,:5 !re are f\r vacant spaces y c pile f\r c>ds 9 sev5 moves1 ) only ?ree spaces y c pile !m 9 n9e moves1 & ) two spaces y _c pile m ?an two c>ds4 ,:5 y h a grasp ( ^! & simil> facts y w be able to remove a numb] ( c>ds bodily & write d[n #g1 #i1 or :atev] ! numb] ( moves may be4 ,! gradual %ort5+ ( play is fasc9at+1 & fir/ attempts are surpris+ly l5g?y4 #chf4--;,a ,,trick ,,) ,,dice4 ,illu/ration ,"h is a neat little trick ) ?ree dice4 ,i ask y to ?r[ ! dice )\t my see+ !m4 ,!n ,i tell y to multiply ! po9ts ( ! fir/ die by #b & add #e2 !n multiply ! result by #e & add ! po9ts ( ! second die2 !n multiply ! result by #aj & add ! po9ts ( ! ?ird die4 ,y !n give me ! total1 & ,i c at once tell y ! po9ts ?r[n ) ! ?ree dice4 ,h[ d ,i d x8 ,z an example1 if y ?rew #a1 #c1 & #f1 z 9 ! illu/ration1 ! result y w\ld give me w\ld be #chf1 f :i* ,i c\ld at once say :at y _h ?r[n4 #chg4--,,! ,,village ,,cricket ,,mat*4 ,9 a cricket mat*1 ,d+ley ,dell v4 ,all ,muggleton1 ! latt] _h ! fir/ inn+s4 ,mr4 ,dumk9s & ,mr4 ,podd] 7 at ! wickets1 :5 ! w>y ,dumk9s made a spl5did late cut1 & ,mr4 ,podd] call$ on him to run4 ,f\r runs 7 app>5tly complet$1 b ! vigilant umpires at ea* end call$1 ?ree %ort1 mak+ six %ort runs 9 all4 ,:at numb] did ,mr4 ,dumk9s score8 ,:5 ,d+ley ,dell took _! turn at ! wickets _! *ampions 7 ,mr4 ,luffey & ,mr4 ,/ruggles4 ,! latt] made a magnific5t (f-drive1 & invit$ 8 colleague to come along1 ) ! result t ! obs]vant spectators applaud$ !m = :at 0 suppos$ to h be5 ?ree %>p runs4 ,b ! umpires decl>$ t !re _h be5 two %ort runs at ea* end--f\r 9 all4 ,to :at ext5t1 if any1 did ?is manoeuvre increase ,mr4 ,/ruggles's total8 #chh4--,,sl[ ,,cricket4 ,9 ! rec5t c\nty mat* betwe5 ,wessex & ,n9com%ire ! =m] team 7 at ! wickets all day1 ! la/ man be+ put out a few m9utes be=e ! "t = draw+ /umps4 ,! play 0 s sl[ t mo/ ( ! spectators 7 fa/ asleep1 &1 on be+ awak5$ by "o ( ! (ficials cle>+ ! gr\nd1 we le>nt t two m5 _h be5 put out leg-be=e-wicket = a comb9$ score ( #ai runs2 f\r m5 7 caught = a comb9$ score or #ag runs2 "o man 0 run out = a duck's egg2 & ! o!rs 7 all b[l$ = #c runs ea*4 ,!re 7 no extras4 ,we 7 n told :i* ( ! m5 0 ! capta91 b he made exactly #ae m ?an ! av]age ( 8 team4 ,:at 0 ! capta9's score8 #chi4--,,! ,,football ,,play]s4 ,x is a glori\s game6 an en?usia/ 0 he>d to exclaim4 ,at ! close ( la/ season1 ( ! football]s ( my acqua9tance f\r _h brok5 _! left arm1 five _h brok5 _! "r arm1 two _h ! "r arm s\nd1 & ?ree _h s\nd left arms4 ,c y discov] f t /atem5t :at is ! smalle/ numb] ( play]s t ! speak] c\ld be acqua9t$ )8 ,x does n at all foll[ t !re 7 z _m z f\rte5 m51 because1 = example1 two ( ! m5 :o _h brok5 ! left arm might also be ! two :o _h s\nd "r arms4 #cij4--,,! ,,horse-,,race ,,puzzle4 ,!re are no morals 9 puzzles4 ,:5 we are solv+ ! old puzzle ( ! capta9 :o1 hav+ to ?r[ half 8 crew ov]bo>d 9 a /orm1 arrang$ to draw lots1 b s plac$ ! m5 t only ! ,turks 7 sacrific$1 & all ! ,*ri/ians left on bo>d1 we d n /op to discuss ! que/ionable morality ( ! proce$+4 ,& :5 we are deal+ ) a measur+ problem1 9 :i* c]ta9 ?ir/y pilgrims are to make an equitable division ( a b>rel ( be]1 we d n object t1 z total ab/a9]s1 x is aga9/ our consci5ce to h any?+ to d ) intoxicat+ liquor4 ,!re=e ,i make no apology = introduc+ a puzzle t deals ) bett+4 ,?ree horses--,acorn1 ,bluebottle1 & ,capsule--/>t 9 a race4 ,! odds are #d to #a1 ,acorn2 #c to #a1 ,bluebottle2 #b to #a1 ,capsule4 ,n[1 h[ mu* mu/ ,i inve/ on ea* horse 9 ord] to w9 #ac1 no matt] :i* horse comes 9 fir/8 ,suppos+1 z an example1 t ,i bett$ #e on ea* horse4 ,!n1 if ,acorn won1 ,i %\ld receive #bj 7f\r times #e71 & h to pay #e ea* = ! o!r two horses2 !reby w9n+ #aj4 ,b x w be f\nd t if ,bluebottle 0 fir/ ,i %\ld only w9 #e1 & if ,capsule won ,i %\ld ga9 no?+ & lose no?+4 ,?is w make ! "q p]fectly cle> to ! novice1 :o1 l myself1 is n int]e/$ 9 ! call+ ( ! frat]nity :o pr(ess to be engag$ 9 ! noble task ( improv+ ! bre$ ( horses4 #cia4--,,! ,,motor-,,c> ,,race4 ,sometimes a q simple /atem5t ( fact1 if word$ 9 an unfamili> mann]1 w cause consid]able p]plexity4 ,"h is an example1 & x w d\btless puzzle "s ( my m y\?ful read]s j a little4 ,i happ5$ to be at a motor-c> race at ,brookl&s1 :5 "o spectator said to ano!r1 :ile a numb] ( c>s 7 :irl+ r\nd & r\nd ! circul> track3-- ,!re's ,gogglesmi?--t man 9 ! :ite c>6 ,yes1 ,i see1 0 ! reply2 b h[ _m c>s are runn+ 9 ?is race8 ,!n came ?is curi\s rejo9d]3-- ,"o-?ird ( ! c>s 9 front ( ,gogglesmi? add$ to ?ree-qu>t]s ( ^? beh9d him w give y ! answ]4 ,n[1 c y tell h[ _m c>s 7 runn+ 9 ! race8 ,,puzzle ,,games4 ,he t is beat5 may be said ,to lie 9 hon\r's truckle b$4 ,,hudibras4 ,x may be said g5]ally t a game is a conte/ ( skill = two or m p]sons1 into :i* we ent] ei!r = amusem5t or to w9 a prize4 ,a puzzle is some?+ to be done or solv$ by ! individual4 ,= example1 if x 7 possible = u s to ma/] ! complexities ( ! game ( *ess t we c\ld be assur$ ( always w9n+ ) ! fir/ or second move1 z ! case might be1 or ( always draw+1 !n x w\ld cease to be a game & w\ld become a puzzle4 ,( c\rse among ! "y & un9=m$1 :5 ! correct w9n+ play is n und]/ood1 a puzzle may well make a v good game4 ,?us !re is no d\bt *ildr5 w cont9ue to play ,n\ghts & ,crosses1 ?\gh ,i h %[n 7,no4 #aji1 ,cant]bury ,puzzles7 t betwe5 two play]s :o bo? ?or\ghly und]/& ! play1 e game %\ld be drawn4 ,nei!r play] c\ld ev] w9 except ?r\gh ! blund]+ ( 8 oppon5t4 ,b ,i am writ+ f ! po9t ( view ( ! /ud5t ( ^! ?+s4 ,! examples t ,i give 9 ?is class are app>5tly games1 b1 s9ce ,i %[ 9 e case h[ "o play] may w9 if he only play correctly1 !y are 9 reality puzzles4 ,_! int]e/1 !re=e1 lies 9 attempt+ to discov] ! lead+ me?od ( play4 #cib4--,,! ,,pebble ,,game4 ,"h is an int]e/+ little puzzle game t ,i us$ to play ) an acqua9tance on ! bea* at ,slocomb-on-,sea4 ,two play]s place an odd numb] ( pebbles1 we w say fifte51 betwe5 !m4 ,!n ea* takes 9 turn "o1 two1 or ?ree pebbles 7z he *ooses71 & ! w9n] is ! "o :o gets ! odd numb]4 ,?us1 if y get sev5 & y\r oppon5t eight1 y w94 ,if y get six & he gets n9e1 he w9s4 ,ought ! fir/ or second play] to w91 & h[8 ,:5 y h settl$ ! "q ) fifte5 pebbles try aga9 )1 say1 ?irte54 #cic4--,,! ,,two ,,rooks4 ,?is is a puzzle game = two play]s4 ,ea* play] has a s+le rook4 ,! fir/ play] places 8 rook on any squ>e ( ! bo>d t he may *oose to select1 & !n ! second play] does ! same4 ,!y n[ play 9 turn1 ! po9t ( ea* play be+ to capture ! oppon5t's rook4 ,b 9 ?is game y _c play ?r\gh a l9e ( attack )\t be+ captur$4 ,t is to say1 if 9 ! diagram x is ,black's turn to play1 he _c move 8 rook to 8 k+'s knight's squ>e1 or to 8 k+'s rook's squ>e1 because he w\ld ent] ! l9e ( fire :5 pass+ 8 k+'s bi%op's squ>e4 ,= ! same reason he _c move to 8 que5's rook's sev5? or eigh? squ>es4 ,n[1 ! game c nev] end 9 a draw4 ,soon] or lat] "o ( ! rooks mu/ fall1 unless1 ( c\rse1 bo? play]s commit ! absurdity ( n try+ to w94 ,! trick ( w9n+ is ridicul\sly simple :5 y "k x4 ,c y solve ! puzzle8 ,illu/ration #cid4--,,puss ,,9 ,,! ,,corn]4 ,illu/ration ,?is v>iation ( ! la/ puzzle is also play$ by two p]sons4 ,"o puts a c\nt] on ,no4 #f1 & ! o!r puts "o on ,no4 #ee1 & !y play alt]nately by remov+ ! c\nt] to any o!r numb] 9 a l9e4 ,if y\r oppon5t moves at any "t on to "o ( ! l9es y occupy1 or ev5 crosses "o ( y\r l9es1 y imm$iately capture him & w94 ,we w take an illu/rative game4 ,a moves f #ee to #eb2 ,b moves f #f to #ac2 ,a advances to #bc2 ,b goes to #ae2 ,a retreats to #bf2 ,b retreats to #ac2 ,a advances to #ba2 ,b retreats to #b2 ,a advances to #g2 ,b goes to #c2 ,a moves to #f2 ,b mu/ n[ g to #d2 ,a e/abli%es himself at #aa1 & ,b mu/ be captur$ next move because he is compell$ to cross a l9e on :i* ,a /&s4 ,play ?is ov] & y w und]/& ! game directly4 ,n[1 ! puzzle "p ( ! game is ?is3 ,:i* play] %\ld w91 & h[ _m moves are necess>y8 #cie4--;,a ,,w> ,,puzzle ,,game4 ,illu/ration ,"h is ano!r puzzle game4 ,"o play]1 repres5t+ ! ,briti% g5]al1 places a c\nt] at ,b1 & ! o!r play]1 repres5t+ ! enemy1 places 8 c\nt] at ,e4 ,! ,briti%] makes ! fir/ advance along "o ( ! roads to ! next t[n1 !n ! enemy moves to "o ( 8 ne>e/ t[ns1 & s on 9 turns1 until ! ,briti% g5]al gets into ! same t[n z ! enemy & captures him4 ,al?\gh ea* mu/ always move along a road to ! next t[n only1 & ! second play] may d 8 utmo/ to avoid capture1 ! ,briti% g5]al 7z we %\ld suppose1 f ! analogy ( real life7 mu/ infallibly w94 ,b h[8 ,t is ! "q4 #cif4--;,a ,,mat* ,,my/]y4 ,"h is a little game t is *ildi%ly simple 9 its conditions4 ,b x is well wor? inve/igation4 ,mr4 ,/ubbs pull$ a small table betwe5 himself & 8 fri5d1 ,mr4 ,wilson1 & took a box ( mat*es1 f :i* he c\nt$ out ?irty4 ,"h are ?irty mat*es1 he said4 ,i divide !m into ?ree unequal heaps4 ,let me see4 ,we h #ad1 #aa1 & #e1 z x happ5s4 ,n[1 ! two play]s draw alt]nately any numb] f any "o heap1 & he :o draws ! la/ mat* loses ! game4 ,t's all6 ,i w play ) y1 ,wilson4 ,i h =m$ ! heaps1 s y h ! fir/ draw4 ,z ,i c draw any numb]1 ,mr4 ,wilson said1 suppose ,i exhibit my usual mod]ation & take all ! #ad heap4 ,t is ! wor/ y c\ld d1 = x loses "r away4 ,i take #f f ! #aa1 leav+ two equal heaps ( #e1 & to leave two equal heaps is a c]ta9 w9 7) ! s+le exception ( #a1 #a71 because :atev] y d 9 "o heap ,i c repeat 9 ! o!r4 ,if y leave #d 9 "o heap1 ,i leave #d 9 ! o!r4 ,if y !n leave #b 9 "o heap1 ,i leave #b 9 ! o!r4 ,if y leave only #a 9 "o heap1 !n ,i take all ! o!r heap4 ,if y take all "o heap1 ,i take all b "o 9 ! o!r4 ,no1 y mu/ nev] leave two heaps1 unless !y are equal heaps & m ?an #a1 #a4 ,let's beg9 aga94 ,v well1 !n1 said ,mr4 ,wilson4 ,i w take #f f ! #ad1 & leave y #h1 #aa1 #e4 ,mr4 ,/ubbs !n left #h1 #aa1 #c2 ,mr4 ,wilson1 #h1 #e1 #c2 ,mr4 ,/ubbs1 #f1 #e1 #c2 ,mr4 ,wilson1#d1 #e1 #c2 ,mr4 ,/ubbs1 #d1 #e1 #a2 ,mr4 ,wilson1 #d1 #c1 #a2 ,mr4 ,/ubbs1 #b1 #c1 #a2 ,mr4 ,wilson1 #b1 #a1 #a2 :i* ,mr4 ,/ubbs r$uc$ to #a1 #a1 #a4 ,x is n[ q cle> t ,i mu/ w91 said ,mr4 ,/ubbs1 because y mu/ take #a1 & !n ,i take #a1 leav+ y ! la/ mat*4 ,y nev] _h a *ance4 ,!re are j ?irte5 diff]5t ways 9 :i* ! mat*es may be gr\p$ at ! />t = a c]ta9 w94 ,9 fact1 ! gr\ps select$1 #ad1 #aa1 #e1 are a c]ta9 w91 because = :atev] y\r oppon5t may play !re is ano!r w9n+ gr\p y c secure1 & s on & on d[n to ! la/ mat*4 #cig4--,,! ,,mont5egr9 ,,dice ,,game4 ,x is said t ! inhabitants ( ,mont5egro h a little dice game t is bo? +5i\s & well wor? inve/igation4 ,! two play]s fir/ select two diff]5t pairs ( odd numb]s 7always high] ?an #c7 & !n alt]nately toss ?ree dice4 ,:i*ev] fir/ ?r[s ! dice s t !y add up to "o ( 8 select$ numb]s w9s4 ,if !y are bo? successful 9 two successive ?r[s x is a draw & !y try aga94 ,= example1 "o play] may select #g & #ae & ! o!r #e & #ac4 ,!n if ! fir/ play] ?r[s s t ! ?ree dice add up #g or #ae he w9s1 unless ! second man gets ei!r #e or #ac on 8 ?r[4 ,! puzzle is to discov] :i* two pairs ( numb]s %\ld be select$ 9 ord] to give bo? play]s an exactly ev5 *ance4 #cih4--,,! ,,cig> ,,puzzle4 ,i once prop\nd$ ! foll[+ puzzle 9 a ,london club1 & = a consid]able p]iod x absorb$ ! att5tion ( ! memb]s4 ,!y c\ld make no?+ ( x1 & consid]$ x q impossible ( solution4 ,& yet1 z ,i %all %[1 ! answ] is rem>kably simple4 ,two m5 are seat$ at a squ>e-topp$ table4 ,"o places an ord9>y cig> 7flat at "o end1 po9t$ at ! o!r7 on ! table1 !n ! o!r does ! same1 & s on alt]nately1 a condition be+ t no cig> %all t\* ano!r4 ,:i* play] %\ld succe$ 9 plac+ ! la/ cig>1 assum+ t !y ea* w play 9 ! be/ possible mann]8 ,! size ( ! table top & ! size ( ! cig> are n giv51 b 9 ord] to exclude ! ridicul\s answ] t ! table might be s dim9utive z only to take "o cig>1 we w say t ! table mu/ n be less ?an #b feet squ>e & ! cig> n m ?an #d in*es long4 ,) ^? re/rictions y may take any dim5sions y l4 ,( c\rse we assume t all ! cig>s are exactly alike 9 e respect4 ,%\ld ! fir/ play]1 or ! second play]1 w98 ,,magic ,,squ>e ,,problems4 ,by magic numb]s4 ,,congreve1 ,! ,m\rn+ ,bride4 ,?is is a v anci5t bran* ( ma!matical puzzl$om1 & x has an imm5se1 ?\gh scatt]$1 lit]ature ( its own4 ,9 _! simple =m ( consecutive :ole numb]s arrang$ 9 a squ>e s t e column1 e r[1 & ea* ( ! two long diagonals %all add up alike1 ^! magic squ>es (f] ?ree ma9 l9es ( inve/igation3 ,con/ruction1 ,enum]ation1 & ,classification4 ,( rec5t ye>s _m +5i\s me?ods h be5 devis$ = ! con/ruction ( magics1 & ! law ( _! =mation is s well und]/ood t all ! anci5t my/]y has evaporat$ & !re is no long] any difficulty 9 mak+ squ>es ( any dim5sions4 ,almo/ ! la/ ^w has be5 said on ?is subject4 ,! "q ( ! enum]ation ( all ! possible squ>es ( a giv5 ord] /&s j :]e x did ov] two hundr$ ye>s ago4 ,ev]ybody kn[s t !re is only "o solution = ! ?ird ord]1 ?ree cells by ?ree2 & ,frnicle publi%$ 9 #afic diagrams ( all ! arrangem5ts ( ! f\r? ord]--#hhj 9 numb]--& 8 results h be5 v]ifi$ ov] & ov] aga94 ,i may "h ref] to ! g5]al solution = ?is ord]1 = numb]s n necess>ily consecutive1 by ,e4 ,b]gholt 9 ,nature1 ,may #bf1 #aiaj1 z x is ( ! greate/ importance to /ud5ts ( ?is subject4 ,! enum]ation ( ! examples ( any high] ord] is a completely unsolv$ problem4 ,z to classification1 x is l>gely a matt] ( individual ta/e--p]haps an /hetic "q1 = !re is beauty 9 ! law & ord] ( numb]s4 ,a man once said t he divid$ ! human race into two great classes3 ^? :o take snuff & ^? :o d n4 ,i am n sure t "s ( our classifications ( magic squ>es are n almo/ z valueless4 ,h[ev]1 lov]s ( ^! ?+s seem some:at agre$ t ,nasik magic squ>es 7s nam$ by ,mr4 ,fro/1 a /ud5t ( !m1 aft] ! t[n 9 ,india :]e he liv$1 & also call$ ,diabolique & ,p&iagonal7 & ,associat$ magic squ>es are ( special int]e/1 s ,i w j expla9 :at ^! are = ! b5efit ( ! novice4 ,illu/ration3 ,,simple ,illu/ration3 ,,semi-,,nasik ,illu/ration3 ,,associat$ ,illu/ration3 ,,nasik ,i publi%$ 9 ,! ,que5 = ,janu>y #ae1 #aiaj1 an article t w\ld enable ! read] to write out1 if he s desir$1 all ! #hhj magics ( ! f\r? ord]1 & ! foll[+ is ! complete classification t ,i gave4 ,! fir/ example is t ( a ,simple squ>e t fulfils ! simple conditions & no m4 ,! second example is a ,semi-,nasik1 :i* has ! additional prop]ty t ! opposite %ort diagonals ( two cells ea* toge!r sum to #cd4 ,?us1 #ad #d #aa #e #cd & #ab #f #ac #c #cd4 ,! ?ird example is n only ,semi-,nasik b also ,associat$1 because 9 x e numb]1 if add$ to ! numb] t is equidi/ant1 9 a /raight l9e1 f ! c5tre gives #ag4 ,?us1 #a #af1 #b #ae1 #c #ad1 etc4 ,! f\r? example1 consid]$ ! mo/ p]fect ( all1 is a ,nasik4 ,"h all ! brok5 diagonals sum to #cd4 ,?us1 = example1 #ae #ad #b #c1 & #aj #d #g #ac1 & #ae #e #b #ab4 ,z a consequ5ce1 its prop]ties are su* t if y repeat ! squ>e 9 all directions y may m>k (f a squ>e1 #d #d1 :]ev] y please1 & x w be magic4 ,! foll[+ table n only gives a complete enum]ation "u ! f\r =ms describ$1 b also a classification "u ! twelve graphic types indicat$ 9 ! diagrams4 ,! dots at ! end ( ea* l9e repres5t ! relative positions ( ^? complem5t>y pairs1 #a #af1 #b #ae1 etc41 :i* sum to #ag4 ,= example1 x w be se5 t ! fir/ & second magic squ>es giv5 are ( ,type ,,vi41 t ! ?ird squ>e is ( ,type ,,iii41 & t ! f\r? is ( ,type ,i4 ,ed\>d ,lucas indicat$ ^! types1 b he dropp$ exactly half ( !m & did n attempt ! classification4 ,,nasik 7,type ,i47 4 4 4 4 4 #dh ,,semi-,,nasik 7,type ,,ii41 ,transpositions ( ,nasik7 4 #dh 7,type ,,iii41 ,associat$7 #dh 7,type ,,iv47 4 4 4 #if 7,type ,v47 4 4 4 #if #aib 7,type ,,vi47 4 4 4 #if #chd ,,simple4 7,type ,,vi47 4 4 4 #bjh 7,type ,,vii47 4 4 4 #ef 7,type ,,viii474 4 4 #ef 7,type ,,ix47 4 4 4 #ef 7,type ,x47 4 4 4 #ef #bbd 7,type ,,xi47 4 4 4 #h 7,type ,,xii47 4 4 4 #h #af #ddh #hhj ,x is h>dly necess>y to say t e "o ( ^! squ>es w produce sev5 o!rs by m]e rev]sals & reflections1 :i* we d n c\nt z diff]5t4 ,s t !re are #g1jdj squ>es ( ?is ord]1 #hhj ( :i* are fundam5tally diff]5t4 ,an inf9ite v>iety ( puzzles may be made introduc+ new conditions into ! magic squ>e4 ,9 ,! ,cant]bury ,puzzles ,i h giv5 examples ( su* squ>es ) co9s1 ) po/age /amps1 ) cutt+-out conditions1 & o!r tricks4 ,i w n[ give a few v>iants involv+ fur!r novel conditions4 #cii4--,,! ,,tr\blesome ,,eight4 ,ne>ly ev]ybody kn[s t a magic squ>e is an arrangem5t ( numb]s 9 ! =m ( a squ>e s t e r[1 e column1 & ea* ( ! two long diagonals adds up alike4 ,= example1 y w\ld f9d little difficulty 9 m]ely plac+ a diff]5t numb] 9 ea* ( ! n9e cells 9 ! illu/ration s t ! r[s1 columns1 & diagonals %all all add up #ae4 ,& at y\r fir/ attempt y w probably f9d t y h an #h 9 "o ( ! corn]s4 ,! puzzle is to con/ruct ! magic squ>e1 "u ! same conditions1 ) ! #h 9 ! position %[n4 ,illu/ration #djj4--,,! ,,magic ,,/rips4 ,illu/ration ,i happ5$ to h ly+ on my table a numb] ( /rips ( c>dbo>d1 ) numb]s pr9t$ on !m f #a upw>ds 9 num]ical ord]4 ,! idea sudd5ly came to me1 z ideas h a way ( unexpect$ly com+1 to make a little puzzle ( ?is4 ,i wond] :e!r _m read]s w arrive at ! same solution t ,i did4 ,take sev5 /rips ( c>dbo>d & lay !m toge!r z above4 ,!n write on ea* ( !m ! numb]s #a1 #b1 #c1 #d1 #e1 #f1 #g1 z %[n1 s t ! numb]s %all =m sev5 r[s & sev5 columns4 ,n[1 ! puzzle is to cut ^! /rips into ! fewe/ possible pieces s t !y may be plac$ toge!r & =m a magic squ>e1 ! sev5 r[s1 sev5 columns1 & two diagonals add+ up ! same numb]4 ,no figures may be turn$ upside d[n or plac$ on _! sides--t is1 all ! /rips mu/ lie 9 _! orig9al direction4 ,( c\rse y c\ld cut ea* /rip into sev5 sep>ate pieces1 ea* piece conta9+ a numb]1 & ! puzzle w\ld !n be v easy1 b ,i ne$ h>dly say t =ty-n9e pieces is a long way f be+ ! fewe/ possible4 #dja4--;,,eight ,,jolly ,,gaol ,,birds4 ,illu/ration ,! illu/ration %[s ! plan ( a prison ( n9e cells all communicat+ ) "o ano!r by doorways4 ,! eight prison]s h _! numb]s on _! backs1 & any "o ( !m is all[$ to ex]cise himself 9 :i*ev] cell may happ5 to be vacant1 subject to ! rule t at no "t %all two prison]s be 9 ! same cell4 ,! m]ry mon>* 9 ^: dom9ions ! prison 0 situat$ (f]$ !m special com=ts "o ,*ri/mas ,eve if1 )\t break+ t rule1 !y c\ld s place !mselves t _! numb]s %\ld =m a magic squ>e4 ,n[1 prison] ,no4 #g happ5$ to "k a good deal ab\t magic squ>es1 s he work$ out a s*eme & naturally select$ ! me?od t 0 mo/ exp$iti\s--t is1 "o involv+ ! fewe/ possible moves f cell to cell4 ,b "o man 0 a surly1 ob/9ate fell[ 7q unfit = ! society ( 8 jovial companions71 & he refus$ to move out ( 8 cell or take any "p 9 ! proce$+s4 ,b ,no4 #g 0 q equal to ! em]g5cy1 & f\nd t he c\ld /ill d :at 0 requir$ 9 ! fewe/ possible moves )\t tr\bl+ ! brute to leave 8 cell4 ,! puzzle is to %[ h[ he did x &1 incid5tally1 to discov] :i* prison] 0 s /upidly ob/9ate4 ,c y f9d ! fell[8 #djb4--,,n9e ,,jolly ,,gaol ,,birds4 ,illu/ration ,%ortly aft] ! episode record$ 9 ! la/ puzzle occurr$1 a n9? prison] 0 plac$ 9 ! vacant cell1 & ! m]ry mon>* !n (f]$ !m all complete lib]ty on ! foll[+ /range conditions4 ,!y 7 requir$ s to re>range !mselves 9 ! cells t _! numb]s =m$ a magic squ>e )\t _! movem5ts caus+ any two ( !m ev] to be 9 ! same cell toge!r1 except t at ! />t "o man 0 all[$ to be plac$ on ! %\ld]s ( ano!r man1 & ?us add _! numb]s toge!r1 & move z "o man4 ,= example1 ,no4 #h might be plac$ on ! %\ld]s ( ,no4 #b1 & !n !y w\ld move ab\t toge!r z #aj4 ,! read] %\ld seek fir/ to solve ! puzzle 9 ! fewe/ possible moves1 & !n see t ! man :o is burd5$ has ! lea/ possible am\nt ( "w to d4 #djc4--,,! ,,spani% ,,dungeon4 ,n fifty miles f ,cadiz /ood 9 ! middle ages a ca/le1 all traces ( :i* h = c5turies disappe>$4 ,among o!r int]e/+ features1 ?is ca/le conta9$ a p>ticul>ly unpleasant dungeon divid$ into sixte5 cells1 all communicat+ ) "o ano!r1 z %[n 9 ! illu/ration4 ,n[1 ! gov]nor 0 a m]ry wight1 & v fond ( puzzles )al4 ,"o day he w5t to ! dungeon & said to ! prison]s1 ,by my halidame6 7or its equival5t 9 ,spani%7 y %all all be set free if y c solve ?is puzzle4 ,y mu/ s arrange y\rselves 9 ! sixte5 cells t ! numb]s on y\r backs %all =m a magic squ>e 9 :i* e column1 e r[1 & ea* ( ! two diagonals %all add up ! same4 ,only rememb] ?is3 t 9 no case may two ( y ev] be toge!r 9 ! same cell4 ,"o ( ! prison]s1 aft] work+ at ! problem = two or ?ree days1 ) a piece ( *alk1 und]took to obta9 ! lib]ty ( himself & 8 fell[-prison]s if !y w\ld foll[ 8 directions & move ?r\gh ! doorway f cell to cell 9 ! ord] 9 :i* he %\ld call out _! numb]s4 ,illu/ration ,he succe$$ 9 8 attempt1 &1 :at is m rem>kable1 x w\ld seem f ! acc\nt ( 8 me?od record$ 9 ! anci5t manuscript ly+ be=e me1 t he did s 9 ! fewe/ possible moves4 ,! read] is ask$ to %[ :at ^! moves 74 #djd4--,,! ,,sib]ian ,,dungeons4 ,illu/ration ,! above is a tru/wor?y plan ( a c]ta9 ,russian prison 9 ,sib]ia4 ,all ! cells are numb]$1 & ! prison]s are numb]$ ! same z ! cells !y occupy4 ,! prison diet is s fatt5+ t ^! political prison]s are 9 p]petual fe> le/1 %\ld _! p>don arrive1 !y might n be able to squeeze !mselves ?r\gh ! n>r[ doorways & get out4 ,& ( c\rse x w\ld be an unreasonable ?+ to ask any gov]nm5t to pull d[n ! walls ( a prison j to lib]ate ! prison]s1 h[ev] innoc5t !y might be4 ,!re=e ^! m5 take all ! heal?y ex]cise !y c 9 ord] to ret>d _! increas+ obesity1 & "o ( _! recreations w s]ve to furni% u ) ! foll[+ puzzle4 ,%[1 9 ! fewe/ possible moves1 h[ ! sixte5 m5 may =m !mselves into a magic squ>e1 s t ! numb]s on _! backs %all add up ! same 9 ea* ( ! f\r columns1 f\r r[s1 & two diagonals )\t two prison]s hav+ be5 at any "t 9 ! same cell toge!r4 ,i _h bett] say1 = ! in=mation ( ^? :o h n yet be5 made acqua9t$ ) ^! places1 t x is a peculi>ity ( prisons t y are n all[$ to g outside _! walls4 ,any prison] may g any di/ance t is possible 9 a s+le move4 #dje4--;,,c>d ,,magic ,,squ>es4 ,illu/ration ,take an ord9>y pack ( c>ds & ?r[ out ! twelve c\rt c>ds4 ,n[1 ) n9e ( ! rema9d] 7diff]5t suits are ( no consequ5ce7 =m ! above magic squ>e4 ,x w be se5 t ! pips add up fifte5 9 e r[ 9 e column1 & 9 ea* ( ! two long diagonals4 ,! puzzle is ) ! rema9+ c>ds 7)\t di/urb+ ?is arrangem5t7 to =m ?ree m su* magic squ>es1 s t ea* ( ! f\r %all add up to a diff]5t sum4 ,!re w1 ( c\rse1 be f\r c>ds 9 ! r$uc$ pack t w n be us$4 ,^! f\r may be any t y *oose4 ,x is n a difficult puzzle1 b requires j a little ?\ght4 #djf4--,,! ,,eighte5 ,,dom9oes4 ,! illu/ration %[s eighte5 dom9oes arrang$ 9 ! =m ( a squ>e s t ! pips 9 e "o ( ! six columns1 six r[s1 & two long diagonals add up #ac4 ,?is is ! smalle/ summation possible ) any selection ( dom9oes f an ord9>y box ( tw5ty-eight4 ,! greate/ possible summation is #bc1 & a solution = ?is numb] may be easily obta9$ by sub/itut+ = e numb] its complem5t to #f4 ,?us = e blank sub/itute a #f1 = e #a a #e1 = e #b a #d1 = #c a #c1 = #d a #b1 = #e a #a1 & = #f a blank4 ,b ! puzzle is to make a selection ( eighte5 dom9oes & arrange !m 79 exactly ! =m %[n7 s t ! summations %all be #ah 9 all ! f\rte5 directions m5tion$4 ,illu/ration ,,subtract+1 ,,multiply+1 ,,& ,,divid+ ,,magics4 ,al?\gh ! add+ magic squ>e is ( su* great antiquity1 curi\sly en\gh ! multiply+ magic does n appe> to h be5 m5tion$ until ! end ( ! eighte5? c5tury1 :5 x 0 ref]r$ to slightly by "o writ] & !n =gott5 until ,i reviv$ x 9 ,tit-,bits 9 #ahig4 ,! divid+ magic 0 app>5tly fir/ discuss$ by me 9 ,! ,weekly ,dispat* 9 ,june #ahih4 ,! subtract+ magic is "h introduc$ = ! fir/ "t4 ,x w n[ be conv5i5t to deal ) all f\r k9ds ( magic squ>es toge!r4 ,illu/ration3 ,,add+ ,,subtract+ ,,multiply+ ,,divid+ ,9 ^! f\r diagrams we h examples 9 ! ?ird ord] ( add+1 subtract+1 multiply+1 & divid+ squ>es4 ,9 ! fir/ ! con/ant1 #ae1 is obta9$ by ! addition ( ! r[s1 columns1 & two diagonals4 ,9 ! second case y get ! con/ant1 #e1 by subtract+ ! fir/ numb] 9 a l9e f ! second1 & ! result f ! ?ird4 ,y c1 ( c\rse1 p]=m ! op]ation 9 ei!r direction2 b1 9 ord] to avoid negative numb]s1 x is m conv5i5t simply to d$uct ! middle numb] f ! sum ( ! two extreme numb]s4 ,?is is1 9 effect1 ! same ?+4 ,x w be se5 t ! con/ant ( ! add+ squ>e is n times t ( ! subtract+ squ>e d]iv$ f x1 :]e n is ! numb] ( cells 9 ! side ( squ>e4 ,& ! mann] ( d]ivation "h is simply to rev]se ! two diagonals4 ,bo? squ>es are associat$--a t]m ,i h expla9$ 9 ! introductory article to ?is dep>tm5t4 ,! ?ird squ>e is a multiply+ magic4 ,! con/ant1 #baf1 is obta9$ by multiply+ toge!r ! ?ree numb]s 9 any l9e4 ,x is associat$ by multiplication1 in/ead ( by addition4 ,x is "h necess>y to rem>k t 9 an add+ squ>e x is n ess5tial t ! n9e numb]s %\ld be consecutive4 ,write d[n any n9e numb]s 9 ?is way-- #a #c #e #d #f #h #g #i #aa s t ! horizontal diff]5ces are all alike & ! v]tical diff]5ces also alike 7"h #b & #c71 & ^! numb]s w =m an add+ magic squ>e4 ,by mak+ ! diff]5ces #a & #c we1 ( c\rse1 get consecutive numb]s--a p>ticul> case1 & no?+ m4 ,n[1 9 ! case ( ! multiply+ squ>e we mu/ take ^! numb]s 9 geometrical in/ead ( ari?metical progression1 ?us-- #a #c #i #b #f #ah #d #ab #cf ,"h ea* successive numb] 9 ! r[s is multipli$ by #c1 & 9 ! columns by #b4 ,_h we multipli$ by #b & #h we %\ld get ! regul> geometrical progression1 #a1 #b1 #d1 #h1 #af1 #cb1 #fd1 #abh1 & #bef1 b ,i wi% to avoid high numb]s4 ,! numb]s are arrang$ 9 ! squ>e 9 ! same ord] z 9 ! add+ squ>e4 ,! f\r? diagram is a divid+ magic squ>e4 ,! con/ant #f is "h obta9$ by divid+ ! second numb] 9 a l9e by ! fir/ 79 ei!r direction7 & ! ?ird numb] by ! quoti5t4 ,b1 aga91 ! process is simplifi$ by divid+ ! product ( ! two extreme numb]s by ! middle numb]4 ,?is squ>e is also associat$ by multiplication4 ,x is d]iv$ f ! multiply+ squ>e by m]ely rev]s+ ! diagonals1 & ! con/ant ( ! multiply+ squ>e is ! cube ( t ( ! divid+ squ>e d]iv$ f x4 ,! next set ( diagrams %[s ! solutions = ! fif? ord] ( squ>e4 ,!y are all associat$ 9 ! same way z be=e4 ,! subtract+ squ>e is d]iv$ f ! add+ squ>e by rev]s+ ! diagonals & ex*ang+ opposite numb]s 9 ! c5tres ( ! bord]s1 & ! con/ant ( "o is aga9 n times t ( ! o!r4 ,! divid+ squ>e is d]iv$ f ! multiply+ squ>e 9 ! same way1 & ! con/ant ( ! latt] is ! #e? p[] 7t is ! n?7 ( t ( ! =m]4 ,illu/ration ,^! squ>es are ?us q easy = odd ord]s4 ,b ! read] w probably f9d "s difficulty ov] ! ev5 ord]s1 conc]n+ :i* ,i w leave him to make 8 own rese>*es1 m]ely prop\nd+ two little problems4 #djg4--,,two ,,new ,,magic ,,squ>es4 ,con/ruct a subtract+ magic squ>e ) ! fir/ sixte5 :ole numb]s t %all be associat$ by subtraction4 ,! con/ant is1 ( c\rse1 obta9$ by subtract+ ! fir/ numb] f ! second 9 l9e1 ! result f ! ?ird1 & ! result aga9 f ! f\r?4 ,also con/ruct a divid+ magic squ>e ( ! same ord] t %all be associat$ by division4 ,! con/ant is obta9$ by divid+ ! second numb] 9 a l9e by ! fir/1 ! ?ird by ! quoti5t1 & ! f\r? by ! next quoti5t4 #djh4--,,magic ,,squ>es ,,( ,,two ,,degrees4 ,:ile read+ a ,fr5* ma!matical "w ,i happ5$ to come across1 ! foll[+ /atem5t3 ,a v rem>kable magic squ>e ( #h1 9 two degrees1 has be5 con/ruct$ by ,m4 ,pfeff]mann4 ,9 o!r words1 he has manag$ to dispose ! sixty-f\r fir/ numb]s on ! squ>es ( a *essbo>d 9 su* a way t ! sum ( ! numb]s 9 e l9e1 e column1 & 9 ea* ( ! two diagonals1 %all be ! same2 & m1 t if "o sub/itutes = all ! numb]s _! squ>es1 ! squ>e /ill rema9s magic4 ,i at once set to "w to solve ?is problem1 &1 al?\gh x prov$ a v h>d nut1 "o 0 rew>d$ by ! discov]y ( "s curi\s & beautiful laws t gov]n x4 ,! read] may l to try 8 h& at ! puzzle4 ,,magic ,,squ>es ,,( ,,primes4 ,! problem ( con/ruct+ magic squ>es ) prime numb]s only 0 fir/ discuss$ by myself 9 ,! ,weekly ,dispat* = #bbnd ,july & #e? ,augu/ #aijj2 b dur+ ! la/ ?ree or f\r ye>s x has receiv$ great att5tion f ,am]ican ma!maticians4 ,fir/1 !y h s\ght to =m ^! squ>es ) ! l[e/ possible con/ants4 ,?us1 ! fir/ n9e prime numb]s1 #a to #bc inclusive1 sum to #ii1 :i* 7be+ divisible by #c7 is !oretically a suitable s]ies2 yet x has be5 demon/rat$ t ! l[e/ possible con/ant is #aaa1 & ! requir$ s]ies z foll[s3 #a1 #g1 #ac1 #ca1 #cg1 #dc1 #fa1 #fg1 & #gc4 ,simil>ly1 9 ! case ( ! f\r? ord]1 ! l[e/ s]ies ( primes t are !oretically suitable w n s]ve4 ,b 9 e o!r ord]1 up to ! #ab? inclusive1 magic squ>es h be5 con/ruct$ ) ! l[e/ s]ies ( primes !oretically possible4 ,& ! #ab? is ! l[e/ ord] 9 :i* a /raight s]ies ( prime numb]s1 unbrok51 f #a upw>ds has be5 made to "w4 ,9 o!r words1 ! fir/ #add odd prime numb]s h actually be5 arrang$ 9 magic =m4 ,! foll[+ summ>y is tak5 f ,! ,moni/ 7,*icago7 = ,octob] #aiac3-- ,ord] ( ,totals ( ,l[e/ ,squ>es ,squ>e4 ,s]ies4 ,con/ants4 made by-- 7,h5ry ,e4 #crd #ccc #aaa ,dud5ey 7 7#aijj74 7,erne/ ,b]gholt #d? #djh #ajb & ,c4 ,d4 7 ,%uldham4 #e? #ajfe #bac ,h4 ,a4 ,sayles4 7,c4 ,d4 ,%uldham #f? #bddh #djh & ,j4 7 ,n4 ,muncey4 #g? #dhic #fii d4 #h? #hiab #aaad d4 #i? #aeabi #afha d4 #aj? #bdafj #bdaf ,j4 ,n4 ,muncey4 #aa? #cfjie #ccee d4 #ab? #edafh #dead d4 ,= fur!r details ! read] %\ld consult ! article itself1 by ,w4 ,s4 ,&rews & ,h4 ,a4 ,sayles4 ,^! same inve/igators h also p]=m$ notable feats 9 con/ruct+ associat$ & bord]$ prime magics1 & ,mr4 ,%uldham has s5t me a rem>kable pap] 9 :i* he gives examples ( ,nasik squ>es con/ruct$ ) primes = all ord]s f ! #d? to ! #aj?1 ) ! exception ( ! #crd 7:i* is cle>ly impossible7 & ! #i?1 :i*1 up to ! "t ( writ+1 has baffl$ all attempts4 #dji4--,,! ,,baskets ,,( ,,plums4 ,illu/ration ,?is is ! =m 9 :i* ,i fir/ introduc$ ! "q ( magic squ>es ) prime numb]s4 ,i w "h w>n ! read] t !re is a little trap4 ,a fruit m]*ant _h n9e baskets4 ,e basket conta9$ plums 7all s\nd & ripe71 & ! numb] 9 e basket 0 diff]5t4 ,:5 plac$ z %[n 9 ! illu/ration !y =m$ a magic squ>e1 s t if he took any ?ree baskets 9 a l9e 9 ! eight possible directions !re w\ld always be ! same numb] ( plums4 ,?is "p ( ! puzzle is easy en\gh to und]/&4 ,b :at foll[s seems at fir/ sight a little que]4 ,! m]*ant told "o ( 8 m5 to di/ribute ! cont5ts ( any basket he *ose among "s *ildr51 giv+ plums to e *ild s t ea* %\ld receive an equal numb]4 ,b ! man f\nd x q impossible1 no matt] :i* basket he select$ & no matt] h[ _m *ildr5 he includ$ 9 ! treat4 ,%[1 by giv+ cont5ts ( ! n9e baskets1 h[ ?is c\ld come ab\t4 #daj4--,,! ,,m&>9',s ,t ,,puzzle4 ,illu/ration ,be=e ,mr4 ,beau*amp ,*olmondely ,m>joribanks set out on 8 t\r 9 ! ,f> ,ea/1 he prid$ himself on 8 k ( magic squ>es1 a subject t he _h made 8 special hobby2 b he soon discov]$ t he _h nev] really t\*$ m ?an ! fr+e ( ! subject1 & t ! wily ,*9ee c\ld beat him easily4 ,i pres5t a little problem t "o le>n$ m&>9 prop\nd$ to our travell]1 z depict$ on ! la/ page4 ,! ,*9aman1 aft] rem>k+ t ! con/ruction ( ! ord9>y magic squ>e ( tw5ty-five cells is too velly mu*ee easy1 ask$ our c\ntryman s to place ! numb]s #a to #be 9 ! squ>e t e column1 e r[1 & ea* ( ! two diagonals %\ld add up #fe1 ) only prime numb]s on ! %ad$ ,t4 ,( c\rse ! prime numb]s available are #a1 #b1 #c1 #e1 #g1 #aa1 #ac1 #ag1 #ai1 & #bc1 s y are at lib]ty to select any n9e ( ^! t w s]ve y\r purpose4 ,c y con/ruct ?is curi\s little magic squ>e8 #daa4--;,a ,,magic ,,squ>e ,,( ,,composites4 ,z we h j discuss$ ! con/ruction ( magic squ>es ) prime numb]s1 ! foll[+ =ms an int]e/+ companion problem4 ,make a magic squ>e ) n9e consecutive composite numb]s--! smalle/ possible4 #dab4--,,! ,,magic ,,knight',s ,,t\r4 ,"h is a problem t has nev] yet be5 solv$1 nor has its impossibility be5 demon/rat$4 ,play ! knight once to e squ>e ( ! *essbo>d 9 a complete t\r1 numb]+ ! squ>es 9 ! ord] visit$1 s t :5 complet$ ! squ>e %all be magic1 add+ up to #bfj 9 e column1 e r[1 & ea* ( ! two long diagonals4 ,i %all give ! be/ answ] t ,i h be5 able to obta91 9 :i* !re is a slight error 9 ! diagonals alone4 ,c a p]fect solution be f\nd8 ,i am conv9c$ t x _c1 b x is only a pi\s op9ion4 ,,mazes ,,& ,,h[ ,,to ,,?read ,,!m4 ,9 w&]+ mazes lo/4 ,p>adise ,lo/4 ,! ,old ,engli% ^w maze1 signify+ a labyr9?1 probably comes f ! ,sc&9avian1 b its orig9 is some:at unc]ta94 ,! late ,pr(essor ,skeat ?\ght t ! sub/antive 0 d]iv$ f ! v]b1 & z 9 old times to be maz$ or amaz$ 0 to be lo/ 9 ?\ght1 ! transition to a maze 9 ^: tortu\s w9d+s we are lo/ is natural & easy4 ,! ^w labyr9? is d]iv$ f a ,greek ^w signify+ ! passages ( a m9e4 ,! anci5t m9es ( ,greece & else:]e inspir$ fe> & awe on acc\nt ( _! d>kness & ! dang] ( gett+ lo/ 9 _! intricate passages4 ,leg5d 0 aft]w>ds built r\nd ^! mazes4 ,! mo/ famili> in/ance is ! labyr9? made by ,ddalus 9 ,crete = ,k+ ,m9os4 ,9 ! c5tre 0 plac$ ! ,m9otaur1 & no "o :o ent]$ c\ld f9d 8 way out aga91 b became ! prey ( ! mon/]4 ,sev5 y\?s & sev5 maid5s 7 s5t regul>ly by ! ,a!nians1 & 7 duly dev\r$1 until ,!seus slew ! mon/] & escap$ f ! maze by aid ( ! clue ( ?read provid$ by ,ariadne2 :i* acc\nts = our us+ to-day ! expression ?read+ a maze4 ,! v>i\s =ms ( con/ruction ( mazes include complicat$ ranges ( cav]ns1 ar*itectural labyr9?s1 or sepul*ral build+s1 tortu\s devices indicat$ by col\r$ m>bles & til$ pavem5ts1 w9d+ pa?s cut 9 ! turf1 & topi>y mazes =m$ by clipp$ h$ges4 ,z a matt] ( fact1 !y may be said to h desc5d$ to u 9 precisely ?is ord] ( v>iety4 ,mazes 7 us$ z ornam5ts on ! /ate robes ( ,*ri/ian emp]ors be=e ! n9? c5tury1 & 7 soon adopt$ 9 ! decoration ( ca!drals & o!r *ur*es4 ,! orig9al idea 0 d\btless to employ !m z symbols ( ! complicat$ folds ( s9 by :i* man is surr\nd$4 ,!y began to ab\nd 9 ! e>ly "p ( ! twelf? c5tury1 & ,i give an illu/ration ( "o ( ?is p]iod 9 ! p>i% *ur* at ,/4 ,qu5t9 7,fig4 #a74 ,x =m$ a pavem5t ( ! nave1 & its diamet] is #cd feet4 ,! pa? "h is ! l9e itself4 ,if y place y\r p5cil at ! po9t ,a & ignore ! enclos+ l9e1 ! l9e leads y to ! c5tre by a long r\te ov] ! entire area2 b y nev] h any option z to direction dur+ y\r c\rse4 ,z we %all f9d 9 simil> cases1 ^! e>ly ecclesia/ical mazes 7 g5]ally n ( a puzzle nature1 b simply long1 w9d+ pa?s t took y ov] practically all ! gr\nd enclos$4 ,illu/ration3 ,,fig4 #a4--,maze at ,/4 ,qu5t94 ,illu/ration3 ,,fig4 #b4--,maze 9 ,*>tres ,ca!dral4 ,9 ! abbey *ur* ( ,/4 ,b]l91 at ,/4 ,om]1 is ano!r ( ^! curi\s floors1 repres5t+ ! ,temple ( ,j]usalem1 ) /ations = pilgrims4 ,^! mazes 7 actually visit$ & trav]s$ by !m z a compromise = n go+ to ! ,holy ,l& 9 fulfilm5t ( a v[4 ,!y 7 also us$ z a means ( p5ance1 ! p5it5t frequ5tly be+ direct$ to g ! :ole c\rse ( ! maze on h&s & knees4 ,illu/ration3 ,,fig4 #c4--,maze 9 ,lucca ,ca!dral4 ,! maze 9 ,*>tres ,ca!dral1 ( :i* ,i give an illu/ration 7,fig4 #b71 is #dj feet across1 & 0 us$ by p5it5ts foll[+ ! procession ( ,calv>y4 ,a labyr9? 9 ,ami5s ,ca!dral 0 octagonal1 simil> to t at ,/4 ,qu5t91 measur+ #db feet across4 ,x bore ! date #abhh1 b 0 de/roy$ 9 #agjh4 ,9 ! *apt]-h\se at ,bayeux is a labyr9? =m$ ( tiles1 r$1 black1 & encau/ic1 ) a patt]n ( br[n & yell[4 ,dr4 ,duc>el1 9 8 ,t\r ?r\gh ,"p ( ,norm&y 7pr9t$ 9 #agfg71 m5tions ! floor ( ! great gu>d-*amb] 9 ! abbey ( ,/4 ,/eph51 at ,ca51 ! middle :]e( repres5ts a maze or labyr9? ab\t #aj feet diamet]1 & s artfully contriv$ t1 7 we to suppose a man foll[+ all ! intricate me&]s ( its volutes1 he c\ld n travel less ?an a mile be=e he got f "o end to ! o!r4 ,illu/ration3 ,,fig4 #d4--,maze at ,saffron ,wald51 ,essex4 ,!n ^! mazes 7 sometimes r$uc$ 9 size & repres5t$ on a s+le tile 7,fig4 #c74 ,i give an example f ,lucca ,ca!dral4 ,x is on "o ( ! por* pi]s1 & is #ai in*es 9 diamet]4 ,a writ] 9 #aheh says t1 f ! cont9ual attrition x has receiv$ f ?\s&s ( trac+ f+]s1 a c5tral gr\p ( ,!seus & ! ,m9otaur has n[ be5 v ne>ly effac$4 ,o!r examples 71 & p]haps /ill are1 to be f\nd 9 ! ,abbey ( ,t\ss>ts1 at ,*lons-sur-,m>ne1 9 ! v anci5t *ur* ( ,/4 ,mi*ele at ,pavia1 at ,aix 9 ,prov5ce1 9 ! ca!drals ( ,poiti]s1 ,rheims1 & ,arras1 9 ! *ur* ( ,santa ,m>ia 9 ,aquiro 9 ,rome1 9 ,san ,vitale at ,rav5na1 9 ! ,roman mosaic pavem5t f\nd at ,salzburg1 & else:]e4 ,^! mazes 7 sometimes call$ ,*em9s de ,j]usalem1 z be+ emblematical ( ! difficulties att5d+ a j\rney to ! e>?ly ,j]usalem & ( ^? enc\nt]$ by ! ,*ri/ian be=e he c rea* ! heav5ly ,j]usalem--:]e ! c5tre 0 frequ5tly call$ ,ciel4 ,common z ^! mazes 7 ^u ! ,cont95t1 x is probable t no example is to be f\nd 9 any ,engli% *ur*2 at lea/ ,i am n aw>e ( ! exi/5ce ( any4 ,b almo/ e c\nty has1 or has _h1 its specim5s ( mazes cut 9 ! turf4 ,?\gh ^! are frequ5tly kn[n z miz-mazes or mize-mazes1 x is n uncommon to f9d !m locally call$ ,troy-t[ns1 %eph]ds' races1 or ,julian's ,b[]s--names t are mislead+1 z sugge/+ a false orig94 ,f ! facts alone t _m ( ^! ,engli% turf mazes are cle>ly copi$ f ^? 9 ! ,cont95tal *ur*es1 & practically all are f\nd close to "s ecclesia/ical build+ or ne> ! site ( an anci5t "o1 we may reg>d x z c]ta9 t !y 7 ( *ur* orig9 & n inv5t$ by ! %eph]ds or o!r ru/ics4 ,& curi\sly en\gh1 ^! turf mazes are app>5tly unkn[n on ! ,cont95t4 ,!y are di/9ctly m5tion$ by ,%akespe>e3-- ,! n9e m5's morris is fill$ up ) mud1 ,& ! qua9t mazes 9 ! wanton gre5 ,= lack ( tread are undi/+ui%able4 ,a ,midsumm] ,night's ,dream1 ii4 #a4 ,my old bones a*e3 "h's a maze trod inde$1 ,?r\gh =?-rights & me&]s6 ,! ,tempe/1 iii4 #c4 ,illu/ration3 ,,fig4 #e4--,maze at ,sne9ton1 ,nott+ham%ire4 ,!re 0 su* a maze at ,comb]ton1 9 ,cambridge%ire1 & ano!r1 locally call$ ! miz-maze1 at ,leigh1 9 ,dorset4 ,! latt] 0 on ! highe/ "p ( a field on ! top ( a hill1 a qu>t] ( a mile f ! village1 & 0 slightly holl[ 9 ! middle & enclos$ by a bank ab\t #c feet high4 ,x 0 circul>1 & 0 ?irty paces 9 diamet]4 ,9 #ahfh ! turf _h gr[n ov] ! little tr5*es1 & x 0 !n impossible to trace ! pa?s ( ! maze4 ,! ,comb]ton "o 0 at ! same date believ$ to be p]fect1 b :e!r ei!r or bo? h n[ disappe>$ ,i _c say4 ,nor h ,i be5 able to v]ify ! exi/5ce or non-exi/5ce ( ! o!r examples ( :i* ,i am able to give illu/rations4 ,i %all !re=e write ( !m all 9 ! pa/ t5se1 reta9+ ! hope t "s are /ill pres]v$4 ,illu/ration3 ,,fig4 #f4--,maze at ,alkbor\gh1 ,l9coln%ire4 ,9 ! next two mazes giv5--t at ,saffron ,wald51 ,essex 7#aaj feet 9 diamet]1 ,fig4 #d71 & ! "o ne> ,/4 ,anne's ,well1 at ,sne9ton1 ,nott+ham%ire 7,fig4 #e71 :i* 0 pl\gh$ up on ,febru>y #bg?1 #agig 7#ea feet 9 diamet]1 ) a pa? #ece y>ds long7--! pa?s mu/ 9 ea* case be und]/ood to be on ! l9es1 black or :ite1 z ! case may be4 ,illu/ration3 ,,fig4 #g4--,maze at ,b\ghton ,gre51 ,nott+ham%ire4 ,i give 9 ,fig4 #f a maze t 0 at ,alkbor\gh1 ,l9coln%ire1 ov]look+ ! ,humb]4 ,?is 0 #dd feet 9 diamet]1 & ! resemblance betwe5 x & ! mazes at ,*>tres & ,lucca 7,figs4 #b & #c7 w be at once p]ceiv$4 ,a maze at ,b\ghton ,gre51 9 ,nott+ham%ire1 a place celebrat$ at "o "t = its fair 7,fig4 #g71 0 #cg feet 9 diamet]4 ,i also include ! plan 7,fig4 #h7 ( "o t us$ to be on ! outskirts ( ! village ( ,w+1 ne> ,upp+ham1 ,rutl&%ire4 ,?is maze 0 #dj feet 9 diamet]4 ,illu/ration3 ,,fig4 #h4--,maze at ,w+1 ,rutl&%ire4 ,illu/ration3 ,,fig4 #i4--,maze on ,/4 ,ca!r9e's ,hill1 ,w9*e/]4 ,! maze t 0 on ,/4 ,ca!r9e's ,hill1 ,w9*e/]1 9 ! p>i% ( ,*ilcombe1 0 a poor specim5 7,fig4 #i71 s9ce1 z w be se51 !re 0 "o %ort direct r\te to ! c5tre1 unless1 z 9 ,fig4 #aj aga91 ! pa? is ! l9e itself f end to end4 ,?is maze 0 #hf feet squ>e1 cut 9 ! turf1 & 0 locally kn[n z ! ,mize-maze4 ,x became v indi/9ct ab\t #aheh1 & 0 !n recut by ! ,w>d5 ( ,w9*e/]1 ) ! aid ( a plan possess$ by a lady liv+ 9 ! neighb\rhood4 ,illu/ration3 ,,fig4 #aj4--,maze on ,ripon ,common4 ,a maze =m]ly exi/$ on ,ripon ,common1 9 ,york%ire 7,fig4 #aj74 ,x 0 pl\gh$ up 9 #ahbg1 b its plan 0 =tunately pres]v$4 ,?is example 0 #bj y>ds 9 diamet]1 & its pa? is said to h be5 #djg y>ds long4 ,illu/ration3 ,,fig4 #aa4--,maze at ,!obalds1 ,h]t=d%ire4 ,9 ! case ( ! maze at ,!obalds1 ,h]t=d%ire1 aft] y h f\nd ! entrance )9 ! f\r enclos+ h$ges1 ! pa? is =c$ 7,fig4 #aa74 ,z fur!r illu/rations ( ?is class ( maze1 ,i give "o tak5 f an ,italian "w on ar*itecture by ,s]lio1 publi%$ 9 #aecg 7,fig4 #ab71 & "o by ,london & ,wise1 ! design]s ( ! ,hampton ,c\rt maze1 f _! book1 ,! ,retir$ ,g>d'n]1 publi%$ 9 #agjf 7,fig4 #ac74 ,also1 ,i add a ,dut* maze 7,fig4 #ad74 ,illu/ration3 ,,fig4 #ab4--;,italian ,maze ( ,sixte5? ,c5tury4 ,illu/ration3 ,,fig4 #ac4--;,by ! ,design]s ( ,hampton ,c\rt ,maze4 ,illu/ration3 ,,fig4 #ad4--;,a ,dut* ,maze4 ,s f> our mazes h be5 ( hi/orical int]e/1 b !y h pres5t$ no difficulty 9 ?read+4 ,aft] ! ,re=mation p]iod we f9d mazes conv]t$ into m$iums = recreation1 & !y g5]ally consi/$ ( labyr9?9e pa?s enclos$ by ?ick & c>efully trimm$ h$ges4 ,^! topi>y h$ges 7 kn[n to ! ,romans1 ) :om ! topi>ius 0 ! ornam5tal g>d5]4 ,?is type ( maze has ( late ye>s deg5]at$ into ! seaside ,puzzle ,g>d5s4 ,teas1 sixp5ce1 includ+ admission to ! ,maze4 ,! ,hampton ,c\rt ,maze1 sometimes call$ ! ,wild]ness1 at ! royal palace1 0 design$1 z ,i h said1 by ,london & ,wise = ,william ,,iii41 :o _h a lik+ = su* ?+s 7,fig4 #ae74 ,i h be=e me "s ?ree or f\r v]sions ( x1 all slightly diff]5t f "o ano!r2 b ! plan ,i select is tak5 f an old guide-book to ! palace1 & !re=e ought to be tru/wor?y4 ,! mean+ ( ! dott$ l9es1 etc41 w be expla9$ lat] on4 ,illu/ration3 ,,fig4 #ae4--,maze at ,hampton ,c\rt ,palace4 ,illu/ration3 ,,fig4 #af4--,maze at ,hatfield ,h\se1 ,h]ts4 ,illu/ration3 ,,fig4 #ag4--,maze =m]ly at ,s\? ,k5s+ton4 ,illu/ration3 ,,fig4 #ah4--;,a ,g]man ,maze4 ,! maze at ,hatfield ,h\se 7,fig4 #af71 ! seat ( ! ,m>quis ( ,salisbury1 l s _m labyr9?s1 is n difficult on pap]2 b bo? ?is & ! ,hampton ,c\rt ,maze may prove v puzzl+ to actually ?read )\t kn[+ ! plan4 ,"o reason is t "o is s apt to g d[n ! same bl9d alleys ov] & ov] aga91 if "o proce$s )\t me?od4 ,! maze plann$ by ! desire ( ! ,pr9ce ,consort = ! ,royal ,horticultural ,society's ,g>d5s at ,s\? ,k5s+ton 0 all[$ to g to ru91 & 0 !n de/roy$--no great loss1 = x 0 a feeble ?+4 ,x w be se5 t !re 7 ?ree entrances f ! outside 7,fig4 #ag71 b ! way to ! c5tre is v easy to discov]4 ,i include a ,g]man maze t is curi\s1 b n difficult to ?read on pap] 7,fig4 #ah74 ,! example ( a labyr9? =m]ly exi/+ at ,pimp]ne1 9 ,dorset1 is 9 a class by itself 7,fig4 #ai74 ,x 0 =m$ ( small ridges ab\t a foot high1 & cov]$ ne>ly an acre ( gr\nd2 b x 01 un=tunately1 pl\gh$ up 9 #agcj4 ,illu/ration3 ,,fig4 #ai4--,maze at ,pimp]ne1 ,dorset4 ,we w n[ pass to ! int]e/+ subject ( h[ to ?read any maze4 ,:ile be+ necess>ily brief1 ,i w try to make ! matt] cle> to read]s :o h no k ( ma!matics4 ,& fir/ ( all we w assume t we are try+ to ent] a maze 7t is1 get to ! c5tre7 ( :i* we h no plan & ab\t :i* we "k no?+4 ,! fir/ rule is ?is3 ,if a maze has no p>ts ( its h$ges deta*$ f ! re/1 !n if we always keep 9 t\* ) ! h$ge ) ! "r h& 7or always t\* x ) ! left71 go+ d[n to ! /op 9 e bl9d alley & com+ back on ! o!r side1 we %all pass ?r\gh e "p ( ! maze & make our exit :]e we w5t 94 ,!re=e we mu/ at "o "t or ano!r ent] ! c5tre1 & e alley w be trav]s$ twice4 ,illu/ration3 ,,fig4 #bj4--,m4 ,tremaux's ,me?od ( ,solution4 ,illu/ration3 ,,fig4 #ba4--;,h[ to ?read ! ,hatfield ,maze4 ,n[ look at ! ,hampton ,c\rt plan4 ,foll[1 say to ! "r1 ! pa? indicat$ by ! dott$ l9e1 & :at ,i h said is cle>ly correct if we oblit]ate ! two deta*$ p>ts1 or isl&s1 situat$ on ea* side ( ! />4 ,b z ^! isl&s are !re1 y _c by ?is me?od trav]se e "p ( ! maze2 & if x _h be5 s plann$ t ! c5tre 01 l ! />1 betwe5 ! two isl&s1 y w\ld nev] pass ?r\gh ! c5tre at all4 ,a glance at ! ,hatfield maze w %[ t !re are ?ree ( ^! deta*$ h$ges or isl&s at ! c5tre1 s ?is me?od w nev] take y to ! c5tre ( t "o4 ,b ! rule w at lea/ always br+ y safely out aga9 unless y blund] 9 ! foll[+ way4 ,suppose1 :5 y 7 go+ 9 ! direction ( ! arr[ 9 ! ,hampton ,c\rt ,maze1 t y c\ld n di/9ctly see ! turn+ at ! bottom1 t y imag9$ y 7 9 a bl9d alley &1 to save "t1 cross$ at once to ! opposite h$ge1 !n y w\ld g r\nd & r\nd t ,u-%ap$ isl& ) y\r "r h& /ill always on ! h$ge--= ev] aft]6 ,illu/ration3 ,,fig4 #bb4 ,! ,philadelphia ,maze1 & its ,solution4 ,?is blund] happ5$ to me a few ye>s ago 9 a little maze on ! isle ( ,caldy1 ,s\? ,wales4 ,i knew ! maze 0 a small "o1 b aft] a v long walk ,i 0 amaz$ to f9d t ,i did n ei!r rea* ! c5tre or get out aga94 ,s ,i ?rew a piece ( pap] on ! gr\nd1 & soon came r\nd to x2 f :i* ,i knew t ,i _h blund]$ ov] a suppos$ bl9d alley & 0 go+ r\nd & r\nd an isl&4 ,cross+ to ! opposite h$ge & us+ m c>e1 ,i 0 quickly at ! c5tre & out aga94 ,n[1 if ,i _h made a simil> mi/ake at ,hampton ,c\rt1 & discov]$ ! error :5 at ! />1 ,i %\ld m]ely h pass$ f "o isl& to ano!r6 ,& if ,i _h aga9 discov]$ t ,i 0 on a deta*$ "p1 ,i might ) ill luck h recross$ to ! fir/ isl& aga96 ,we ?us see t ?is t\*+ ! h$ge me?od %\ld always br+ u safely out ( a maze t we h ent]$2 x may happ5 to take u ?r\gh ! c5tre1 & if we miss ! c5tre we %all "k !re mu/ be isl&s4 ,b x has to be done ) a little c>e1 & 9 no case c we be sure t we h trav]s$ e alley or t !re are no deta*$ p>ts4 ,illu/ration3 ,,fig4 #bc4--,simplifi$ ,diagram ( ,fig4 #bb4 ,if ! maze has _m isl&s1 ! trav]s+ ( ! :ole ( x may be a matt] ( consid]able difficulty4 ,"h is a me?od = solv+ any maze1 due to ,m4 ,trmaux1 b x necessitates c>efully m>k+ 9 "s way y\r entrances & exits :]e ! gall]ies =k4 ,i give a diagram ( an imag9>y maze ( a v simple "* t w s]ve our purpose j z well z some?+ m complex 7,fig4 #bj74 ,! circles at ! regions :]e we h a *oice ( turn+s we may call nodes4 ,a new pa? or node is "o t has n be5 ent]$ be=e on ! r\te2 an old pa? or node is "o t has already be5 ent]$1 #a4 ,no pa? may be trav]s$ m ?an twice4 #b4 ,:5 y come to a new node1 take any pa? y l4 #c4 ,:5 by a new pa? y come to an old node or to ! /op ( a bl9d alley1 return by ! pa? y came4 #d4 ,:5 by an old pa? y come to an old node1 take a new pa? if !re is "o2 if n1 an old pa?4 ,! r\te indicat$ by ! dott$ l9e 9 ! diagram is tak5 9 accordance ) ^! simple rules1 & x w be se5 t x leads u to ! c5tre1 al?\gh ! maze consi/s ( f\r isl&s4 ,illu/ration3 ,,fig4 #bd4--;,c y f9d ! ,%orte/ ,way to ,c5tre8 ,nei!r ( ! me?ods ,i h giv5 w disclose to u ! %orte/ way to ! c5tre1 nor ! numb] ( ! diff]5t r\tes4 ,b we c easily settle ^! po9ts ) a plan4 ,let u take ! ,hatfield maze 7,fig4 #ba74 ,x w be se5 t ,i h suppress$ all ! bl9d alleys by ! %ad+4 ,i beg9 at ! /op & "w backw>ds until ! pa? =ks4 ,^! %ad$ p>ts1 !re=e1 c nev] be ent]$ )\t our hav+ to retrace our /eps4 ,!n x is v cle>ly se5 t if we ent] at ,a we mu/ come out at ,b2 if we ent] at ,c we mu/ come out at ,d4 ,!n we h m]ely to det]m9e :e!r ,a1 ,b1 ,e1 or ,c1 ,d1 ,e1 is ! %ort] r\te4 ,z a matt] ( fact1 x w be f\nd by r\gh measurem5t or calculation t ! %orte/ r\te to ! c5tre is by way ( ,c1 ,d1 ,e1 ,f4 ,illu/ration3 ,,fig4 #be4--,rosamund's ,b[]4 ,i w n[ give ?ree mazes t are simply puzzles on pap]1 =1 s f> z ,i "k1 !y h nev] be5 con/ruct$ 9 any o!r way4 ,! fir/ ,i w call ! ,philadelphia maze 7,fig4 #bb74 ,f\rte5 ye>s ago a travell+ salesman1 liv+ 9 ,philadelphia1 ,u4,s4,a41 develop$ a curi\sly unre/ra9$ passion = puzzles4 ,he neglect$ 8 bus9ess1 & soon 8 position 0 tak5 f him4 ,8 days & nights 7 n[ pass$ ) ! subject t fasc9at$ him1 & ?is little maze seems to h driv5 him into insanity4 ,he _h be5 puzzl+ ov] x = "s "t1 & f9ally x s5t him mad & caus$ him to fire a bullet ?r\gh 8 bra94 ,goodness kn[s :at 8 difficulties c\ld h be56 ,b !re c be little d\bt t he _h a disord]$ m9d1 & t if ?is little puzzle _h n caus$ him to lose 8 m5tal balance "s o!r m or less trivial ?+ w\ld 9 "t h done s4 ,!re is no moral 9 ! /ory1 unless x be t ( ! ,iri% maxim1 :i* applies to e occupation ( life z mu* z to ! solv+ ( puzzles3 ,take ?+s aisy2 if y c't take !m aisy1 take !m z aisy z y c4 ,& x is a bad & empirical way ( solv+ any puzzle--by bl[+ y\r bra9s out4 ,n[1 h[ _m diff]5t r\tes are !re f ,a to ,b 9 ?is maze if we mu/ nev] 9 any r\te g along ! same passage twice8 ,! f\r op5 spaces :]e f\r passages end are n reckon$ z passages4 ,9 ! diagram 7,fig4 #bb7 x w be se5 t ,i h aga9 suppress$ ! bl9d alleys4 ,x w be f\nd t1 9 any case1 we mu/ g f ,a to ,c1 & also f ,f to ,b4 ,b :5 we h arriv$ at ,c !re are ?ree ways1 m>k$ #a1 #b1 #c1 ( gett+ to ,d4 ,simil>ly1 :5 we get to ,e !re are ?ree ways1 m>k$ #d1 #e1 #f1 ( gett+ to ,f4 ,we h also ! dott$ r\te f ,c to ,e1 ! o!r dott$ r\te f ,d to ,f1 & ! passage f ,d to ,e1 indicat$ by />s4 ,we c1 !re=e1 express ! position ( affairs by ! little diagram annex$ 7,fig4 #bc74 ,"h e condition ( r\te exactly corresponds to t 9 ! circul> maze1 only x is mu* less confus+ to ! eye4 ,n[1 ! numb] ( r\tes1 "u ! conditions1 f ,a to ,b on ?is simplifi$ diagram is #fdj1 & t is ! requir$ answ] to ! maze puzzle4 ,f9ally1 ,i w leave two easy maze puzzles 7,figs4 #bd1 #be7 = my read]s to solve = !mselves4 ,! puzzle 9 ea* case is to f9d ! %orte/ possible r\te to ! c5tre4 ,ev]ybody kn[s ! /ory ( ,fair ,rosamund & ! ,wood/ock maze4 ,:at ! maze 0 l or :e!r x ev] exi/$ except 9 imag9ation is n kn[n1 _m writ]s believ+ t x 0 simply a badly-con/ruct$ h\se ) a l>ge numb] ( confus+ rooms & passages4 ,at any rate1 my sket* lacks ! au?ority ( ! o!r mazes 9 ?is article4 ,my ,rosamund's ,b[] is simply design$ to %[ t :]e y h ! plan be=e y x (t5 happ5s t ! easie/ way to f9d a r\te into a maze is by work+ backw>ds & fir/ f9d+ a way out4 ,,! ,,p>adox ,,p>ty4 ,is n life itself a p>adox8 ,c4,l4 ,,dodgson1 ,pill[ ,problems4 ,x is a wond]ful age6 said ,mr4 ,allgood1 & ev]ybody at ! table turn$ t[>ds him & assum$ an attitude ( expectancy4 ,?is 0 an ord9>y ,*ri/mas d9n] ( ! ,allgood family1 ) a spr9kl+ ( local fri5ds4 ,nobody w\ld h suppos$ t ! above rem>k w\ld lead1 z x did1 to a succession ( curi\s puzzles & p>adoxes1 to :i* e memb] ( ! p>ty contribut$ some?+ ( int]e/4 ,! little symposium 0 q unprem$itat$1 s we mu/ n be too critical respect+ a few ( ! pos]s t 7 =?com+4 ,! v>i$ "* ( ! contributions is j :at we w\ld expect on su* an occasion1 = x 0 a ga!r+ n ( exp]t ma!maticians & logicians1 b ( q ord9>y folk4 ,x is a wond]ful age6 repeat$ ,mr4 ,allgood4 ,a man has j design$ a squ>e h\se 9 su* a cunn+ mann] t all ! w9d[s on ! f\r sides h a s\? aspect4 ,t w\ld appeal to me1 said ,mrs4 ,allgood1 = ,i _c endure a room ) a nor? aspect4 ,i _c conceive h[ x is done1 ,uncle ,john confess$4 ,i suppose he puts bay w9d[s on ! ea/ & we/ sides2 b h[ on e>? c be contrive to look s\? f ! nor? side8 ,does he use mirrors1 or some?+ ( t k9d8 ,no1 repli$ ,mr4 ,allgood1 no?+ ( ! sort4 ,all ! w9d[s are flu% ) ! walls1 & yet y get a s\!rly prospect f e "o ( !m4 ,y see1 !re is no real difficulty 9 design+ ! h\se if y select ! prop] spot = its erection4 ,n[1 ?is h\se is design$ = a g5tleman :o proposes to build x exactly at ! ,nor? ,pole4 ,if y ?9k a mom5t y w realize t :5 y /& at ! ,nor? ,pole x is impossible1 no matt] :i* way y may turn1 to look else:]e ?an due s\?6 ,!re are no su* directions z nor?1 ea/1 or we/ :5 y are exactly at ! ,nor? ,pole4 ,ev]y?+ is due s\?6 ,i am afraid1 "m1 said h] son ,george1 aft] ! laught] _h subsid$1 t1 h[ev] mu* y might l ! aspect1 ! situation w\ld be a little too brac+ = y4 ,ah1 well6 %e repli$4 ,y\r ,uncle ,john fell also into ! trap4 ,i am no good at cat*es & puzzles4 ,i suppose ,i hav5't ! "r sort ( bra94 ,p]haps "s "o w expla9 ?is to me4 ,only la/ week ,i rem>k$ to my hairdress] t x _h be5 said t !re are m p]sons 9 ! _w ?an any "o ( !m has hairs on 8 head4 ,he repli$1 ',!n x foll[s1 madam1 t two p]sons1 at lea/1 mu/ h exactly ! same numb] ( hairs on _! heads4' ,if ?is is a fact1 ,i confess ,i _c see x4 ,h[ d ! bald-head$ affect ! "q8 ask$ ,uncle ,john4 ,if !re are su* p]sons 9 exi/5ce1 repli$ ,mrs4 ,allgood1 :o hav5't a solit>y hair on _! heads discov]able "u a magnify+-glass1 we w leave !m out ( ! "q4 ,/ill1 ,i don't see h[ y are to prove t at lea/ two p]sons h exactly ! same numb] to a hair4 ,i ?9k ,i c make x cle>1 said ,mr4 ,filk9s1 :o _h dropp$ 9 = ! ev5+4 ,assume ! population ( ! _w to be only "o million4 ,any numb] w d z well z ano!r4 ,!n y\r /atem5t 0 to ! effect t no p]son has m ?an n9e hundr$ & n9ety-n9e ?\s& n9e hundr$ & n9ety-n9e hairs on 8 head4 ,is t s8 ,let me ?9k1 said ,mrs4 ,allgood4 ,yes--yes--t is correct4 ,v well1 !n4 ,z !re are only n9e hundr$ & n9ety-n9e ?\s& n9e hundr$ & n9ety-n9e diff]5t ways ( be>+ hair1 x is cle> t ! million? p]son mu/ repeat "o ( ^? ways4 ,d y see8 ,yes2 ,i see t--at lea/ ,i ?9k ,i see x4 ,!re=e two p]sons at lea/ mu/ h ! same numb] ( hairs on _! heads2 & z ! numb] ( p on ! e>? s greatly exce$s ! numb] ( hairs on any "o p]son's head1 !re mu/1 ( c\rse1 be an imm5se numb] ( ^! repetitions4 ,b1 ,mr4 ,filk9s1 said little ,willie ,allgood1 :y c\ld n ! million? man h1 say1 t5 ?\s& hairs & a half8 ,t is m]e hair-splitt+1 ,willie1 & does n come into ! "q4 ,"h is a curi\s p>adox1 said ,george4 ,if a ?\s& soldi]s are drawn up 9 battle array on a plane--!y und]/ood him to mean pla9--only "o man w /& upright4 ,nobody c\ld see :y4 ,b ,george expla9$ t1 accord+ to ,euclid1 a plane c t\* a sph]e only at "o po9t1 & t p]son only :o /&s at t po9t1 ) respect to ! c5tre ( ! e>?1 w /& upright4 ,9 ! same way1 he rem>k$1 if a billi>d-table 7 q level--t is1 a p]fect plane--! balls ought to roll to ! c5tre4 ,?\gh he tri$ to expla9 ?is by plac+ a visit+-c>d on an orange & exp\nd+ ! law ( gravitation1 ,mrs4 ,allgood decl9$ to accept ! /atem5t4 ,%e c\ld n see t ! top ( a true billi>d-table mu/1 !oretically1 be sph]ical1 j l a portion ( ! orange-peel t ,george cut out4 ,( c\rse1 ! table is s small 9 proportion to ! surface ( ! e>? t ! curvature is n appreciable1 b x is nev]!less true 9 !ory4 ,a surface t we call level is n ! same z our idea ( a true geometrical plane4 ,uncle ,john1 broke 9 ,willie ,allgood1 !re is a c]ta9 isl& situat$ betwe5 ,engl& & ,france1 & yet t isl& is f>!r f ,france ?an ,engl& is4 ,:at is ! isl&8 ,t seems absurd1 my boy2 because if ,i place ?is tumbl]1 to repres5t ! isl&1 betwe5 ^! two plates1 x seems impossible t ! tumbl] c be f>!r f ei!r ( ! plates ?an !y are f ea* o!r4 ,b isn't ,gu]nsey betwe5 ,engl& & ,france8 ask$ ,willie4 ,yes1 c]ta9ly4 ,well1 !n1 ,i ?9k y w f9d1 uncle1 t ,gu]nsey is ab\t tw5ty-six miles f ,france1 & ,engl& is only tw5ty-"o miles f ,france1 betwe5 ,calais & ,dov]4 ,my ma!matical ma/]1 said ,george1 has be5 try+ to induce me to accept ! axiom t 'if equals be multipli$ by equals ! products are equal4' ,x is self-evid5t1 po9t$ out ,mr4 ,filk9s4 ,= example1 if #c feet equal #a y>d1 !n twice #c feet w equal #b y>ds4 ,d y see8 ,b1 ,mr4 ,filk9s1 ask$ ,george1 is ?is tumbl] half full ( wat] equal to a simil> glass half empty8 ,c]ta9ly1 ,george4 ,!n x foll[s f ! axiom t a glass full mu/ equal a glass empty4 ,is t correct8 ,no1 cle>ly n4 ,i nev] ?\ght ( x 9 t light4 ,p]haps1 sugge/$ ,mr4 ,allgood1 ! rule does n apply to liquids4 ,j :at ,i 0 ?9k+1 ,allgood4 ,x w\ld seem t we mu/ make an exception 9 ! case ( liquids4 ,b x w\ld be awkw>d1 said ,george1 ) a smile1 if we also _h to except ! case ( solids4 ,= in/ance1 let u take ! solid e>?4 ,"o mile squ>e equals "o squ>e mile4 ,!re=e two miles squ>e mu/ equal two squ>e miles4 ,is ?is s8 ,well1 let me see6 ,no1 ( c\rse n1 ,mr4 ,filk9s repli$1 because two miles squ>e is f\r squ>e miles4 ,!n1 said ,george1 if ! axiom is n true 9 ^! cases1 :5 is x true8 ,mr4 ,filk9s promis$ to look into ! matt]1 & p]haps ! read] w also l to give x consid]ation at leisure4 ,look "h1 ,george1 said 8 c\s9 ,reg9ald ,woolley3 by :at fractional "p does f\r-f\r?s exce$ ?ree-f\r?s8 ,by "o-f\r?6 %\t$ ev]ybody at once4 ,try ano!r "o1 ,george sugge/$4 ,) pleasure1 :5 y h answ]$ t "o correctly1 0 ,reg9ald's reply4 ,d y mean to say t x isn't "o-f\r?8 ,c]ta9ly ,i d4 ,sev]al memb]s ( ! company fail$ to see t ! correct answ] is "o-?ird1 al?\gh ,reg9ald tri$ to expla9 t ?ree ( any?+1 if increas$ by "o-?ird1 becomes f\r4 ,uncle ,john1 h[ d y pron\nce 't-o-o'8 ask$ ,willie4 ',too1 my boy4 ,& h[ d y pron\nce 't-w-o'8 ,t is also 'too4' ,!n h[ d y pron\nce ! second day ( ! week8 ,well1 t ,i %\ld pron\nce ',tuesday1' n ',toosday4' ,w\ld y really8 ,i %\ld pron\nce x ',monday4' ,if y g on l ?is1 ,willie1 said ,uncle ,john1 ) mock sev]ity1 y w soon be )\t a fri5d 9 ! _w4 ,c any ( y write d[n quickly 9 figures 'twelve ?\s& twelve hundr$ & twelve p\nds'8 ask$ ,mr4 ,allgood4 ,8 elde/ daught]1 ,miss ,mildr$1 0 ! only p]son :o happ5$ to h a p5cil at h&4 ,x c't be done1 %e decl>$1 aft] mak+ an attempt on ! :ite table-clo?2 b ,mr4 ,allgood %[$ h] t x %\ld be writt51 #ac1bab4 ,n[ x is my turn1 said ,mildr$4 ,i h be5 wait+ to ask y all a "q4 ,9 ! ,massacre ( ! ,innoc5ts "u ,h]od1 a numb] ( poor little *ildr5 7 buri$ 9 ! s& ) only _! feet /ick+ out4 ,h[ might y di/+ui% ! boys f ! girls8 ,i suppose1 said ,mrs4 ,allgood1 x is a conundrum--some?+ to d ) _! poor little 's\ls4' ,b aft] ev]ybody _h giv5 x up1 ,mildr$ rem9d$ ! company t only boys 7 put to dea?4 ,once ^u a "t1 began ,george1 ,a*illes _h a race ) a tortoise-- ,/op1 ,george6 int]pos$ ,mr4 ,allgood4 ,we won't h t "o4 ,i knew two m5 9 my y\? :o 7 once ! be/ ( fri5ds1 b !y qu>rell$ ov] t inf]nal ?+ ( ,z5o's1 & !y nev] spoke to "o ano!r aga9 = ! re/ ( _! lives4 ,i draw ! l9e at t1 & ! o!r /upid ?+ by ,z5o ab\t ! fly+ arr[4 ,i don't believe anybody und]/&s !m1 because ,i c\ld nev] d s myself4 ,oh1 v well1 !n1 fa!r4 ,"h is ano!r4 ,! ,po/-,(fice p 7 ab\t to erect a l9e ( telegraph-po/s ov] a high hill f ,turmitville to ,wurzleton2 b z x 0 f\nd t a railway company 0 mak+ a deep level cutt+ 9 ! same direction1 !y arrang$ to put up ! po/s beside ! l9e4 ,n[1 ! po/s 7 to be a hundr$ y>ds ap>t1 ! l5g? ( ! road ov] ! hill be+ five miles1 & ! l5g? ( ! level cutt+ only f\r & a half miles4 ,h[ _m po/s did !y save by erect+ !m on ! level8 ,t is a v simple matt] ( calculation1 said ,mr4 ,filk9s4 ,f9d h[ _m times "o hundr$ y>ds w g 9 five miles1 & h[ _m times 9 f\r & a half miles4 ,!n d$uct "o f ! o!r1 & y h ! numb] ( po/s sav$ by ! %ort] r\te4 ,q "r1 confirm$ ,mr4 ,allgood4 ,no?+ c\ld be easi]4 ,t is j :at ! ,po/-,(fice p said1 repli$ ,george1 b x is q wrong4 ,if y look at ?is sket* t ,i h j made1 y w see t !re is no diff]5ce :atev]4 ,if ! po/s are a hundr$ y>ds ap>t1 j ! same numb] w be requir$ on ! level z ov] ! surface ( ! hill4 ,illu/ration ,surely y mu/ be wrong1 ,george1 said ,mrs4 ,allgood1 = if ! po/s are a hundr$ y>ds ap>t & x is half a mile f>!r ov] ! hill1 y h to put up po/s on t extra half-mile4 ,look at ! diagram1 "m4 ,y w see t ! di/ance f po/ to po/ is n ! di/ance f base to base measur$ along ! gr\nd4 ,i am j ! same di/ance f y if ,i /& on ?is spot on ! c>pet or /& imm$iately above x on ! *air4 ,b ,mrs4 ,allgood 0 n conv9c$4 ,mr4 ,smoo?ly1 ! curate1 at ! end ( ! table1 said at ?is po9t t he _h a little "q to ask4 ,suppose ! e>? 7 a p]fect sph]e ) a smoo? surface1 & a girdle ( /eel 7 plac$ r\nd ! ,equator s t x t\*$ at e po9t4 ',i'll put a girdle r\nd ab\t ! e>? 9 =ty m9utes1' mutt]$ ,george1 quot+ ! words ( ,puck 9 ,a ,midsumm] ,night's ,dream4 ,n[1 if six y>ds 7 add$ to ! l5g? ( ! girdle1 :at w\ld !n be ! di/ance betwe5 ! girdle & ! e>?1 suppos+ t di/ance to be equal all r\nd8 ,9 su* a great l5g?1 said ,mr4 ,allgood1 ,i d n suppose ! di/ance w\ld be wor? m5tion+4 ,:at d y say1 ,george8 ask$ ,mr4 ,smoo?ly4 ,well1 )\t calculat+ ,i %\ld imag9e x w\ld be a v m9ute fraction ( an in*4 ,reg9ald & ,mr4 ,filk9s 7 ( ! same op9ion4 ,i ?9k x w surprise y all1 said ! curate1 to le>n t ^? extra six y>ds w\ld make ! di/ance f ! e>? all r\nd ! girdle v ne>ly a y>d6 ,v ne>ly a y>d6 ev]ybody exclaim$1 ) a/oni%m5t2 b ,mr4 ,smoo?ly 0 q correct4 ,! increase is indep5d5t ( ! orig9al l5g? ( ! girdle1 :i* may be r\nd ! e>? or r\nd an orange2 9 any case ! additional six y>ds w give a di/ance ( ne>ly a y>d all r\nd4 ,?is is apt to surprise ! non-ma!matical m9d4 ,did y he> ! /ory ( ! extraord9>y precocity ( ,mrs4 ,p]k9s's baby t di$ la/ week8 ask$ ,mrs4 ,allgood4 ,x 0 only ?ree mon?s old1 & ly+ at ! po9t ( dea?1 :5 ! grief-/rick5 "m ask$ ! doctor if no?+ c\ld save x4 ',absolutely no?+6' said ! doctor4 ,!n ! infant look$ up pitifully into its "m's face & said--absolutely no?+6 ,impossible6 insi/$ ,mildr$4 ,& only ?ree mon?s old6 ,!re h be5 extraord9>y cases ( infantile precocity1 said ,mr4 ,filk9s1 ! tru? ( :i* has (t5 be5 c>efully atte/$4 ,b are y sure ?is really happ5$1 ,mrs4 ,allgood8 ,positive1 repli$ ! lady4 ,b d y really ?9k x a/oni%+ t a *ild ( ?ree mon?s %\ld say absolutely no?+8 ,:at w\ld y expect x to say8 ,speak+ ( dea?1 said ,mr4 ,smoo?ly1 solemnly1 ,i knew two m51 fa!r & son1 :o di$ 9 ! same battle dur+ ! ,s\? ,african ,w>4 ,!y 7 bo? nam$ ,&rew ,johnson & buri$ side by side1 b !re 0 "s difficulty 9 di/+ui%+ !m on ! head/ones4 ,:at w\ld y h done8 ,q simple1 said ,mr4 ,allgood4 ,!y %\ld h describ$ "o z ',&rew ,johnson1 ,s5ior1' & ! o!r z ',&rew ,johnson1 ,junior4' ,b ,i =got to tell y t ! fa!r di$ fir/4 ,:at diff]5ce c t make8 ,well1 y see1 !y want$ to be absolutely exact1 & t 0 ! difficulty4 ,b ,i don't see any difficulty1 said ,mr4 ,allgood1 nor c\ld anybody else4 ,well1 expla9$ ,mr4 ,smoo?ly1 x is l ?is4 ,if ! fa!r di$ fir/1 ! son 0 !n no long] ',junior4' ,is t s8 ,to be /rictly exact1 yes4 ,t is j :at !y want$--to be /rictly exact4 ,n[1 if he 0 no long] ',junior1' !n he did n die ',junior4 ,consequ5tly x mu/ be incorrect s to describe him on ! head/one4 ,d y see ! po9t8 ,"h is a r curi\s ?+1 said ,mr4 ,filk9s1 t ,i h j rememb]$4 ,a man wrote to me ! o!r day t he _h rec5tly discov]$ two old co9s :ile digg+ 9 8 g>d54 ,"o 0 dat$ '#ea ,b4,c41' & ! o!r "o m>k$ ',george ,i4' ,h[ d ,i "k t he 0 n writ+ ! tru?8 ,p]haps y "k ! man to be addict$ to ly+1 said ,reg9ald4 ,b t w\ld be no pro( t he 0 n tell+ ! tru? 9 ?is in/ance4 ,p]haps1 sugge/$ ,mildr$1 y "k t !re 7 no co9s made at ^? dates4 ,on ! contr>y1 !y 7 made at bo? p]iods4 ,7 !y silv] or copp] co9s8 ask$ ,willie4 ,my fri5d did n /ate1 & ,i really _c see1 ,willie1 t x makes any diff]5ce4 ,i see x6 %\t$ ,reg9ald4 ,! lett]s ',b4,c4' w\ld nev] be us$ on a co9 made be=e ! bir? ( ,*ri/4 ,!y nev] anticipat$ ! ev5t 9 t way4 ,! lett]s 7 only adopt$ lat] to d5ote dates previ\s to ^? :i* we call ',a4,d4' ,t is v good2 b ,i _c see :y ! o!r /atem5t c\ld n be correct4 ,reg9ald is q "r1 said ,mr4 ,filk9s1 ab\t ! fir/ co94 ,! second "o c\ld n exi/1 because ! fir/ ,george w\ld nev] be describ$ 9 8 lifetime z ',george ,i4' ,:y n8 ask$ ,mrs4 ,allgood4 ,he 0 ,george ,i4 ,yes2 b !y w\ld n "k x until !re 0 a ,george ,,ii4 ,!n !re 0 no ,george ,,ii4 until ,george ,,iii4 came to ! ?rone8 ,t does n foll[4 ,! second ,george becomes ',george ,,ii4' on acc\nt ( !re hav+ be5 a ',george ,i4' ,!n ! fir/ ,george 0 ',george ,i4' on acc\nt ( !re hav+ be5 no k+ ( t "n be=e him4 ,don't y see1 "m1 said ,george ,allgood1 we did n call ,que5 ,victoria ',victoria ,i42' b if !re is ev] a ',victoria ,,ii41' !n %e w be kn[n t way4 ,b !re h be5 sev]al ,georges1 & !re=e he 0 ',george ,i4' ,!re hav5't be5 sev]al ,victorias1 s ! two cases are n simil>4 ,!y gave up ! attempt to conv9ce ,mrs4 ,allgood1 b ! read] w1 ( c\rse1 see ! po9t cle>ly4 ,"h is a "q1 said ,mildr$ ,allgood1 t ,i %\ld l "s ( y to settle = me4 ,i am accu/om$ to buy f our gre5groc] bundles ( asp>agus1 ea* #ab in*es 9 circumf]5ce4 ,i always put a tape measure r\nd !m to make sure ,i am gett+ ! full quantity4 ,! o!r day ! man _h no l>ge bundles 9 /ock1 b h&$ me in/ead two small ones1 ea* #f in*es 9 circumf]5ce4 ',t is ! same ?+1' ,i said1 '&1 ( c\rse1 ! price w be ! same2' b he insi/$ t ! two bundles toge!r conta9$ m ?an ! l>ge "o1 & *>g$ me a few p5ce extra4 ,n[1 :at ,i want to "k is1 :i* ( u 0 correct8 ,w\ld ! two small bundles conta9 ! same quantity z ! l>ge "o8 ,or w\ld !y conta9 m8 ,t is ! anci5t puzzle1 said ,reg9ald1 laugh+1 ( ! sack ( corn t ,sempronius borr[$ f ,caius1 :i* y\r gre5groc]1 p]haps1 _h be5 read+ ab\t some:]e4 ,he caught y beautifully4 ,!n !y 7 equal8 ,on ! contr>y1 y 7 bo? wrong1 & y 7 badly *eat$4 ,y only got half ! quantity t w\ld h be5 conta9$ 9 a l>ge bundle1 & !re=e ought to h be5 *>g$ half ! orig9al price1 in/ead ( m4 ,yes1 x 0 a bad sw9dle1 und\bt$ly4 ,a circle ) a circumf]5ce half t ( ano!r mu/ h its area a qu>t] t ( ! o!r4 ,!re=e ! two small bundles conta9$ toge!r only half z mu* asp>agus z a l>ge "o4 ,mr4 ,filk9s1 c y answ] ?is8 ask$ ,willie4 ,!re is a man 9 ! next village :o eats two eggs = breakfa/ e morn+4 ,no?+ v extraord9>y 9 t1 ,george broke 94 ,if y told u t ! two eggs ate ! man x w\ld be int]e/+4 ,don't int]rupt ! boy1 ,george1 said 8 "m4 ,well1 ,willie cont9u$1 ?is man nei!r buys1 borr[s1 b>t]s1 begs1 /eals1 nor f9ds ! eggs4 ,he doesn't keep h5s1 & ! eggs are n giv5 to him4 ,h[ does he get ! eggs8 ,does he take !m 9 ex*ange = some?+ else8 ask$ ,mildr$4 ,t w\ld be b>t]+ !m1 ,willie repli$4 ,p]haps "s fri5d s5ds !m to him1 sugge/$ ,mrs4 ,allgood4 ,i said t !y 7 n giv5 to him4 ,i "k1 said ,george1 ) confid5ce4 ,a /range h5 comes into 8 place & lays !m4 ,b t w\ld be f9d+ !m1 w\ldn't x8 ,does he hire !m8 ask$ ,reg9ald4 ,if s1 he c\ld n return !m aft] !y 7 eat51 s t w\ld be /eal+ !m4 ,p]haps x is a pun on ! ^w 'lay1' ,mr4 ,filk9s said4 ,does he lay !m on ! table8 ,he w\ld h to get !m fir/1 w\ldn't he8 ,! "q 01 ,h[ does he get !m8 ,give x up6 said ev]ybody4 ,!n little ,willie crept r\nd to ! protection ( 8 "m1 = ,george 0 apt to be r\gh on su* occasions4 ,! man keeps ducks6 he cri$1 & 8 s]vant collects ! eggs e morn+4 ,b y said he doesn't keep birds6 ,george prote/$4 ,i didn't1 did ,i1 ,mr4 ,filk9s8 ,i said he doesn't keep h5s4 ,b he f9ds !m1 said ,reg9ald4 ,no2 ,i said 8 s]vant f9ds !m4 ,well1 !n1 ,mildr$ int]pos$1 8 s]vant gives !m to him4 ,y _c give a man 8 own prop]ty1 c y8 ,all agre$ t ,willie's answ] 0 q satisfactory4 ,!n ,uncle ,john produc$ a little fallacy t br\ght ! proce$+s to a close1 z ! newspap]s say4 #dac4--;,a ,,*essbo>d ,,fallacy4 ,illu/ration ,"h is a diagram ( a *essbo>d1 he said4 ,y see !re are sixty-f\r squ>es--eight by eight4 ,n[ ,i draw a /raight l9e f ! top left-h& corn]1 :]e ! fir/ & second squ>es meet1 to ! bottom "r-h& corn]4 ,i cut along ?is l9e ) ! scissors1 slide up ! piece t ,i h m>k$ ,b1 & !n clip (f ! little corn] ,c by a cut along ! fir/ upright l9e4 ,?is little piece w exactly fit into its place at ! top1 & we n[ h an oblong ) sev5 squ>es on "o side & n9e squ>es on ! o!r4 ,!re are1 !re=e1 n[ only sixty-?ree squ>es1 because sev5 multipli$ by n9e makes sixty-?ree4 ,:]e on e>? does t lo/ squ>e g to8 ,i h tri$ ov] & ov] aga9 to cat* ! little begg>1 b he always eludes me4 ,= ! life ( me ,i _c discov] :]e he hides himself4 ,x seems to be l ! o!r old *essbo>d fallacy1 & p]haps ! explanation is ! same1 said ,reg9ald--t ! pieces d n exactly fit4 ,b !y d fit1 said ,uncle ,john4 ,try x1 & y w see4 ,lat] 9 ! ev5+ ,reg9ald & ,george1 7 se5 9 a corn] ) _! heads toge!r1 try+ to cat* t elusive little squ>e1 & x is only fair to record t be=e !y retir$ = ! night !y succe$$ 9 secur+ _! prey1 ?\gh "s o!rs ( ! company fail$ to see x :5 captur$4 ,c ! read] solve ! little my/]y8 ,,unclassifi$ ,,problems4 ,a snapp] up ( unconsid]$ trifles4 ,w9t]'s ,tale1 iv4 #b4 #dad4--,,:o ,,0 ,,fir/8 ,&]son1 ,biggs1 & ,c>p5t] 7 /ay+ toge!r at a place by ! seaside4 ,"o day !y w5t out 9 a boat & 7 a mile at sea :5 a rifle 0 fir$ on %ore 9 _! direction4 ,:y or by :om ! %ot 0 fir$ =tunately does n conc]n u1 z no in=mation on ^! po9ts is obta9able1 b f ! facts ,i pick$ up we c get mat]ial = a curi\s little puzzle = ! novice4 ,x seems t ,&]son only he>d ! report ( ! gun1 ,biggs only saw ! smoke1 & ,c>p5t] m]ely saw ! bullet /rike ! wat] ne> !m4 ,n[1 ! "q arises3 ,:i* ( !m fir/ knew ( ! dis*>ge ( ! rifle8 #dae4--;,a ,,wond]ful ,,village4 ,!re is a c]ta9 village 9 ,japan1 situat$ 9 a v l[ valley1 & yet ! sun is ne>] to ! inhabitants e noon1 by #c1jjj miles & upw>ds1 ?an :5 he ei!r rises or sets to ^! p4 ,9 :at "p ( ! c\ntry is ! village situat$8 #daf4--;,a ,,cal5d> ,,puzzle4 ,if ! end ( ! _w %\ld come on ! fir/ day ( a new c5tury1 c y say :at are ! *ances t x w happ5 on a ,sunday8 #dag4--,,! ,,tir+ ,,irons4 ,illu/ration ,! illu/ration repres5ts "o ( ! mo/ anci5t ( all me*anical puzzles4 ,its orig9 is unkn[n4 ,c>dan1 ! ma!matician1 wrote ab\t x 9 #aeej1 & ,wallis 9 #afic2 :ile x is said /ill to be f\nd 9 obscure ,engli% villages 7sometimes deposit$ 9 /range places1 su* z a *ur* belfry71 made ( iron1 & appropriately call$ tir+-irons1 & to be us$ by ! ,norwegians to-day z a lock = boxes & bags4 ,9 ! toy%ops x is sometimes call$ ! ,*9ese r+s1 ?\gh !re seems to be no au?ority = ! description1 & x m frequ5tly goes by ! unsatisfactory "n ( ! puzzl+ r+s4 ,! ,fr5* call x ,bagu5audi]4 ,! puzzle w be se5 to consi/ ( a simple loop ( wire fix$ 9 a h&le to be held 9 ! left h&1 & a c]ta9 numb] ( r+s secur$ by wires :i* pass ?r\gh holes 9 ! b> & are kept !re by _! blunt$ ends4 ,! wires "w freely 9 ! b>1 b _c come ap>t f x1 nor c ! wires be remov$ f ! r+s4 ,! g5]al puzzle is to deta* ! loop completely f all ! r+s1 & !n to put !m all on aga94 ,n[1 x w be se5 at a glance t ! fir/ r+ 7to ! "r7 c be tak5 (f at any "t by slid+ x ov] ! end & dropp+ x ?r\gh ! loop2 or x may be put on by rev]s+ ! op]ation4 ,) ?is exception1 ! only r+ t c ev] be remov$ is ! "o t happ5s to be a contigu\s second on ! loop at ! "r-h& end4 ,?us1 ) all ! r+s on1 ! second c be dropp$ at once2 ) ! fir/ r+ d[n1 y _c drop ! second1 b may remove ! ?ird2 ) ! fir/ ?ree r+s d[n1 y _c drop ! f\r?1 b may remove ! fif?2 & s on4 ,x w be f\nd t ! fir/ & second r+s c be dropp$ toge!r or put on toge!r2 b to prev5t confusion we w ?r\gh\t disall[ ?is exceptional d\ble move1 & say t only "o r+ may be put on or remov$ at a "t4 ,we c ?us take (f "o r+ 9 #a move2 two r+s 9 #b moves2 ?ree r+s 9 #e moves2 f\r r+s 9 #aj moves2 five r+s 9 #ba moves2 & if we keep on d\bl+ 7& add+ "o :]e ! numb] ( r+s is odd7 we may easily asc]ta9 ! numb] ( moves = completely remov+ any numb] ( r+s4 ,to get (f all ! sev5 r+s requires #he moves4 ,let u look at ! five moves made 9 remov+ ! fir/ ?ree r+s1 ! circles above ! l9e /&+ = r+s on ! loop & ^? "u = r+s (f ! loop4 ,drop ! fir/ r+2 drop ! ?ird2 put up ! fir/2 drop ! second2 & drop ! fir/--#e moves1 z %[n cle>ly 9 ! diagrams4 ,! d>k circles %[ at ea* /age1 f ! />t+ position to ! f9i%1 :i* r+s x is possible to drop4 ,aft] move #b x w be notic$ t no r+ c be dropp$ until "o has be5 put on1 because ! fir/ & second r+s f ! "r n[ on ! loop are n toge!r4 ,aft] ! fif? move1 if we wi% to remove all sev5 r+s we mu/ n[ drop ! fif?4 ,b be=e we c !n remove ! f\r? x is necess>y to put on ! fir/ ?ree & remove ! fir/ two4 ,we %all !n h #g1 #f1 #d1 #c on ! loop1 & may !re=e drop ! f\r?4 ,:5 we h put on #b & #a & remov$ #c1 #b1 #a1 we may drop ! sev5? r+4 ,! next op]ation !n w be to get #f1 #e1 #d1 #c1 #b1 #a on ! loop & remove #d1 #c1 #b1 #a1 :5 #f w come (f2 !n get #e1 #d1 #c1 #b1 #a on ! loop1 & remove #c1 #b1 #a1 :5 #e w come (f2 !n get #d1 #c1 #b1 #a on ! loop & remove #b1 #a1 :5 #d w come (f2 !n get #c1 #b1 #a on ! loop & remove #a1 :5 #c w come (f2 !n get #b1 #a on ! loop1 :5 #b w come (f2 & #a w fall ?r\gh on ! #he? move1 leav+ ! loop q free4 ,! read] %\ld n[ be able to und]/& ! puzzle1 :e!r or n he has x 9 8 h& 9 a practical =m4 ,illu/ration ,illu/ration3 o o o o o 99 99 ------------- o o o o 99 o #a------------- o o o o o o #b------------- o o o o o o 99 99 #c------------- o o o o o 99 #d------------- o o o o 99 o #e------------- o o o ,! p>ticul> problem ,i propose is simply ?is4 ,suppose !re are altoge!r f\rte5 r+s on ! tir+-irons1 & we proce$ to take !m all (f 9 ! correct way s z n to wa/e any moves4 ,:at w be ! position ( ! r+s aft] ! #i1iii? move has be5 made8 #dah4--,,su* ,a ,,gett+ ,,up/airs4 ,9 a suburban villa !re is a small /aircase ) eight /eps1 n c\nt+ ! l&+4 ,! little puzzle ) :i* ,tommy ,sm>t p]plex$ 8 family is ?is4 ,y are requir$ to />t f ! bottom & l& twice on ! floor above 7/opp+ !re at ! f9i%71 hav+ return$ once to ! gr\nd floor4 ,b y mu/ be c>eful to use e tread ! same numb] ( times4 ,9 h[ few /eps c y make ! asc5t8 ,x seems a v simple matt]1 b x is m ?an likely t at y\r fir/ attempt y w make a great _m m /eps ?an are necess>y4 ,( c\rse y mu/ n g m ?an "o ris] at a "t4 ,tommy kn[s ! trick1 & has %[n x to 8 fa!r1 :o pr(esses to h a contempt = su* ?+s2 b :5 ! *ildr5 are 9 b$ ! pat] w (t5 take fri5ds out into ! hall & enjoy a good laugh at _! bewild]m5t4 ,& yet x is all s v simple :5 y "k h[ x is done4 #dai4--,,! ,,five ,,p5nies4 ,"h is a really h>d puzzle1 & yet its conditions are s absurdly simple4 ,e read] kn[s h[ to place f\r p5nies s t !y are equidi/ant f ea* o!r4 ,all y h to d is to arrange ?ree ( !m flat on ! table s t !y t\* "o ano!r 9 ! =m ( a triangle1 & lay ! f\r? p5ny on top 9 ! c5tre4 ,!n1 z e p5ny t\*es e o!r p5ny1 !y are all at equal di/ances f "o ano!r4 ,n[ try to d ! same ?+ ) five p5nies--place !m s t e p5ny %all t\* e o!r p5ny--& y w f9d x a diff]5t matt] altoge!r4 #dbj4--,,! ,,indu/ri\s ,,bookworm4 ,illu/ration ,our fri5d ,pr(essor ,rackbrane is se5 9 ! illu/ration to be prop\nd+ ano!r ( 8 little pos]s4 ,he is expla9+ t s9ce he la/ _h occasion to take d[n ^? ?ree volumes ( a le>n$ book f _! place on 8 %elves a bookworm has actually bor$ a hole /raight ?r\gh f ! fir/ page to ! la/4 ,he says t ! leaves are toge!r ?ree in*es ?ick 9 ea* volume1 & t e cov] is exactly "o-eigh? ( an in* ?ick1 & he asks h[ long a tunnel _h ! indu/ri\s worm to bore 9 prep>+ 8 new tube railway4 ,c y tell him8 #dba4--;,a ,,*a9 ,,puzzle4 ,illu/ration ,?is is a puzzle bas$ on a pretty little idea fir/ dealt ) by ! late ,mr4 ,sam ,loyd4 ,a man _h n9e pieces ( *a91 z %[n 9 ! illu/ration4 ,he want$ to jo9 ^! fifty l9ks into "o endless *a94 ,x w co/ a p5ny to op5 any l9k & twop5ce to weld a l9k toge!r aga91 b he c\ld buy a new endless *a9 ( ! same "* & quality = #bs4 #b;d4 ,:at 0 ! *eape/ c\rse = him to adopt8 ,unless ! read] is cunn+ he may f9d himself a good way out 9 8 answ]4 #dbb4--,,! ,,sabba? ,,puzzle4 ,i h come across ! foll[+ little pos] 9 an old book4 ,i wond] h[ _m read]s w see ! au?or's int5d$ solution to ! riddle4 ,*ri/ians ! week's fir/ day = ,sabba? hold2 ,! ,jews ! sev5?1 z !y did ( old2 ,! ,turks ! six?1 z we h (t be5 told4 ,h[ c ^! ?ree1 9 ! same place & day1 ,h ea* 8 own true ,sabba?8 tell1 ,i pray4 #dbc4--,,! ,,ruby ,,broo*4 ,! annals ( ,scotl& ,y>d conta9 "s rem>kable cases ( jewel robb]ies1 b "o ( ! mo/ p]plex+ 0 ! !ft ( ,lady ,littlewood's rubies4 ,!re h1 ( c\rse1 be5 _m great] robb]ies 9 po9t ( value1 b few s artfully conceiv$4 ,lady ,littlewood1 ( ,romley ,manor1 _h a beautiful b r ecc5tric heirloom 9 ! =m ( a ruby broo*4 ,:ile /ay+ at h] t[n h\se e>ly 9 ! eighties %e took ! jewel to a %op 9 ,brompton = "s slight repairs4 ,a f9e collection ( rubies1 madam1 said ! %opkeep]1 to :om h] lady%ip 0 a /rang]4 ,yes1 %e repli$2 b curi\sly en\gh ,i h nev] actually c\nt$ !m4 ,my "m once po9t$ out to me t if y />t f ! c5tre & c\nt up "o l9e1 along ! outside & d[n ! next l9e1 !re are always eight rubies4 ,s ,i %\ld always "k if a /one 7 miss+4 ,illu/ration ,six mon?s lat] a bro!r ( ,lady ,littlewood's1 :o _h return$ f 8 regim5t 9 ,india1 notic$ t 8 si/] 0 we>+ ! ruby broo* "o night at a c\nty ball1 & on _! return home ask$ to look at x m closely4 ,he imm$iately detect$ ! fact t f\r ( ! /ones 7 gone4 ,h[ c t possibly be8 said ,lady ,littlewood4 ,if y c\nt up "o l9e f ! c5tre1 along ! edge1 & d[n ! next l9e1 9 any direction1 !re are always eight /ones4 ,?is 0 always s & is s n[4 ,h[1 !re=e1 w\ld x be possible to remove a /one )\t my detect+ x8 ,no?+ c\ld be simpl]1 repli$ ! bro!r4 ,i "k ! broo* well4 ,x orig9ally conta9$ =ty-five /ones1 & !re are n[ only =ty-"o4 ,somebody has /ol5 f\r rubies1 & !n reset z small a numb] ( ! o!rs z possible 9 su* a way t !re %all always be eight 9 any ( ! directions y h m5tion$4 ,!re 0 n ! slighte/ d\bt t ! ,brompton jewell] 0 ! ?ief1 & ! matt] 0 plac$ 9 ! h&s ( ! police4 ,b ! man 0 want$ = o!r robb]ies1 & _h left ! neighb\rhood "s "t be=e4 ,to ?is day he has nev] be5 f\nd4 ,! int]e/+ little po9t t at fir/ baffl$ ! police1 & :i* =ms ! subject ( our puzzle1 is ?is3 ,h[ 7 ! =ty-five rubies orig9ally arrang$ on ! broo*8 ,! illu/ration %[s exactly h[ ! =ty-"o 7 arrang$ aft] x came back f ! jewell]2 b al?\gh !y c\nt eight correctly 9 any ( ! directions m5tion$1 !re are f\r /ones miss+4 #dbd4--,,! ,,dovetail$ ,,block4 ,illu/ration ,"h is a curi\s me*anical puzzle t 0 giv5 to me "s ye>s ago1 b ,i _c say :o fir/ inv5t$ x4 ,x consi/s ( two solid blocks ( wood securely dovetail$ toge!r4 ,on ! o!r two v]tical sides t are n visible ! appe>ance is precisely ! same z on ^? %[n4 ,h[ 7 ! pieces put toge!r8 ,:5 ,i publi%$ ?is little puzzle 9 a ,london newspap] ,i receiv$ 7?\gh !y 7 unsolicit$7 q a /ack ( models1 9 oak1 9 teak1 9 mahogany1 rosewood1 sat9wood1 elm1 & deal2 "s half a foot 9 l5g?1 & o!rs v>y+ 9 size "r d[n to a delicate little model ab\t half an in* squ>e4 ,x seem$ to create consid]able int]e/4 #dbe4--;,,jack ,,& ,,! ,,bean/alk4 ,illu/ration ,! illu/ration1 by a ,briti% arti/1 is a sket* ( ,jack climb+ ! bean/alk4 ,n[1 ! arti/ has made a s]i\s blund] 9 ?is draw+4 ,c y f9d out :at x is8 #dbf4--,,! ,,hymn-,,bo>d ,,pos]4 ,! wor?y vic> ( ,*umpley ,/4 ,w9ifr$ is 9 great di/ress4 ,a little *ur* difficulty has aris5 t all ! comb9$ intellig5ce ( ! p>i% seems unable to surm\nt4 ,:at ?is difficulty is ,i w /ate h]eaft]1 b x may add to ! int]e/ ( ! problem if ,i fir/ give a %ort acc\nt ( ! curi\s position t has be5 br\ght ab\t4 ,x all has to d ) ! *ur* hymn-bo>ds1 ! plates ( :i* h become s damag$ t !y h ceas$ to fulfil ! purpose = :i* !y 7 devis$4 ,a g5]\s p>i%ion] has promis$ to pay = a new set ( plates at a c]ta9 rate ( co/2 b /range z x may seem1 no agreem5t c be come to z to :at t co/ %\ld be4 ,! propos$ mak] ( ! plates has nam$ a price :i* ! donor decl>es to be absurd4 ,! good vic> ?9ks !y are bo? wrong1 s he asks ! s*oolma/] to "w out ! little sum4 ,b ?is individual decl>es t he c f9d no rule be>+ on ! subject 9 any ( 8 ari?metic books4 ,an application hav+ be5 made to ! local m$ical practition]1 z a man ( m ?an av]age intellect at ,*umpley1 he has assur$ ! vic> t 8 practice is s heavy t he has n _h "t ev5 to look at x1 ?\gh 8 assi/ant :isp]s t ! doctor has be5 sitt+ up unusually late = sev]al nights pa/4 ,wid[ ,wilson has a sm>t son1 :o is reput$ to h once won a prize = puzzle-solv+4 ,he ass]ts t z he _c f9d any solution to ! problem x mu/ h some?+ to d ) ! squ>+ ( ! circle1 ! duplication ( ! cube1 or ! trisection ( an angle2 at any rate1 he has nev] be=e se5 a puzzle on ! pr9ciple1 & he gives x up4 ,illu/ration ,?is 0 ! /ate ( affairs :5 ! assi/ant curate 7:o1 ,i %\ld say1 _h frankly confess$ f ! fir/ t a pr(\nd /udy ( !ology _h knock$ out ( 8 head all ! k ( ma!matics he ev] possess$7 k9dly s5t me ! puzzle4 ,a *ur* has ?ree hymn-bo>ds1 ea* to indicate ! numb]s ( five diff]5t hymns to be sung at a s]vice4 ,all ! bo>ds are 9 use at ! same s]vice4 ,! hymn-book conta9s #gjj hymns4 ,a new set ( numb]s is requir$1 & a k9d p>i%ion] (f]s to pres5t a set pa9t$ on metal plates1 b /ipulates t only ! smalle/ numb] ( plates necess>y %all be pur*as$4 ,! co/ ( ea* plate is to be #f;d41 & = ! pa9t+ ( ea* plate ! *>ges are to be3 ,= "o plate1 #as42 = two plates alike1 #aa;d4 ea*2 = ?ree plates alike1 #aa;d4 ea*1 & s on1 ! *>ge be+ "o f>?+ less p] plate = ea* simil>ly pa9t$ plate4 ,n[1 :at %\ld be ! l[e/ co/8 ,read]s w note t !y are requir$ to use e legitimate & practical me?od ( economy4 ,! illu/ration w make cle> ! nature ( ! ?ree hymn-bo>ds & plates4 ,! five hymns are "h indicat$ by means ( twelve plates4 ,^! plates slide 9 sep>ately at ! back1 & 9 ! illu/ration !re is room1 ( c\rse1 = ?ree m plates4 #dbg4--,,pheasant-,,%oot+4 ,a ,cockney fri5d1 :o is v apt to draw ! long b[1 & is evid5tly less ( a sportsman ?an he pret5ds to be1 relates to me ! foll[+ n v cr$ible y>n3-- ,i've j be5 pheasant-%oot+ ) my fri5d ! duke4 ,we _h spl5did sport1 & ,i made "s wond]ful %ots4 ,:at d y ?9k ( ?is1 = in/ance8 ,p]haps y c twi/ x into a puzzle4 ,! duke & ,i 7 cross+ a field :5 sudd5ly tw5ty-f\r pheasants rose on ! w+ "r 9 front ( u4 ,i fir$1 & two-?irds ( !m dropp$ dead at my feet4 ,!n ! duke _h a %ot at :at 7 left1 & br\ght d[n ?ree-tw5ty-f\r?s ( !m1 w\nd$ 9 ! w+4 ,n[1 out ( ^? tw5ty-f\r birds1 h[ _m /ill rema9$8 ,x seems a simple en\gh "q1 b c ! read] give a correct answ]8 #dbh4--,,! ,,g>d5] ,,& ,,! ,,cook4 ,a correspond5t1 sign+ himself ,simple ,simon1 sugge/$ t ,i %\ld give a special cat* puzzle 9 ! issue ( ,! ,weekly ,dispat* = ,all ,fools' ,day1 #aijj4 ,s ,i gave ! foll[+1 & x caus$ consid]able amusem5t2 = out ( a v l>ge body ( competitors1 _m q exp]t1 n a s+le p]son solv$ x1 ?\gh x ran = ne>ly a mon?4 ,illu/ration ,! illu/ration is a fancy sket* ( my correspond5t1 ',simple ,simon1' 9 ! act ( try+ to solve ! foll[+ innoc5t little ari?metical puzzle4 ,a race betwe5 a man & a woman t ,i happ5$ to witness "o ,all ,fools' ,day has fix$ itself indelibly on my memory4 ,x happ5$ at a c\ntry-h\se1 :]e ! g>d5] & ! cook decid$ to run a race to a po9t #ajj feet /raight away & return4 ,i f\nd t ! g>d5] ran #c feet at e b\nd & ! cook only #b feet1 b !n %e made ?ree b\nds to 8 two4 ,n[1 :at 0 ! result ( ! race8 ,a =tnight aft] publication ,i add$ ! foll[+ note3 ,x has be5 sugge/$ t p]haps !re is a cat* 9 ! 'return1' b !re is n4 ,! race is to a po9t #ajj feet away & home aga9--t is1 a di/ance ( #bjj feet4 ,"o correspond5t asks :e!r !y take exactly ! same "t 9 turn+1 to :i* ,i reply t !y d4 ,ano!r seems to suspect t x is really a conundrum1 & t ! answ] is t '! result ( ! race 0 a 7matrimonial7 tie4' ,b ,i _h no su* int5tion4 ,! puzzle is an ari?metical "o1 z x purports to be4 #dbi4--,,plac+ ,,halfp5nies4 ,illu/ration ,"h is an int]e/+ little puzzle sugge/$ to me by ,mr4 ,w4 ,t4 ,:yte4 ,m>k (f on a %eet ( pap] a rectangul> space #e in*es by #c in*es1 & !n f9d ! greate/ numb] ( halfp5nies t c be plac$ )9 ! enclosure "u ! foll[+ conditions4 ,a halfp5ny is exactly an in* 9 diamet]4 ,place y\r fir/ halfp5ny :]e y l1 !n place y\r second co9 at exactly ! di/ance ( an in* f ! fir/1 ! ?ird an in* di/ance f ! second1 & s on4 ,no halfp5ny may t\* ano!r halfp5ny or cross ! b\nd>y4 ,our illu/ration w make ! matt] p]fectly cle>4 ,no4 #b co9 is an in* f ,no4 #a2 ,no4 #c an in* f ,no4 #b2 ,no4 #d an in* f ,no4 #c2 b aft] ,no4 #aj is plac$ we c g no fur!r 9 ?is attempt4 ,yet sev]al m halfp5nies might h be5 got 94 ,h[ _m c ! read] place8 #dcj4--;,,f9d ,,! ,,man',s ,,wife4 ,illu/ration ,"o summ] day 9 #aijc ,i 0 loit]+ on ! ,brighton front1 wat*+ ! p /roll+ ab\t on ! bea*1 :5 ! fri5d :o 0 ) me sudd5ly drew my att5tion to an individual :o 0 /&+ alone1 & said1 ,c y po9t out t man's wife8 ,!y are /opp+ at ! same hotel z ,i am1 & ! lady is "o ( ^? 9 view4 ,aft] a few m9utes' obs]vation1 ,i 0 successful 9 indicat+ ! lady correctly4 ,my fri5d 0 curi\s to "k by :at me?od ( reason+ ,i _h arriv$ at ! result4 ,?is 0 my answ]3-- ,we may at once exclude t ,si/] ( ,m]cy & ! girl 9 ! %ort frock2 also ! woman sell+ oranges4 ,x _c be ! lady 9 wid[s' we$s4 ,x is n ! lady 9 ! ba? *air1 because %e is n /ay+ at y\r hotel1 = ,i happ5$ to see h] come out ( a private h\se ?is morn+ assi/$ by h] maid4 ,! two ladies 9 r$ breakfa/$ at my hotel ?is morn+1 & z !y 7 n we>+ outdoor dress ,i conclude !y are /ay+ !re4 ,x !re=e re/s betwe5 ! lady 9 blue & ! "o ) ! gre5 p>asol4 ,b ! left h& t holds ! p>asol is1 y see1 unglov$ & be>s no w$d+-r+4 ,consequ5tly ,i am driv5 to ! conclusion t ! lady 9 blue is ! man's wife--& y say ?is is correct4 ,n[1 z my fri5d 0 an arti/1 & z ,i ?\ght an amus+ puzzle might be devis$ on ! l9es ( 8 "q1 ,i ask$ him to make me a draw+ accord+ to "s directions t ,i gave him1 & ,i h pleasure 9 pres5t+ 8 production to my read]s4 ,x w be se5 t ! picture %[s six m5 & six ladies3 ,nos4 #a1 #c1 #e1 #g1 #i1 & #aa are ladies1 & ,nos4 #b1 #d1 #f1 #h1 #aj1 & #ab are m54 ,^! twelve individuals repres5t six m>ri$ c\ples1 all /rang]s to "o ano!r1 :o1 9 walk+ aimlessly ab\t1 h got mix$ up4 ,b we are only conc]n$ ) ! man t is we>+ a /raw hat--,numb] #aj4 ,! puzzle is to f9d ?is man's wife4 ,exam9e ! six ladies c>efully1 & see if y c det]m9e :i* "o ( !m x is4 ,i %[$ ! picture at ! "t to a few fri5ds1 & !y express$ v diff]5t op9ions on ! matt]4 ,"o said1 ,i don't believe he w\ld m>ry a girl l ,numb] #g4 ,ano!r said1 ,i am sure a nice girl l ,numb] #c w\ld n m>ry su* a fell[6 ,ano!r said1 ,x mu/ be ,numb] #a1 because %e has got z f> away z possible f ! brute6 ,x 0 sugge/$1 aga91 t x mu/ be ,numb] #aa1 because he seems to be look+ t[>ds h]2 b a cynic retort$1 ,= t v reason1 if he is really look+ at h]1 ,i %\ld say t %e is n 8 wife6 ,i n[ leave ! "q 9 ! h&s ( my read]s4 ,:i* is really ,numb] #aj's wife8 ,! illu/ration is ( necessity consid]ably r$uc$ f ! l>ge scale on :i* x orig9ally appe>$ 9 ,! ,weekly ,dispat* 7#bd? ,may #aijc71 b x is hop$ t ! details w be suffici5tly cle> to all[ ! read] to d]ive ent]ta9m5t f its exam9ation4 ,9 any case ! solution giv5 w enable him to foll[ ! po9ts ) int]e/4 ,,solutions4 #a4--;,a ,,po/-,,(fice ,,p]plexity4 ,! "y lady suppli$ #e twop5ny /amps1 #cj p5ny /amps1 & #h twop5ce-halfp5ny /amps1 :i* deliv]y exactly fulfils ! conditions & repres5ts a co/ ( five %ill+s4 #b4--,,y\?ful ,,precocity4 ,! price ( ! banana mu/ h be5 "o p5ny f>?+4 ,?us1 #ifj bananas w\ld co/ #e1 & #dhj sixp5ces w\ld buy #b1cjd bananas4 #c4--;,,at ,a ,,cattle ,,m>ket4 ,jakes mu/ h tak5 #g animals to m>ket1 ,hodge mu/ h tak5 #aa1 & ,durrant mu/ h tak5 #ba4 ,!re 7 ?us #ci animals altoge!r4 #d4--,,! ,,beanfea/ ,,puzzle4 ,! cobbl]s sp5t #ces41 ! tailors sp5t also #ces41 ! hatt]s sp5t #dbs41 & ! glov]s sp5t #bas4 ,?us1 !y sp5t altoge!r #f1acs41 :ile x w be f\nd t ! five cobbl]s sp5t z mu* z f\r tailors1 twelve tailors z mu* z n9e hatt]s1 & six hatt]s z mu* z eight glov]s4 #e4--;,a ,,que] ,,co9cid5ce4 ,puzzles ( ?is class are g5]ally solv$ 9 ! old books by ! t$i\s process ( work+ backw>ds4 ,b a simple g5]al solution is z foll[s3 ,if !re are n play]s1 ! am\nt held by e play] at ! end w be m7#bn71 ! la/ w9n] mu/ h held m7n #a7 at ! />t1 ! next m7#bn #a71 ! next m7#dn #a71 ! next m7#hn #a71 & s on to ! fir/ play]1 :o mu/ h held m7#bn - #an #a74 ,?us1 9 ?is case1 n #g1 & ! am\nt held by e play] at ! end 0 #bg f>?+s4 ,!re=e m #a1 & ,g />t$ ) #h f>?+s1 ,f ) #ae1 ,e ) #bi1 ,d ) #eg1 ,c ) #aac1 ,b ) #bbe1 & ,a ) #ddi f>?+s4 #f4--;,a ,,*>itable ,,beque/4 ,!re are sev5 diff]5t ways 9 :i* ! money may be di/ribut$3 #e wom5 & #ai m51 #aj wom5 & #af m51 #ae wom5 & #ac m51 #bj wom5 & #aj m51 #be wom5 & #g m51 #cj wom5 & #d m51 & #ce wom5 & #a man4 ,b ! la/ case mu/ n be c\nt$1 because ! condition 0 t !re %\ld be m51 & a s+le man is n m54 ,!re=e ! answ] is six ye>s4 #g4--,,! ,,wid[',s ,,legacy4 ,! wid['s %>e ( ! legacy mu/ be #bje1 #bs4 #f;d4 & #aj_/ac ( a p5ny4 #h4--;,,indiscrim9ate ,,*>ity ,! g5tleman mu/ h _h #cs4 #f;d4 9 8 pocket :5 he set out = home4 #i4--,,! ,,two ,,a]oplanes4 ,! man mu/ h paid #ejj & #gej = ! two ma*9es1 mak+ toge!r #a1bej2 b z he sold !m = only #a1bjj1 he lo/ #ej by ! transaction4 #aj4--;,,buy+ ,,pres5ts4 ,jork9s _h orig9ally #ai1 #ahs4 9 8 pocket1 & sp5t #i1 #ais4 #aa4--,,! ,,cycli/s' ,,fea/4 ,!re 7 t5 cycli/s at ! fea/4 ,!y %\ld h paid #hs4 ea*2 b1 ow+ to ! dep>ture ( two p]sons1 ! rema9+ eight w\ld pay #ajs4 ea*4 #ab4--;,a ,,que] ,,?+ ,,9 ,,money4 ,! answ] is z foll[s3 #dd1ddd1 #ds4 #d;d4 #bh1 &1 r$uc$ to p5ce1 #aj1fff1fabbh4 ,x is a curi\s little co9cid5ce t 9 ! answ] #aj1fff1fab ! f\r c5tral figures indicate ! only o!r answ]1 #ff1 #fs4 #f;d4 #ac4--;,a ,,new ,,money ,,puzzle4 ,! smalle/ sum ( money1 9 p\nds1 %ill+s1 p5ce1 & f>?+s1 conta9+ all ! n9e digits once1 & once only1 is #b1efg1 #ahs4 #i;d4 #ad4--,,squ>e ,,money4 ,! answ] is #a;d4 & #c;d4 ,add$ toge!r !y make #d;d41 & #a;d4 multipli$ by #c is also #d;d4 #ae4--,,pocket ,,money4 ,! l>ge/ possible sum is #aes4 #i;d41 compos$ ( a cr[n & a half-cr[n 7or ?ree half-cr[ns71 f\r flor9s1 & a ?reep5ny piece4 #af4--,,! ,,millionaire',s ,,p]plexity4 ,! answ] to ?is q easy puzzle may1 ( c\rse1 be readily obta9$ by trial1 d$uct+ ! l>ge/ p[] ( #g t is conta9$ 9 "o million doll>s1 !n ! next l>ge/ p[] f ! rema9d]1 & s on4 ,b ! little problem is int5d$ to illu/rate a simple direct me?od4 ,! answ] is giv5 at once by conv]t+ #a1jjj1jjj to ! sept5>y scale1 & x is on ?is subject ( scales ( notation t ,i propose to write a few words = ! b5efit ( ^? :o h nev] suffici5tly consid]$ ! matt]4 ,our mann] ( figur+ is a sort ( p]fect$ ari?metical %or?&1 a sy/em devis$ to enable u to manipulate numb]s z rapidly & correctly z possible by means ( symbols4 ,if we write ! numb] #b1cda to repres5t two ?\s& ?ree hundr$ & =ty-"o doll>s1 we wi% to imply #a doll>1 add$ to f\r times #aj doll>s1 add$ to ?ree times #ajj doll>s1 add$ to two times #a1jjj doll>s4 ,f ! numb] 9 ! units place on ! "r1 e figure to ! left is und]/ood to repres5t a multiple ( ! p>ticul> p[] ( #aj t its position indicates1 :ile a ciph] 7#j7 mu/ be ins]t$ :]e necess>y 9 ord] to prev5t confusion1 = if in/ead ( #bjg we wrote #bg x w\ld be obvi\sly mislead+4 ,we ?us only require t5 figures1 because directly a numb] exce$s #i we put a second figure to ! left1 directly x exce$s #ii we put a ?ird figure to ! left1 & s on4 ,x w be se5 t ?is is a purely arbitr>y me?od4 ,x is work+ 9 ! d5>y 7or t57 scale ( notation1 a sy/em und\bt$ly d]iv$ f ! fact t our =efa!rs :o devis$ x _h t5 f+]s ^u :i* !y 7 accu/om$ to c\nt1 l our *ildr5 ( to-day4 ,x is unnecess>y = u ord9>ily to /ate t we are us+ ! d5>y scale1 because ?is is always und]/ood 9 ! common affairs ( life4 ,b if a man said t he _h #f1eec doll>s 9 ! sept5>y 7or sev57 scale ( notation1 y w f9d t ?is is precisely ! same am\nt z #b1cda 9 our ord9>y d5>y scale4 ,in/ead ( us+ p[]s ( t51 he uses p[]s ( #g1 s t he nev] ne$s any figure high] ?an #f1 & #f1eec really /&s = #c1 add$ to five times #g1 add$ to five times #di1 add$ to six times #cdc 79 ! ord9>y notation71 or #b1cda4 ,to rev]se ! op]ation1 & conv]t #b1cda f ! d5>y to ! sept5>y scale1 we divide x by #g1 & get #ccd & rema9d] #c2 divide #ccd by #g1 & get #dg & rema9d] #e2 & s keep on divid+ by #g z long z !re is any?+ to divide4 ,! rema9d]s1 read backw>ds1 #f1 #e1 #e1 #c1 give u ! answ]1 #f1eec4 ,n[1 z ,i h said1 our puzzle may be solv$ at once by m]ely conv]t+ #a1jjj1jjj doll>s to ! sept5>y scale4 ,keep on divid+ ?is numb] by #g until !re is no?+ m left to divide1 & ! rema9d]s w be f\nd to be #aaccccaa :i* is #a1jjj1jjj express$ 9 ! sept5>y scale4 ,!re=e1 #a gift ( #a doll>1 #a gift ( #g doll>s1 #c gifts ( #di doll>s1 #c gifts ( #cdc doll>s1 #c gifts ( #b1dja doll>s1 #c gifts ( #af1hjg doll>s1 #a gift ( #aag1fdi doll>s1 & "o sub/antial gift ( #hbc1edc doll>s1 satisfactorily solves our problem4 ,& x is ! only possible solution4 ,x is ?us se5 t no trials are necess>y2 by conv]t+ to ! sept5>y scale ( notation we g direct to ! answ]4 #ag4--,,! ,,puzzl+ ,,money ,,boxes4 ,! correct answ] to ?is puzzle is z foll[s3 ,john put into 8 money-box two d\ble flor9s 7#hs471 ,william a half-sov]eign & a flor9 7#abs471 ,*>les a cr[n 7#es471 & ,?omas a sov]eign 7#bjs474 ,!re are six co9s 9 all1 ( a total value ( #des4 ,if ,john _h #bs4 m1 ,william #bs4 less1 ,*>les twice z mu*1 & ,?omas half z mu* z !y really possess$1 !y w\ld ea* h _h exactly #ajs4 #ah4--,,! ,,m>ket ,,wom54 ,! price receiv$ 0 9 e case #aje f>?+s4 ,!re=e ! greate/ numb] ( wom5 is eight1 z ! goods c\ld only be sold at ! foll[+ rates3 #aje lbs4 at #a f>?+1 #ce at #c1 #ba at #e1 #ae at #g1 #g at #ae1 #e at #ba1 #c at #ce1 & #a lb4 at #aje f>?+s4 #ai4--,,! ,,new ,,ye>',s ,,eve ,,supp]s4 ,! company pres5t on ! occasion mu/ h consi/$ ( sev5 pairs1 t5 s+le m51 & "o s+le lady4 ,?us1 !re 7 tw5ty-five p]sons 9 all1 & at ! prices /at$ !y w\ld pay exactly #e toge!r4 #bj4--;,,beef ,,& ,,sausages4 ,! lady b\ght #dh lbs4 ( beef at #bs41 & ! same quantity ( sausages at #as4 #f;d41 ?us sp5d+ #h1 #hs4 ,_h %e b\ght #db lbs4 ( beef & #ef lbs4 ( sausages %e w\ld h sp5t #d1 #ds4 on ea*1 & h obta9$ #ih lbs4 in/ead ( #if lbs4--a ga9 9 weight ( #b lbs4 #ba4--;,a ,,deal ,,9 ,,apples4 ,i 0 fir/ (f]$ sixte5 apples = my %ill+1 :i* w\ld be at ! rate ( n9ep5ce a doz54 ,! two extra apples gave me eighte5 = a %ill+1 :i* is at ! rate ( eightp5ce a doz51 or "o p5ny a doz5 less ?an ! fir/ price ask$4 #bb4--;,a ,,deal ,,9 ,,eggs4 ,! man mu/ h b\ght t5 eggs at fivep5ce1 t5 eggs at "o p5ny1 & eighty eggs at a halfp5ny4 ,he w\ld !n h "o hundr$ eggs at a co/ ( eight %ill+s & f\rp5ce1 & ! same numb] ( eggs ( two ( ! qualities4 #bc4--,,! ,,*ri/mas-,,boxes4 ,! di/ribution took place "s ye>s ago1 :5 ! f\rp5ny-piece 0 9 circulation4 ,n9ete5 p]sons mu/ ea* h receiv$ n9ete5 p5ce4 ,!re are five diff]5t ways 9 :i* ?is sum may h be5 paid 9 silv] co9s4 ,we ne$ only use two ( ^! ways4 ,?us if f\rte5 m5 ea* receiv$ f\r f\r-p5ny-pieces & "o ?reep5ny-piece1 & five m5 ea* receiv$ five ?reep5ny-pieces & "o f\rp5ny-piece1 ea* man w\ld receive n9ete5 p5ce1 & !re w\ld be exactly "o hundr$ co9s ( a total value ( #a1 #ajs4 #a;d4 #bd4--;,a ,,%opp+ ,,p]plexity4 ,! fir/ pur*ase am\nt$ to #as4 #e;d41 ! second to #as4 #aa;d41 & toge!r !y make #cs4 #e;d4 ,n "o ( ^! ?ree am\nts c be paid 9 few] ?an six curr5t co9s ( ! realm4 #be4--;,,*9ese ,,money4 ,z a *+-*ang is wor? twop5ce & f\r-fifte5?s ( a *+-*ang1 ! rema9+ elev5-fifte5?s ( a *+-*ang mu/ be wor? twop5ce4 ,!re=e elev5 *+-*angs are wor? exactly ?irty p5ce1 or half a cr[n4 ,n[1 ! ex*ange mu/ be made ) sev5 r\nd-hol$ co9s & "o squ>e-hol$ co94 ,?us x w be se5 t #g r\nd-hol$ co9s are wor? sev5-elev5?s ( #ae *+-*angs1 & #a squ>e-hol$ co9 is wor? "o-elev5? ( #af *+-*angs--t is1 #gg r\nds equal #aje *+-*angs & #aa squ>es equal #af *+-*angs4 ,!re=e #gg r\nds add$ to #aa squ>es equal #aba *+-*angs2 or #g r\nds & #a squ>e equal #aa *+-*angs1 or its equival5t1 half a cr[n4 ,?is is m simple 9 practice ?an x looks "h4 #bf4--,,! ,,junior ,,cl]ks' ,,puzzle4 ,al?\gh ,snoggs's reason = wi%+ to take 8 rise at #b1 #ajs4 half-ye>ly did n conc]n our puzzle1 ! fact t he 0 dup+ 8 employ] into pay+ him m ?an 0 int5d$ did conc]n x4 ,_m read]s w be surpris$ to f9d t1 al?\gh ,moggs only receiv$ #cej 9 five ye>s1 ! artful ,snoggs actually obta9$ #cfb1 #ajs4 9 ! same "t4 ,! re/ is simplicity itself4 ,x is evid5t t if ,moggs sav$ #hg1 #ajs4 & ,snoggs #aha1 #es41 ! latt] w\ld be sav+ twice z great a proportion ( 8 sal>y z ! =m] 7namely1 "o-half z aga9/ "o-qu>t]71 & ! two sums add$ toge!r make #bfh1 #aes4 #bg4--;,,giv+ ,,*ange4 ,! way to help ! ,am]ican tradesman out ( 8 dilemma is ?is4 ,describ+ ! co9s by ! numb] ( c5ts t !y repres5t1 ! tradesman puts on ! c\nt] #ej & #be2 ! buy] puts d[n #ajj1 #c1 & #b2 ! /rang] adds 8 #aj1 #aj1 #e1 #b1 & #a4 ,n[1 consid]+ t ! co/ ( ! pur*ase am\nt$ to #cd c5ts1 x is cle> t out ( ?is pool$ money ! tradesman has to receive #aji1 ! buy] #ga1 & ! /rang] 8 #bh c5ts4 ,!re=e x is obvi\s at a glance t ! #ajj-piece mu/ g to ! tradesman1 & x !n foll[s t ! #ej-piece mu/ g to ! buy]1 & !n ! #be-piece c only g to ! /rang]4 ,ano!r glance w n[ make x cle> t ! two #aj-;c5t pieces mu/ g to ! buy]1 because ! tradesman n[ only wants #i & ! /rang] #c4 ,!n x becomes obvi\s t ! buy] mu/ take ! #a c5t1 t ! /rang] mu/ take ! #c c5ts1 & ! tradesman ! #e1 #b1 & #b4 ,to sum up1 ! tradesman takes #ajj1 #e1 #b1 & #b2 ! buy]1 #ej1 #aj1 #aj1 & #a2 ! /rang]1 #be & #c4 ,x w be se5 t n "o ( ! ?ree p]sons reta9s any "o ( 8 own co9s4 #bh4--;,,defective ,,obs]vation4 ,( c\rse ! date on a p5ny is on ! same side z ,britannia--! tail side4 ,six p5nies may be laid ar\nd ano!r p5ny1 all flat on ! table1 s t e "o ( !m t\*es ! c5tral "o4 ,! numb] ( ?reep5ny-pieces t may be laid on ! surface ( a half-cr[n1 s t no piece lies on ano!r or ov]laps ! edge ( ! half-cr[n1 is "o4 ,a second ?reep5ny-piece w ov]lap ! edge ( ! l>g] co94 ,few p guess few] ?an ?ree1 & _m p]sons give an absurdly high numb]4 #bi4--,,! ,,brok5 ,,co9s4 ,if ! ?ree brok5 co9s :5 p]fect 7 wor? #bec p5ce1 & are n[ 9 _! brok5 condition wor? #bdj p5ce1 x %\ld be obvi\s t #ac_/bec ( ! orig9al value has be5 lo/4 ,& z ! same fraction ( ea* co9 has be5 brok5 away1 ea* co9 has lo/ #ac_/bec ( its orig9al bulk4 #cj4--,,two ,,que/ions ,,9 ,,probabilities4 ,9 toss+ ) ! five p5nies all at ! same "t1 x is obvi\s t !re are #cb diff]5t ways 9 :i* ! co9s may fall1 because ! fir/ co9 may fall 9 ei!r ( two ways1 !n ! second co9 may also fall 9 ei!r ( two ways1 & s on4 ,!re=e five #b's multipli$ toge!r make #cb4 ,n[1 h[ are ^! #cb ways made up8 ,"h !y are3-- 7a7 #e heads #a way 7b7 #e tails #a way 7c7 #d heads & #a tail #e ways 7d7 #d tails & #a head #e ways 7e7 #c heads & #b tails #aj ways 7f7 #c tails & #b heads #aj ways ,n[1 x w be se5 t ! only fav\rable cases are a1 b1 c1 & d--#ab cases4 ,! rema9+ #bj cases are unfav\rable1 because !y d n give at lea/ f\r heads or f\r tails4 ,!re=e ! *ances are only #ab to #bj 9 y\r fav\r1 or 7:i* is ! same ?+7 #c to #e4 ,put ano!r way1 y h only #c *ances out ( #h4 ,! am\nt t %\ld be paid = a draw f ! bag t conta9s ?ree sov]eigns & "o %ill+ is #aes4 #c;d4 ,_m p]sons w say t1 z "o's *ances ( draw+ a sov]eign 7 #c out ( #d1 "o %\ld pay ?ree-f\r?s ( a p\nd1 or #aes41 ov]look+ ! fact t "o mu/ draw at lea/ a %ill+--!re be+ no blanks4 #ca4--;,,dome/ic ,,economy4 ,)\t ! h9t t ,i gave1 my read]s w\ld probably h be5 unanim\s 9 decid+ t ,mr4 ,p]k9s's income mu/ h be5 #a1gaj4 ,b ?is is q wrong4 ,mrs4 ,p]k9s says1 ,we h sp5t a ?ird ( 8 ye>ly income 9 r5t1 etc41 etc4--t is1 9 two ye>s !y h sp5t an am\nt 9 r5t1 etc41 equal to "o-?ird ( 8 ye>ly income4 ,note t %e does n say t !y h sp5t ea* ye> ?is sum1 :atev] x is1 b t dur+ ! two ye>s t am\nt has be5 sp5t4 ,! only possible answ]1 accord+ to ! exact read+ ( h] words1 is1 !re=e1 t 8 income 0 #ahj p] annum4 ,?us ! am\nt sp5t 9 two ye>s1 dur+ :i* 8 income has am\nt$ to #cfj1 w be #fj 9 r5t1 etc41 #ij 9 dome/ic exp5ses1 #bj 9 o!r ways1 leav+ ! balance ( #aij 9 ! bank z /at$4 #cb4--,,! ,,excursion ,,ticket ,,puzzle4 ,n9ete5 %ill+s & n9ep5ce may be paid 9 #deh1ijh1fbb diff]5t ways4 ,i d n propose to give my me?od ( solution4 ,any su* explanation w\ld occupy an am\nt ( space out ( proportion to its int]e/ or value4 ,if ,i c\ld give )9 reasonable limits a g5]al solution = all money paym5ts1 ,i w\ld /ra9 a po9t to f9d room2 b su* a solution w\ld be extremely complex & cumb]some1 & ,i d n consid] x wor? ! lab\r ( work+ out4 ,j to give an idea ( :at su* a solution w\ld involve1 ,i w m]ely say t ,i f9d t1 deal+ only ) ^? sums ( money t are multiples ( ?reep5ce1 if we only use bronze co9s any sum c be paid 9 7n #a7 ways :]e n always repres5ts ! numb] ( p5ce4 ,if ?reep5ny-pieces are admitt$1 !re are #bn #aen #ccn --------------------- #a ways4 #ah ,if sixp5ces are also us$ !re are n#d #bbn #aein #dadn #baf --------------------------------- #baf ways1 :5 ! sum is a multiple ( sixp5ce1 & ! con/ant1 #baf1 *anges to #cbd :5 ! money is n su* a multiple4 ,& s ! =mulas increase 9 complexity 9 an accel]at+ ratio z we g on to ! o!r co9s4 ,i w1 h[ev]1 add an int]e/+ little table ( ! possible ways ( *ang+ our curr5t co9s :i* ,i believe has nev] be5 giv5 9 a book be=e4 ,*ange may be giv5 = a ,f>?+ 9 #j way4 ,halfp5ny 9 #a way4 ,p5ny 9 #c ways4 ,?reep5ny-piece 9 #af ways4 ,sixp5ce 9 #ff ways4 ,%ill+ 9 #djb ways4 ,flor9 9 #c1hah ways4 ,half-cr[n 9 #h1gji ways4 ,d\ble flor9 9 #fj1bci ways4 ,cr[n 9 #aff1fea ways4 ,half-sov]eign 9 #f1bfa1fbb ways4 ,sov]eign 9 #ejj1bia1hcc ways4 ,x is a little surpris+ to f9d t a sov]eign may be *ang$ 9 ov] five hundr$ million diff]5t ways4 ,b ,i h no d\bt z to ! correctness ( my figures4 #cc4--;,a ,,puzzle ,,9 ,,rev]sals4 7i7 #ac4 7#b7 #bc1 #ais4 #aa;d4 ,! words ! numb] ( p\nds exce$s t ( ! p5ce exclude su* sums ( money z #b1 #afs4 #b;d4 & all sums "u #a4 #cd4--,,! ,,groc] ,,& ,,drap]4 ,! groc] 0 delay$ half a m9ute & ! drap] eight m9utes & a half 7sev5te5 times z long z ! groc]71 mak+ toge!r n9e m9utes4 ,n[1 ! groc] took tw5ty-f\r m9utes to weigh out ! sug>1 &1 ) ! half-m9ute delay1 sp5t #bd m94 #cj sec4 ov] ! task2 b ! drap] _h only to make =ty-sev5 cuts to divide ! roll ( clo?1 conta9+ =ty-eight y>ds1 into y>d pieces6 ,?is took him #ae m94 #dj sec41 & :5 we add ! eight m9utes & a half delay we get #bd m94 #aj sec41 f :i* x is cle> t ! drap] won ! race by tw5ty seconds4 ,! majority ( solv]s make =ty-eight cuts to divide ! roll into =ty-eight pieces6 #ce4--;,,judk9s',s ,,cattle4 ,z !re 7 five droves ) an equal numb] ( animals 9 ea* drove1 ! numb] mu/ be divisible by #e2 & z e "o ( ! eight deal]s b\ght ! same numb] ( animals1 ! numb] mu/ be divisible by #h4 ,!re=e ! numb] mu/ be a multiple ( #dj4 ,! highe/ possible multiple ( #dj t w "w w be f\nd to be #abj1 & ?is numb] c\ld be made up 9 "o ( two ways--#a ox1 #bc pigs1 & #if %eep1 or #c ox51 #h pigs1 & #aji %eep4 ,b ! fir/ is exclud$ by ! /atem5t t ! animals consi/$ ( ox51 pigs1 & %eep1 because a s+le ox is n ox54 ,!re=e ! second gr\p+ is ! correct answ]4 #cf4--;,,buy+ ,,apples4 ,z !re 7 ! same numb] ( boys z girls1 x is cle> t ! numb] ( *ildr5 mu/ be ev51 &1 ap>t f a c>eful & exact read+ ( ! "q1 !re w\ld be ?ree diff]5t answ]s4 ,!re might be two1 six1 or f\rte5 *ildr54 ,9 ! fir/ ( ^! cases !re are t5 diff]5t ways 9 :i* ! apples c\ld be b\ght4 ,b we 7 told !re 0 an equal numb] ( boys & girls1 & "o boy & "o girl are n boys & girls1 s ?is case has to be exclud$4 ,9 ! case ( f\rte5 *ildr51 ! only possible di/ribution is t ea* *ild receives "o halfp5ny apple4 ,b we 7 told t ea* *ild 0 to receive an equal di/ribution ( apples1 & "o apple is n apples1 s ?is case has also to be exclud$4 ,we are !re=e driv5 back on our ?ird case1 :i* exactly fits 9 ) all ! conditions4 ,?ree boys & ?ree girls ea* receive #a halfp5ny apple & #b ?ird-p5ny apples4 ,! value ( ^! #c apples is "o p5ny & "o-six?1 :i* multipli$ by six makes sev5p5ce4 ,consequ5tly1 ! correct answ] is t !re 7 six *ildr5--?ree girls & ?ree boys4 #cg4--;,,buy+ ,,*e/nuts4 ,9 solv+ ?is little puzzle we are conc]n$ ) ! exact int]pretation ( ! words us$ by ! buy] & sell]4 ,i w give ! "q aga91 ?is "t add+ a few words to make ! matt] m cle>4 ,! add$ words are pr9t$ 9 italics4 ,a man w5t into a %op to buy *e/nuts4 ,he said he want$ a p5nywor?1 & 0 giv5 five *e/nuts4 ',x is n en\gh2 ,i ought to h a six? ( a *e/nut m1' he rem>k$4 ',b if ,i give y "o *e/nut m1' ! %opman repli$1 'y w h five-six?s too _m4' ,n[1 /range to say1 !y 7 bo? "r4 ,h[ _m *e/nuts %\ld ! buy] receive = half a cr[n8 ,! answ] is t ! price 0 #aee *e/nuts = half a cr[n4 ,divide ?is numb] by #cj1 & we f9d t ! buy] 0 entitl$ to #ea_/f *e/nuts 9 ex*ange = 8 p5ny4 ,he 01 !re=e1 "r :5 he said1 aft] receiv+ five only1 t he /ill want$ a six?4 ,& ! salesman 0 also correct 9 say+ t if he gave "o *e/nut m 7t is1 six *e/nuts 9 all7 he w\ld be giv+ five-six?s ( a *e/nut 9 excess4 #ch4--,,! ,,bicycle ,,?ief4 ,p give all sorts ( absurd answ]s to ?is "q1 & yet x is p]fectly simple if "o j consid]s t ! salesman _c possibly h lo/ m ?an ! cycli/ actually /ole4 ,! latt] rode away ) a bicycle :i* co/ ! salesman elev5 p\nds1 & ! t5 p\nds *ange2 he ?us made (f ) tw5ty-"o p\nds1 9 ex*ange = a wor?less bit ( pap]4 ,?is is ! exact am\nt ( ! salesman's loss1 & ! o!r op]ations ( *ang+ ! *eque & borr[+ f a fri5d d n affect ! "q 9 ! slighte/4 ,! loss ( prospective pr(it on ! sale ( ! bicycle is1 ( c\rse1 n direct loss ( money out ( pocket4 #ci4--,,! ,,co/]mong]',s ,,puzzle4 ,bill mu/ h paid #hs4 p] hundr$ = 8 oranges--t is1 #abe = #ajs4 ,at #hs4 #d;d4 p] hundr$1 he w\ld only h receiv$ #abj oranges = #ajs4 ,?is exactly agrees ) ,bill's /atem5t4 #dj4--,,mamma',s ,,age4 ,! age ( ,mamma mu/ h be5 #bi ye>s #b mon?s2 t ( ,papa1 #ce ye>s2 & t ( ! *ild1 ,tommy1 #e ye>s #aj mon?s4 ,add$ toge!r1 ^! make sev5ty ye>s4 ,! fa!r is six times ! age ( ! son1 &1 aft] #bc ye>s #d mon?s h elaps$1 _! unit$ ages w am\nt to #adj ye>s1 & ,tommy w be j half ! age ( 8 fa!r4 #da4--,,_! ,,ages4 ,! g5tleman's age mu/ h be5 #ed ye>s & t ( 8 wife #de ye>s4 #db4--,,! ,,family ,,ages4 ,! ages 7 z foll[s3 ,billie1 #c ye>s2 ,g]trude1 #a ye>2 ,h5rietta1 #e ye>s2 ,*>lie1 #aj2 ye>s2 & ,janet1 #ba ye>s4 #dc4--,,mrs4 ,,timpk9s',s ,,age4 ,! age ( ! y\ng] at m>riage is always ! same z ! numb] ( ye>s t expire be=e ! eld] becomes twice h] age1 if he 0 ?ree times z old at m>riage4 ,9 our case x 0 eighte5 ye>s aft]w>ds2 !re=e ,mrs4 ,timpk9s 0 eighte5 ye>s ( age on ! w$d+-day1 & h] husb& fifty-f\r4 #dd4--;,a ,,c5sus ,,puzzle4 ,miss ,ada ,jork9s mu/ h be5 tw5ty-f\r & h] little bro!r ,johnnie ?ree ye>s ( age1 ) ?irte5 bro!rs & si/]s betwe54 ,!re 0 a trap = ! solv] 9 ! words sev5 times old] ?an little ,johnnie4 ,( c\rse1 sev5 times old] is equal to eight times z old4 ,x is surpris+ h[ _m p ha/ily assume t x is ! same z sev5 times z old4 ,"s ( ! be/ writ]s h committ$ ?is blund]4 ,probably _m ( my read]s ?\ght t ! ages #bd & #c 7 correct4 #de4--,,"m ,,& ,,daught]4 ,9 f\r & a half ye>s1 :5 ! daught] w be sixte5 ye>s & a half & ! "m =ty-n9e & a half ye>s ( age4 #df4--,,m>y ,,& ,,m>maduke4 ,m>maduke's age mu/ h be5 tw5ty-n9e ye>s & two-fif?s1 & ,m>y's n9ete5 ye>s & ?ree-fif?s4 ,:5 ,m>maduke 0 ag$ n9ete5 & ?ree-fif?s1 ,m>y 0 only n9e & f\r-fif?s2 s ,m>maduke 0 at t "t twice h] age4 #dg4--,,rov]',s ,,age4 ,rov]'s pres5t age is t5 ye>s & ,mildr$'s ?irty ye>s4 ,five ye>s ago _! respective ages 7 five & tw5ty-five4 ,rememb] t we said f\r times old] ?an ! dog1 :i* is ! same z five times z old4 7,see answ] to ,no4 #dd47 #dh4--;,,conc]n+ ,,tommy',s ,,age4 ,tommy ,sm>t's age mu/ h be5 n9e ye>s & ?ree-fif?s4 ,ann's age 0 sixte5 & f\r-fif?s1 ! "m's ?irty-eight & two-fif?s1 & ! fa!r's fifty & two-fif?s4 #di4--,,next-,,door ,,neighb\rs4 ,mr4 ,jupp #ci1 ,mrs4 ,jupp #cd1 ,julia #ad1 & ,joe #ac2 ,mr4 ,simk9 #db2 ,mrs4 ,simk9 #dj2 ,sophy #aj2 & ,sammy #h4 #ej4--,,! ,,bag ,,( ,,nuts4 ,x w be f\nd t :5 ,h]b]t takes twelve1 ,rob]t & ,*ri/oph] w take n9e & f\rte5 respectively1 & t !y w h toge!r tak5 ?irty-five nuts4 ,z #ce is conta9$ 9 #ggj tw5ty-two times1 we h m]ely to multiply #ab1 #i1 & #ad by #bb to discov] t ,h]b]t's %>e 0 #bfd1 ,rob]t's #aih1 & ,*ri/oph]'s #cjh4 ,!n1 z ! total ( _! ages is #ag ye>s or half ! sum ( #ab1 #i1 & #ad1 _! respective ages mu/ be #f1 #d1 & #g ye>s4 #ea4--;,,h[ ,,old ,,0 ,,m>y8 ,! age ( ,m>y to t ( ,ann mu/ be z #e to #c4 ,& z ! sum ( _! ages 0 #dd1 ,m>y 0 #bg & ,ann #af4 ,"o is exactly #aa ye>s old] ?an ! o!r4 ,i w n[ ins]t 9 brackets 9 ! orig9al /atem5t ! v>i\s ages specifi$3 ,m>y is 7#bg7 twice z old z ,ann 0 7#ac7 :5 ,m>y 0 half z old 7#bd7 z ,ann w be 7#di7 :5 ,ann is ?ree times z old 7#di7 z ,m>y 0 7#af7 :5 ,m>y 0 7#af7 ?ree times z old z ,ann 7#e74 ,n[1 *eck ?is backw>ds4 ,:5 ,m>y 0 ?ree times z old z ,ann1 ,m>y 0 #af & ,ann #e 7#aa ye>s y\ng]74 ,!n we get #di = ! age ,ann w be :5 %e is ?ree times z old z ,m>y 0 !n4 ,:5 ,m>y 0 half ?is %e 0 #bd4 ,& at t "t ,ann mu/ h be5 #ac 7#aa ye>s y\ng]74 ,!re=e ,m>y is n[ twice z old--#bg1 & ,ann #aa ye>s y\ng]--#af4 #eb4--,,que] ,,relation%ips4 ,if a man m>ries a woman1 :o dies1 & he !n m>ries 8 deceas$ wife's si/] & himself dies1 x may be correctly said t he _h 7previ\sly7 m>ri$ ! si/] ( 8 wid[4 ,! y\? 0 n ! nephew ( ,jane ,br[n1 because he happ5$ to be h] son4 ,h] surname 0 ! same z t ( h] bro!r1 because %e _h m>ri$ a man ( ! same "n z h]self4 #ec4--;,,he>d ,,on ,,! ,,tube ,,railway4 ,! g5tleman 0 ! second lady's uncle4 #ed4--;,a ,,family ,,p>ty4 ,! p>ty consi/$ ( two little girls & a boy1 _! fa!r & "m1 & _! fa!r's fa!r & "m4 #ee4--;,a ,,mix$ ,,p$igree4 ,illu/ration3 ,?os4 ,bloggs m 4 4 4 4 4 ------------------------------------ ,w4 ,snoggs m ,kate ,bloggs4 4 4 m ,h5ry ,bloggs4 ,joseph ,bloggs m --------------------- ,jane ,john ,alf4 ,m>y ,bloggs m ,snoggs ,snoggs m ,bloggs ,! lett] m /&s = m>ri$4 ,x w be se5 t ,john ,snoggs c say to ,joseph ,bloggs1 ,y are my fa!r's bro!r-9-law1 because my fa!r m>ri$ y\r si/] ,kate2 y are my bro!r's fa!r-9-law1 because my bro!r ,alfr$ m>ri$ y\r daught] ,m>y2 & y are my fa!r-9-law's bro!r1 because my wife ,jane 0 y\r bro!r ,h5ry's daught]4 #ef4--,,wilson',s ,,pos]4 ,if !re are two m51 ea* ( :om m>ries ! "m ( ! o!r1 & !re is a son ( ea* m>riage1 !n ea* ( su* sons w be at ! same "t uncle & nephew ( ! o!r4 ,!re are o!r ways 9 :i* ! relation%ip may be br\ght ab\t1 b ?is is ! simple/4 #eg4--,,:at ,,0 ,,! ,,"t8 ,! "t mu/ h be5 #i4cf p4m4 ,a qu>t] ( ! "t s9ce noon is #b hr4 #bd m941 & a half ( ! "t till noon next day is #g hr4 #ab m94 ,^! add$ toge!r make #i hr4 #cf m94 #eh4--;,a ,,"t ,,puzzle4 ,tw5ty-six m9utes4 #ei4--;,a ,,puzzl+ ,,wat*4 ,if ! #fe m9utes be c\nt$ on ! face ( ! same wat*1 !n ! problem w\ld be impossible3 = ! h&s mu/ co9cide e #fee_/aa m9utes z %[n by its face1 & x matt]s n :e!r x runs fa/ or sl[2 b if x is measur$ by true "t1 x ga9s #e_/aa ( a m9ute 9 #fe m9utes1 or #fj_/adc ( a m9ute p] h\r4 #fj4--,,! ,,wap%aw',s ,,:>f ,,my/]y4 ,!re are elev5 diff]5t times 9 twelve h\rs :5 ! h\r & m9ute h&s ( a clock are exactly "o above ! o!r4 ,if we divide #ab h\rs by #aa we get #a hr4 #e m94 #bgc_/aa sec41 & ?is is ! "t aft] twelve o'clock :5 !y are fir/ toge!r1 & also ! "t t elapses betwe5 "o occasion ( ! h&s be+ toge!r & ! next4 ,!y are toge!r = ! second "t at #b hr4 #aj m94 #edf_/aa sec4 7twice ! above "t72 next at #c hr4 #af m94 #bai_/aa sec42 next at #d hr4 #ba m94 #dia_/aa sec4 ,?is la/ is ! only occasion on :i* ! two h&s are toge!r ) ! second h& j pa/ ! =ty-n9? second4 ,?is1 !n1 is ! "t at :i* ! wat* mu/ h /opp$4 ,guy ,boo?by1 9 ! op5+ s5t5ce ( 8 ,across ! ,_w = a ,wife1 says1 ,x 0 a cold1 dre>y w9t]'s aft]noon1 & by ! "t ! h&s ( ! clock on my mantelpiece jo9$ =ces & /ood at tw5ty m9utes pa/ f\r1 my *amb]s 7 well-nigh z d>k z midnight4 ,x is evid5t t ! au?or "h made a slip1 =1 z we h se5 above1 he is #a m94 #dia_/aa sec4 out 9 8 reckon+4 #fa4--;,,*ang+ ,,places4 ,!re are ?irty-six pairs ( times :5 ! h&s exactly *ange places betwe5 ?ree p4m4 & midnight4 ,! numb] ( pairs ( times f any h\r 7n7 to midnight is ! sum ( #ab - 7n #a7 natural numb]s4 ,9 ! case ( ! puzzle n #c2 !re=e #ab - 7#c #a7 #h & #a #b #c #d #e #f #g #h #cf1 ! requir$ answ]4 ,! fir/ pair ( times is #c hr4 #baeg_/adc m94 & #d hr4 #afaab_/adc m941 & ! la/ pair is #aj hr4 #eihc_/adc m94 & #aa hr4 #edach_/adc m94 ,i w n give all ! rema9d] ( ! ?irty-six pairs ( times1 b supply a =mula by :i* any ( ! sixty-six pairs t occur f midday to midnight may be at once f\nd3-- #gbj;b #fj;a #gbj;a #fj;b m94 a hr ---------- m94 & b hr4 --------------- #adc #adc ,= ! lett] a may be sub/itut$ any h\r f #j1 #a1 #b1 #c up to #aj 7:]e n\ght /&s = #ab o'clock midday72 & b may repres5t any h\r1 lat] ?an a1 up to #aa4 ,by ! aid ( ?is =mula !re is no difficulty 9 discov]+ ! answ] to ! second "q3 a #h & b #aa w give ! pair #h hr4 #ehajf_/adc m94 & #aa hr4 #ddabh_/adc m941 ! latt] be+ ! "t :5 ! m9ute h& is ne>e/ ( all to ! po9t ,,ix--9 fact1 x is only #ae_/adc ( a m9ute di/ant4 ,read]s may f9d x in/ructive to make a table ( all ! sixty-six pairs ( times :5 ! h&s ( a clock *ange places4 ,an easy way is z foll[s3 ,make a column = ! fir/ times & a second column = ! second times ( ! pairs4 ,by mak+ a #j & b #a 9 ! above expressions we f9d ! fir/ case1 & ent] hr4 #ee_/adc m94 at ! head ( ! fir/ column1 & #a hr4 #jfj_/adc m94 at ! head ( ! second column4 ,n[1 by successively add+ #ee_/adc m94 9 ! fir/1 & #a hr4 #jfj_/adc m94 9 ! second column1 we get all ! elev5 pairs 9 :i* ! fir/ "t is a c]ta9 numb] ( m9utes aft] n\ght1 or mid-day4 ,!n !re is a jump 9 ! times1 b y c f9d ! next pair by mak+ a #a & b #b1 & !n by successively add+ ^! two times z be=e y w get all ! t5 pairs aft] #a o'clock4 ,!n !re is ano!r jump1 & y w be able to get by addition all ! n9e pairs aft] #b o'clock4 ,& s on to ! end4 ,i w leave read]s to inve/igate = !mselves ! nature & cause ( ! jumps4 ,9 ?is way we get "u ! successive h\rs1 #aa #aj #i #h #g #f #e #d #c #b #a #ff pairs ( times1 :i* result agrees ) ! =mula 9 ! fir/ p>agraph ( ?is article4 ,"s "t ago ! pr9cipal ( a ,civil ,s]vice ,tra9+ ,college1 :o conducts a ,civil ,s]vice ,column 9 "o ( ! p]iodicals1 _h ! qu]y address$ to him1 ,h[ soon aft] ,,xii o'clock w a clock ) bo? h&s ( ! same l5g? be ambigu\s8 ,8 fir/ answ] 01 ,"s "t pa/ "o o'clock1 b he v>i$ ! answ] f issue to issue4 ,at l5g? "s ( 8 read]s conv9c$ him t ! answ] is1 ,at #ee_/adc m94 pa/ ,,xii2 & ?is he f9ally gave z correct1 toge!r ) ! reason = x t at t "t ! "t indicat$ is ! same :i*ev] h& y may assume z h\r h&6 #fb4--,,! ,,club ,,clock4 ,! positions ( ! h&s %[n 9 ! illu/ration c\ld only indicate t ! clock /opp$ at #dd m94 #eaaadc_/adbg sec4 aft] elev5 o'clock4 ,! second h& w\ld next be exactly midway betwe5 ! o!r two h&s at #de m94 #ebdif_/adbg sec4 aft] elev5 o'clock4 ,if we _h be5 deal+ ) ! po9ts on ! circle to :i* ! ?ree h&s are direct$1 ! answ] w\ld be #de m94 #bbajf_/adbg sec4 aft] elev52 b ! "q appli$ to ! h&s1 & ! second h& w\ld n be betwe5 ! o!rs at t "t1 b outside !m4 #fc4--,,! ,,/op-,,wat*4 ,! "t indicat$ on ! wat* 0 #ee_/aa m94 pa/ #i1 :5 ! second h& w\ld be at #bgc_/aa sec4 ,! next "t ! h&s w\ld be simil> di/ances ap>t w\ld be #edf_/aa m94 pa/ #b1 :5 ! second h& w\ld be at #cbh_/aa sec4 ,b y ne$ only hold ! wat* 7or our previ\s illu/ration ( x7 9 front ( a mirror1 :5 y w see ! second "t reflect$ 9 x6 ,( c\rse1 :5 reflect$1 y w read ,,xi z ,i1 ,x z ,,ii1 & s on4 #fd4--,,! ,,?ree ,,clocks4 ,z a m]e ari?metical problem ?is "q pres5ts no difficulty4 ,9 ord] t ! h&s %all all po9t to twelve o'clock at ! same "t1 x is necess>y t ,b %all ga9 at lea/ twelve h\rs & t ,c %all lose twelve h\rs4 ,z ,b ga9s a m9ute 9 a day ( tw5ty-f\r h\rs1 & ,c loses a m9ute 9 precisely ! same "t1 x is evid5t t "o w h ga9$ #gbj m9utes 7j twelve h\rs7 9 #gbj days1 & ! o!r w h lo/ #gbj m9utes 9 #gbj days4 ,clock ,a keep+ p]fect "t1 all ?ree clocks mu/ indicate twelve o'clock simultane\sly at noon on ! #gbj? day f ,april #a1 #ahih4 ,:at day ( ! mon? w t be8 ,i publi%$ ?is little puzzle 9 #ahih to see h[ _m p 7 aw>e ( ! fact t #aijj w\ld n be a leap ye>4 ,x 0 surpris+ h[ _m 7 !n ignorant on ! po9t4 ,e ye> t c be divid$ by f\r )\t a rema9d] is bissextile or leap ye>1 ) ! exception t "o leap ye> is cut (f 9 ! c5tury4 #ahjj 0 n a leap ye>1 nor 0 #aijj4 ,on ! o!r h&1 h[ev]1 to make ! cal5d> m ne>ly agree ) ! sun's c\rse1 e f\r? hundr$ ye> is /ill consid]$ bissextile4 ,consequ5tly1 #bjjj1 #bdjj1 #bhjj1 #cbjj1 etc41 w all be leap ye>s4 ,may my read]s live to see !m4 ,we !re=e f9d t #gbj days f noon ( ,april #a1 #ahih1 br+s u to noon ( ,m>* #bb1 #aijj4 #fe4--,,! ,,railway ,,/ation ,,clock4 ,! "t mu/ h be5 #dcg_/aa m94 pa/ two o'clock4 #ff4--,,! ,,village ,,simpleton4 ,! day ( ! week on :i* ! conv]sation took place 0 ,sunday4 ,= :5 ! day aft] to-morr[ 7,tuesday7 is ye/]day1 to-day w be ,w$nesday2 & :5 ! day be=e ye/]day 7,friday7 0 to-morr[1 to-day 0 ,?ursday4 ,!re are two days betwe5 ,?ursday & ,sunday1 & betwe5 ,sunday & ,w$nesday4 #fg4--;,,av]age ,,spe$4 ,! av]age spe$ is twelve miles an h\r1 n twelve & a half1 z mo/ p w ha/ily decl>e4 ,take any di/ance y l1 say sixty miles4 ,?is w\ld h tak5 six h\rs go+ & f\r h\rs return+4 ,! d\ble j\rney ( #abj miles w\ld ?us take t5 h\rs1 & ! av]age spe$ is cle>ly twelve miles an h\r4 #fh4--,,! ,,two ,,tra9s4 ,"o tra9 0 runn+ j twice z fa/ z ! o!r4 #fi4--,,! ,,?ree ,,villages4 ,call+ ! ?ree villages by _! initial lett]s1 x is cle> t ! ?ree roads =m a triangle1 ,a1 ,b1 ,c1 ) a p]p5dicul>1 measur+ twelve miles1 dropp$ f ,c to ! base ,a1 ,b4 ,?is divides our triangle into two "r-angl$ triangles ) a twelve-mile side 9 common4 ,x is !n f\nd t ! di/ance f ,a to ,c is #ae miles1 f ,c to ,b #bj miles1 & f ,a to ,b #be 7t is #i & #af7 miles4 ,^! figures are easily prov$1 = ! squ>e ( #ab add$ to ! squ>e ( #i equals ! squ>e ( #ae1 & ! squ>e ( #ab add$ to ! squ>e ( #af equals ! squ>e ( #bj4 #gj4--;,,draw+ ,,h] ,,p5sion4 ,! di/ance mu/ be #f miles4 #ga4--,,sir ,,edwyn ,,de ,,tudor4 ,! di/ance mu/ h be5 sixty miles4 ,if ,sir ,edwyn left at noon & rode #ae miles an h\r1 he w\ld arrive at f\r o'clock--an h\r too soon4 ,if he rode #aj miles an h\r1 he w\ld arrive at six o'clock--an h\r too late4 ,b if he w5t at #ab miles an h\r1 he w\ld rea* ! ca/le ( ! wick$ b>on exactly at five o'clock--! "t appo9t$4 #gb4--,,! ,,hydroplane ,,"q4 ,! ma*9e mu/ h gone at ! rate ( sev5-tw5ty-f\r?s ( a mile p] m9ute & ! w9d travell$ five-tw5ty-f\r?s ( a mile p] m9ute4 ,?us1 go+1 ! w9d w\ld help1 & ! ma*9e w\ld d twelve-tw5ty-f\r?s1 or half a mile a m9ute1 & return+ only two-tw5ty-f\r?s1 or "o-twelf? ( a mile p] m9ute1 ! w9d be+ aga9/ x4 ,! ma*9e )\t any w9d c\ld !re=e d ! t5 miles 9 ?irty-f\r & two-sev5?s m9utes1 s9ce x c\ld d sev5 miles 9 tw5ty-f\r m9utes4 #gc4--;,,donkey ,,rid+4 ,! complete mile 0 run 9 n9e m9utes4 ,f ! facts /at$ we _c det]m9e ! "t tak5 ov] ! fir/ & second qu>t]-miles sep>ately1 b toge!r !y1 ( c\rse1 took f\r & a half m9utes4 ,! la/ two qu>t]s 7 run 9 two & a qu>t] m9utes ea*4 #gd4--,,! ,,basket ,,( ,,potatoes4 ,multiply toge!r ! numb] ( potatoes1 ! numb] less "o1 & twice ! numb] less "o1 !n divide by #c4 ,?us #ej1 #di1 & #ii multipli$ toge!r make #bdb1eej1 :i*1 divid$ by #c1 gives u #hj1hej y>ds z ! correct answ]4 ,! boy w\ld ?us h to travel #de miles & fifte5-sixte5?s--a nice little recreation aft] a day's "w4 #ge4--,,! ,,pass5g]',s ,,f>e4 ,mr4 ,tompk9s %\ld h paid fifte5 %ill+s z 8 correct %>e ( ! motor-c> f>e4 ,he only %>$ half ! di/ance travell$ = #c1 & !re=e %\ld pay half ( ?irty %ill+s1 or fifte5 %ill+s4 #gf4--,,! ,,b>rel ,,( ,,be]4 ,"h ! digital roots ( ! six numb]s are #f1 #d1 #a1 #b1 #g1 #i1 :i* toge!r sum to #bi1 ^: digital root is #b4 ,z ! cont5ts ( ! b>rels sold mu/ be a numb] divisible by #c1 if "o buy] pur*as$ twice z mu* z ! o!r1 we mu/ f9d a b>rel ) root #b1 #e1 or #h to set on "o side4 ,!re is only "o b>rel1 t conta9+ #bj gallons1 t fulfils ^! conditions4 ,s ! man mu/ h kept ^! #bj gallons ( be] = 8 own use & sold "o man #cc gallons 7! #ah-;gallon & #ae-;gallon b>rels7 & sold ! o!r man #ff gallons 7! #af1 #ai1 & #ca gallon b>rels74 #gg4--;,,digits ,,& ,,squ>es4 ,! top r[ mu/ be "o ( ! f\r foll[+ numb]s3 #aib1 #bai1 #bgc1 #cbg4 ,! fir/ 0 ! example giv54 #gh4--,,odd ,,& ,,ev5 ,,digits4 ,z we h to exclude complex & improp] fractions & recurr+ decimals1 ! simple/ solution is ?is3 #gi #ea_/c & #hd #b_/f1 bo? equal #hda_/c4 ,)\t any use ( fractions x is obvi\sly impossible4 #gi4--,,! ,,lock]s ,,puzzle4 ,! smalle/ possible total is #cef #ajg #bdi1 & ! l>ge/ sum possible is #iha #bce #gdf1 or #fegcbd4 ,! middle sum may be ei!r #gbj #acd #ehf1 or #gjb #acd #efh1 or #djg #ach #bfi4 ,! total 9 ?is case mu/ be made up ( ?ree ( ! figures #j1 #b1 #d1 #g1 b no sum o!r ?an ! ?ree giv5 c possibly be obta9$4 ,we h !re=e no *oice 9 ! case ( ! fir/ lock]1 an alt]native 9 ! case ( ! ?ird1 & any "o ( ?ree arrangem5ts 9 ! case ( ! middle lock]4 ,"h is "o solution3-- #ajg #acd #bce #bdi #ehf #gdf --- --- --- #cef #gbj #iha ,( c\rse1 9 ea* case figures 9 ! fir/ two l9es may be ex*ang$ v]tically )\t alt]+ ! total1 & z a result !re are j #c1jgb diff]5t ways 9 :i* ! figures might be actually plac$ on ! lock] doors4 ,i mu/ cont5t myself ) %[+ "o little pr9ciple involv$ 9 ?is puzzle4 ,! sum ( ! digits 9 ! total is always gov]n$ by ! digit omitt$4 #i_/i - #g_/aj - #e_/aa -#c_/ab - #a_/ac - #h_/ad - #f_/ae - #d_/af - #b_/ag - #j_/ah4 ,:i*ev] digit %[n "h 9 ! upp] l9e we omit1 ! sum ( ! digits 9 ! total w be f\nd b5ea? x4 ,?us 9 ! case ( lock] ,a we omitt$ #h1 & ! figures 9 ! total sum up to #ad4 ,if1 !re=e1 we want$ to get #cef1 we may "k at once to a c]ta9ty t x c only be obta9$ 7if at all7 by dropp+ ! #h4 #hj4--,,! ,,?ree ,,gr\ps4 ,!re are n9e solutions to ?is puzzle1 z foll[s1 & no m3-- #ab #dhc #e1gif #bg #aih #e1cdf #db #ach #e1gif #ci #ahf #g1bed #ah #big #e1cdf #dh #aei #g1fcb #bh #aeg #d1cif #d #a1gch #f1ieb #d #a1ifc #g1heb ,! sev5? answ] is ! "o t is mo/ likely to be ov]look$ by solv]s ( ! puzzle4 #ha4--,,! ,,n9e ,,c\nt]s4 ,9 ?is case a c]ta9 am\nt ( m]e trial is unavoidable4 ,b !re are two k9ds ( trials--^? t are purely haphaz>d1 & ^? t are me?odical4 ,! true puzzle lov] is nev] satisfi$ ) m]e haphaz>d trials4 ,! read] w f9d t by j rev]s+ ! figures 9 #bc & #df 7mak+ ! multipli]s #cb & #fd7 bo? products w be #e1jef4 ,?is is an improvem5t1 b x is n ! correct answ]4 ,we c get z l>ge a product z #e1efh if we multiply #agd by #cb & #if by #eh1 b ?is solution is n to be f\nd )\t ! ex]cise ( "s judgm5t & pati5ce4 #hb4--,,! ,,t5 ,,c\nt]s4 ,z ,i po9t$ out1 x is q easy s to arrange ! c\nt]s t !y %all =m a pair ( simple multiplication sums1 ea* ( :i* w give ! same product--9 fact1 ?is c be done by anybody 9 five m9utes ) a little pati5ce4 ,b x is q ano!r matt] to f9d t pair :i* gives ! l>ge/ product & t :i* gives ! smalle/ product4 ,n[1 9 ord] to get ! smalle/ product1 x is necess>y to select z multipli]s ! two smalle/ possible numb]s4 ,if1 !re=e1 we place #a & #b z multipli]s1 all we h to d is to arrange ! rema9+ eight c\nt]s 9 su* a way t !y %all =m two numb]s1 "o ( :i* is j d\ble ! o!r2 & 9 do+ ?is we mu/1 ( c\rse1 try to make ! small] numb] z l[ z possible4 ,( c\rse ! l[e/ numb] we c\ld get w\ld be #c1jde2 b ?is w n "w1 nei!r w #c1dje1 #c1de,o1 etc41 & x may be asc]ta9$ t #c1dhe is ! l[e/ possible4 ,"o ( ! requir$ answ]s is #c1dhe #b #f1igj1 & #f1igj #a #f1igj4 ,! o!r "p ( ! puzzle 7f9d+ ! pair ) ! highe/ product7 is1 h[ev]1 ! real knotty po9t1 = x is n at all easy to discov] :e!r we %\ld let ! multipli] consi/ ( "o or ( two figures1 ?\gh x is cle> t we mu/ keep1 s f> z we c1 ! l>ge/ figures to ! left 9 bo? multipli] & multiplic&4 ,x w be se5 t by ! foll[+ arrangem5t s high a numb] z #eh1efj may be obta9$4 ,?us1 #iae #fd #eh1efj1 & #gcb #hj #eh1efj4 #hc4--;,,digital ,,multiplication4 ,! solution t gives ! smalle/ possible sum ( digits 9 ! common product is #bc #agd #eh #fi #d1jjb1 & ! solution t gives ! l>ge/ possible sum ( digits1 #ifed #ahcbge1hhf4 ,9 ! fir/ case ! digits sum to #f & 9 ! second case to #bg4 ,!re is no way ( obta9+ ! solution b by actual trial4 #hd4--,,! ,,pi]rot',s ,,puzzle4 ,!re are j six diff]5t solutions to ?is puzzle1 z foll[s3-- #h multipli$ by #dgc equals #cghd #i #cea #caei #ae #ic #acie #ba #hg #abhg #bg #ha #bahg #ce #da #adce ,x w be se5 t 9 e case ! two multipli]s conta9 exactly ! same figures z ! product4 #he4--,,! ,,cab ,,numb]s4 ,! highe/ product is1 ,i ?9k1 obta9$ by multiply+ #h1gde1bca by #if--namely1 #hci1edb1agf4 ,deal+ "h ) ! problem g5]ally1 ,i h %[n 9 ! la/ puzzle t ) ?ree digits !re are only two possible solutions1 & ) f\r digits only six diff]5t solutions4 ,^! cases h all be5 giv54 ,) five digits !re are j tw5ty-two solutions1 z foll[s3-- #c #dabh #abchd #c #dbha #abhdc #c #gabe #bacge #c #gbea #bagec #beda #f #aebdf #fea #bd #aefbd #fgh #db #bhdgf #bdf #ea #abedf #eg #hcd #dgech #ge #bca #agcbe #fbd #gh #dhfgb #dce #hg #cghde ------ #i #gdfa #fgadi #gb #icf #fgcib ------ #b #hgad #agdbh #b #hgda #agdhb #fe #bha #ahbfe #fe #ihc #fcihe ------ #digc #h #cighd #feba #h #ebafh #ad #ibf #abifd #hf #bea #baehf ,n[1 if we took e possible comb9ation & te/$ x by multiplication1 we %\ld ne$ to make no few] ?an #cj1bdj trials1 or1 if we at once reject$ ! numb] #a z a multipli]1 #bh1efj trials--a task t ,i ?9k mo/ p w\ld be incl9$ to %irk4 ,b let u consid] :e!r !re be no %ort] way ( gett+ at ! results requir$4 ,i h already expla9$ t if y add toge!r ! digits ( any numb] & !n1 z (t5 z necess>y1 add ! digits ( ! result1 y mu/ ultimately get a numb] compos$ ( "o figure4 ,?is la/ numb] ,i call ! digital root4 ,x is necess>y 9 e solution ( our problem t ! root ( ! sum ( ! digital roots ( our multipli]s %all be ! same z ! root ( _! product4 ,!re are only f\r ways 9 :i* ?is c happ53 :5 ! digital roots ( ! multipli]s are #c & #f1 or #i & #i1 or #b & #b1 or #e & #h4 ,i h divid$ ! tw5ty-two answ]s above into ^! f\r classes4 ,x is ?us evid5t t ! digital root ( any product 9 ! fir/ two classes mu/ be #i1 & 9 ! second two classes #d4 ,ow+ to ! fact t no numb] ( five figures c h a digital sum less ?an #ae or m ?an #ce1 we f9d t ! figures ( our product mu/ sum to ei!r #ah or #bg to produce ! root #i1 & to ei!r #bb or #ca to produce ! root #d4 ,!re are #c ways ( select+ five diff]5t figures t add up to #ah1 !re are #aa ways ( select+ five figures t add up to #bg1 !re are #i ways ( select+ five figures t add up to #bb1 & #e ways ( select+ five figures t add up to #ca4 ,!re are1 !re=e1 #bh diff]5t gr\ps1 & no m1 f any "o ( :i* a product may be =m$4 ,we next write out 9 a column ^! #bh sets ( five figures1 & proce$ to tabulate ! possible factors1 or multipli]s1 into :i* !y may be split4 ,r\ghly speak+1 !re w\ld n[ appe> to be ab\t #b1jjj possible cases to be tri$1 in/ead ( ! #cj1bdj m5tion$ above2 b ! process ( elim9ation n[ beg9s1 & if ! read] has a quick eye & a cle> head he c rapidly dispose ( ! l>ge bulk ( ^! cases1 & !re w be comp>atively few te/ multiplications necess>y4 ,x w\ld take f> too mu* space to expla9 my own me?od 9 detail1 b ,i w take ! fir/ set ( figures 9 my table & %[ h[ easily x is done by ! aid ( little tricks & dodges t %\ld occur to ev]ybody z he goes along4 ,my fir/ product gr\p ( five figures is #hd1cba4 ,"h1 z we h se51 ! root ( ea* factor mu/ be #c or a multiple ( #c4 ,z !re is no #f or #i1 ! only s+le multipli] is #c4 ,n[1 ! rema9+ f\r figures c be arrang$ 9 #bd diff]5t ways1 b !re is no ne$ to make #bd multiplications4 ,we see at a glance t1 9 ord] to get a five-figure product1 ei!r ! #h or ! #d mu/ be ! fir/ figure to ! left4 ,b unless ! #b is prec$$ on ! "r by ! #h1 x w produce :5 multipli$ ei!r a #f or a #g1 :i* mu/ n occur4 ,we are1 !re=e1 r$uc$ at once to ! two cases1 #c #d1abh & #c x #d1bha1 bo? ( :i* give correct solutions4 ,suppose next t we are try+ ! two-figure factor1 #ba4 ,"h we see t if ! numb] to be multipli$ is "u #ejj ! product w ei!r h only f\r figures or beg9 ) #aj4 ,!re=e we h only to exam9e ! cases #ba #hdc & #ba #hcd4 ,b we "k t ! fir/ figure w be repeat$1 & t ! second figure w be twice ! fir/ figure add$ to ! second4 ,consequ5tly1 z twice #c add$ to #d produces a n\ght 9 our product1 ! fir/ case is at once reject$4 ,x only rema9s to try ! rema9+ case by multiplication1 :5 we f9d x does n give a correct answ]4 ,if we are next try+ ! factor #ab1 we see at ! />t t nei!r ! #h nor ! #c c be 9 ! units place1 because !y w\ld produce a #f1 & s on4 ,a %>p eye & an al]t judgm5t w enable u ?us to run ?r\gh our table 9 a mu* %ort] "t ?an w\ld be expect$4 ,! process took me a little m ?an ?ree h\rs4 ,i h n attempt$ to enum]ate ! solutions 9 ! cases ( six1 sev51 eight1 & n9e digits1 b ,i h record$ ne>ly fifty examples ) n9e digits alone4 #hf4--,,que] ,,multiplication4 ,if we multiply #cbedghia by #f1 we get ! product1 #aiebhgcdf4 ,9 bo? cases all ! n9e digits are us$ once & once only4 #hg4--,,! ,,numb] ,,*ecks ,,puzzle4 ,divide ! t5 *ecks into ! foll[+ ?ree gr\ps3 #g #a #e--d #f--c #b #h #i #j1 & ! fir/ multipli$ by ! second produces ! ?ird4 #hh4--;,,digital ,,division4 ,x is conv5i5t to consid] ! digits z arrang$ to =m fractions ( ! respective values1 "o-half1 "o-?ird1 "o-f\r?1 "o-fif?1 "o-six?1 "o-sev5?1 "o-eigh?1 & "o-n9?4 ,i w fir/ give ! eight answ]s1 z foll[s3-- #fgbi_/acdeh #a_/b #ehbc_/agdfi #a_/c #cidb_/aegfh #a_/d #bfig_/acdhe #a_/e #bidc_/agfeh #a_/f #bcid_/afgeh #a_/g #cahg_/bedif #a_/h #fcha_/egdbi #a_/i ,! sum ( ! num]ator digits & ! d5om9ator digits w1 ( c\rse1 always be #de1 & ! digital root is #i4 ,n[1 if we sep>ate ! n9e digits into any two gr\ps1 ! sum ( ! two digital roots w always be #i4 ,9 fact1 ! two digital roots mu/ be ei!r #i--i1 #h--a1 #g--b1 #f--c1 or #e--d4 ,9 ! fir/ case ! actual sum is #ah1 b !n ! digital root ( ?is numb] is itself #i4 ,! solutions 9 ! cases ( "o-?ird1 "o-f\r?1 "o-six?1 "o-sev5?1 & "o-n9? mu/ be ( ! =m #i--i2 t is to say1 ! digital roots ( bo? num]ator & d5om9ator w be #i4 ,9 ! cases ( "o-half & "o-fif?1 h[ev]1 ! digital roots are #f--c1 b ( c\rse ! high] root may occur ei!r 9 ! num]ator or 9 ! d5om9ator2 ?us #bfig_/acdhe1 #bgfi_/achde1 #bigc_/adhfe1 #cgbi_/ahfde1 :]e1 9 ! fir/ two arrangem5ts1 ! roots ( ! num]ator & d5om9ator are respectively #f--c1 & 9 ! la/ two #c--f4 ,! mo/ curi\s case ( all is1 p]haps1 "o-eigh?1 = "h ! digital roots may be ( any "o ( ! five =ms giv5 above4 ,! d5om9ators ( ! fractions be+ reg>d$ z ! num]ators multipli$ by #b1 #c1 #d1 #e1 #f1 #g1 #h1 & #i respectively1 we mu/ pay att5tion to ! c>ry+s ov]4 ,9 ord] to get five figures 9 ! product !re w1 ( c\rse1 always be a c>ry-ov] aft] multiply+ ! la/ figure to ! left1 & 9 e case high] ?an #d we mu/ c>ry ov] at lea/ ?ree times4 ,consequ5tly 9 cases f "o-fif? to "o-n9? we _c produce diff]5t solutions by a m]e *ange ( position ( pairs ( figures1 z1 = example1 we may ) #ehcb_/agdif & #ehbc_/agdfi1 :]e ! #b_/f & #c_/i *ange places4 ,x is true t ! same figures may (t5 be diff]5tly arrang$1 z %[n 9 ! two pairs ( values = "o-fif? t ,i h giv5 9 ! la/ p>agraph1 b "h x w be f\nd !re is a g5]al readju/m5t ( figures & n a simple *ang+ ( ! positions ( pairs4 ,!re are o!r little po9ts t w\ld occur to e solv]--su* z t ! figure #e _c ev] appe> to ! extreme "r ( ! num]ator1 z ?is w\ld result 9 our gett+ ei!r a n\ght or a second #e 9 ! d5om9ator4 ,simil>ly #a _c ev] appe> 9 ! same position1 nor #f 9 ! fraction "o-six?1 nor an ev5 figure 9 ! fraction "o-fif?1 & s on4 ,! prelim9>y consid]ation ( su* po9ts z ,i h t\*$ ^u w n only prev5t our wa/+ a lot ( "t 9 try+ to produce impossible =ms1 b w lead u m or less directly to ! desir$ solutions4 #hi4--;,,add+ ,,! ,,digits4 ,! smalle/ possible sum ( money is #a1 #hs4 #i;d41 ! digits ( :i* add to #be4 #ij4--,,! ,,c5tury ,,puzzle4 ,! problem ( express+ ! numb] #ajj z a mix$ numb] or fraction1 us+ all ! n9e digits once1 & once only1 has1 l all ^! digital puzzles1 a fasc9at+ side to x4 ,! m]e/ tyro c by pati5t trial obta9 correct results1 & !re is a s+ul> pleasure 9 discov]+ & record+ ea* new arrangem5t ak9 to ! delight ( ! botani/ 9 f9d+ "s long-s\ght plant4 ,x is simply a matt] ( arrang+ ^? n9e figures correctly1 & yet ) ! ?\s&s ( possible comb9ations t confront u ! task is n s easy z might at fir/ appe>1 if we are to get a consid]able numb] ( results4 ,"h are elev5 answ]s1 includ+ ! "o ,i gave z a specim53-- #badh #ageb #adbh #aegh #if ----1 #if ----1 #if ----1 #id ----1 #ecg #dch #ceg #bfc #gebd #ehbc #egdb #cedf #ia ----1 #ia ----1 #ia ----1 #hb ----1 #hcf #fdg #fch #aig #gebd #efdc #fibeh #ha ----1 #ha ----1 #c -----4 #cif #big #gad ,n[1 z all ! fractions necess>ily repres5t :ole numb]s1 x w be conv5i5t to deal ) !m 9 ! foll[+ =m3 #if #d1 #id #f1 #ia #i1 #hb #ah1 #ha #ai1 & #c #ig4 ,) any :ole numb] ! digital roots ( ! fraction t br+s x up to #ajj w always be ( "o p>ticul> =m4 ,?us1 9 ! case ( #if #d1 "o c say at once t if any answ]s are obta9able1 !n ! roots ( bo? ! num]ator & ! d5om9ator ( ! fraction w be #f4 ,exam9e ! fir/ ?ree arrangem5ts giv5 above1 & y w f9d t ?is is s4 ,9 ! case ( #id #f ! roots ( ! num]ator & d5om9ator w be respectively #c--b1 9 ! case ( #ia #i & ( #hb #ah !y w be #i--h1 9 ! case ( #ha #ai !y w be #i--i1 & 9 ! case ( #c #ig !y w be #c--c4 ,e fraction t c be employ$ has1 !re=e1 its p>ticul> digital root =m1 & y are only wa/+ y\r "t 9 unconsci\sly attempt+ to break ?r\gh ?is law4 ,e read] w h p]ceiv$ t c]ta9 :ole numb]s are evid5tly impossible4 ,?us1 if !re is a #e 9 ! :ole numb]1 !re w also be a n\ght or a second #e 9 ! fraction1 :i* are b>r$ by ! conditions4 ,!n multiples ( #aj1 su* z #ij & #hj1 _c ( c\rse occur1 nor c ! :ole numb] conclude ) a #i1 l #hi & #gi1 because ! fraction1 equal to #aa or #ba1 w h #a 9 ! la/ place1 & w !re=e repeat a figure4 ,:ole numb]s t repeat a figure1 su* z #hh & #gg1 are also cle>ly useless4 ,^! cases1 z ,i h said1 are all obvi\s to e read]4 ,b :5 ,i decl>e t su* comb9ations z #ih #b1 #ib #h1 #hf #ad1 #hc #ag1 #gd #bf1 etc41 etc41 are to be at once dismiss$ z impossible1 ! reason is n s evid5t1 & ,i un=tunately _c sp>e space to expla9 x4 ,b :5 all ^? comb9ations h be5 /ruck out t are kn[n to be impossible1 x does n foll[ t all ! rema9+ possible =ms w actually "w4 ,! elem5tal =m may be "r en\gh1 b !re are o!r & deep] consid]ations t creep 9 to defeat our attempts4 ,= example1 #ih #b is an impossible comb9ation1 because we are able to say at once t !re is no possible =m = ! digital roots ( ! fraction equal to #b4 ,b 9 ! case ( #ig #c !re is a possible =m = ! digital roots ( ! fraction1 namely1 #f--e1 & x is only on fur!r inve/igation t we are able to det]m9e t ?is =m _c 9 practice be obta9$1 ow+ to curi\s consid]ations4 ,! work+ is greatly simplifi$ by a process ( elim9ation1 bas$ on su* consid]ations z t c]ta9 multiplications produce a repetition ( figures1 & t ! :ole numb] _c be f #ab to #bc inclusive1 s9ce 9 e su* case suffici5tly small d5om9ators are n available = =m+ ! fractional "p4 #ia4--,,m ,,mix$ ,,fractions4 ,! po9t ( ! pres5t puzzle lies 9 ! fact t ! numb]s #ae & #ah are n capable ( solution4 ,!re is no way ( det]m9+ ?is )\t trial4 ,"h are answ]s = ! t5 possible numb]s3-- #iedgb_/acfh #ac2 #ifdce_/abhg #ad2 #abcegf_/hid #af2 #facbeh_/idg #bj2 #aeidcb_/ghf #bg2 #bdigef_/hac #cf2 #bgeadh_/cif #dj2 #feahib_/dgc #fi2 #eicfad_/bgh #gb2 #gecfdh_/aib #id4 ,i h only f\nd ! "o arrangem5t = ea* ( ! numb]s #af1 #bj1 & #bg2 b ! o!r numb]s are all capable ( be+ solv$ 9 m ?an "o way4 ,z = #ae & #ah1 ?\gh ^! may be easily solv$ z a simple fraction1 yet a mix$ fraction assumes ! pres5ce ( a :ole numb]2 & ?\gh my own idea = dodg+ ! conditions is ! foll[+1 :]e ! fraction is bo? complex & mix$1 x w be fair] to keep exactly to ! =m indicat$3-- #cieb ---- #gdf #ae2 #c ---- #a #egdb ---- #fch #ah4 #i ---- #a ,i h prov$ ! possibility ( solution = all numb]s up to #ajj1 except #a1 #b1 #c1 #d1 #ae1 & #ah4 ,! fir/ ?ree are easily %[n to be impossible4 ,i h also notic$ t numb]s ^: digital root is #h--su* z #bf1 #ce1 #dd1 #ec1 etc4--seem to l5d !mselves to ! greate/ numb] ( answ]s4 ,= ! numb] #bf alone ,i h record$ no few] ?an tw5ty-five diff]5t arrangem5ts1 & ,i h no d\bt t !re are _m m4 #ib4--;,,digital ,,squ>e ,,numb]s4 ,s f> z ,i "k1 !re are no publi%$ tables ( squ>e numb]s t g suffici5tly high to be available = ! purposes ( ?is puzzle4 ,! l[e/ squ>e numb] conta9+ all ! n9e digits once1 & once only1 is #aci1hed1bgf1 ! squ>e ( #aa1hbf4 ,! highe/ squ>e numb] "u ! same conditions is1 #ibc1ahg1def1 ! squ>e ( #cj1chd4 #ic4--,,! ,,my/ic ,,elev54 ,mo/ p "k t if ! sum ( ! digits 9 ! odd places ( any numb] is ! same z ! sum ( ! digits 9 ! ev5 places1 !n ! numb] is divisible by #aa )\t rema9d]4 ,?us 9 #hifgdcjab ! odd digits1 #bjdfh1 add up #bj1 & ! ev5 digits1 #acgi1 also add up #bj4 ,!re=e ! numb] may be divid$ by #aa4 ,b few seem to "k t if ! diff]5ce betwe5 ! sum ( ! odd & ! ev5 digits is #aa1 or a multiple ( #aa1 ! rule equally applies4 ,?is law enables u to f9d1 ) a v little trial1 t ! smalle/ numb] conta9+ n9e ( ! t5 digits 7call+ n\ght a digit7 t is divisible by #aa is #ajb1cdg1ehf1 & ! highe/ numb] possible1 #ihg1feb1dac4 #id4--,,! ,,digital ,,c5tury4 ,!re is a v l>ge numb] ( diff]5t ways 9 :i* ari?metical signs may be plac$ betwe5 ! n9e digits1 arrang$ 9 num]ical ord]1 s z to give an expression equal to #ajj4 ,9 fact1 unless ! read] inve/igat$ ! matt] v closely1 he might n suspect t s _m ways are possible4 ,x 0 = ?is reason t ,i add$ ! condition t n only mu/ ! fewe/ possible signs be us$1 b also ! fewe/ possible /rokes4 ,9 ?is way we limit ! problem to a s+le solution1 & arrive at ! simple/ & !re=e 79 ?is case7 ! be/ result4 ,j z 9 ! case ( magic squ>es !re are me?ods by :i* we may write d[n ) ! greate/ ease a l>ge numb] ( solutions1 b n all ! solutions1 s !re are sev]al ways 9 :i* we may quickly arrive at doz5s ( arrangem5ts ( ! ,digital ,c5tury1 )\t f9d+ all ! possible arrangem5ts4 ,!re is1 9 fact1 v little pr9ciple 9 ! ?+1 & !re is no c]ta9 way ( demon/rat+ t we h got ! be/ possible solution4 ,all ,i c say is t ! arrangem5t ,i %all give z ! be/ is ! be/ ,i h up to ! pres5t succe$$ 9 discov]+4 ,i w give ! read] a few int]e/+ specim5s1 ! fir/ be+ ! solution usually publi%$1 & ! la/ ! be/ solution t ,i "k4 ,signs4 ,/rokes4 #a #b #c #d #e #f #g 7#h #i7 #ajj 7 #i #ah7 - 7#a #b7 - #c - #d - #e 7#f #g7 7#h #i7 #ajj 7#ab #bj7 #a 7#b #c7 7#d #e7 - #f #g 7#h #i7 #ajj 7#aa #ba7 7#a #b - #c - #d77e - #f - #g - #h - #i7 #ajj 7 #i #ab7 #a 7#b #c7 #d #e #fg #h #i #ajj 7#h #af7 7#a #b7 #cd #ef #g - #h #i #ajj 7#g #ac7 #ab #c - #d #e #fg #h #i #ajj 7#f #aa7 #abc - #d - #e - #f - #g #h - #i #ajj 7#f #g7 #abc #d - #e #fg - #h - #i #ajj 7#d #f7 #abc #de - #fg #h - #i #ajj 7#d #f7 #abc - #de - #fg #hi #ajj 7#c #d7 ,x w be notic$ t 9 ! above ,i h c\nt$ ! bracket z "o sign & two /rokes4 ,! la/ solution is s+ul>ly simple1 & ,i d n ?9k x w ev] be beat54 #ie4--,,! ,,f\r ,,sev5s4 ,! way to write f\r sev5s ) simple ari?metical signs s t !y repres5t #ajj is z foll[s3-- #g #g -- -- #ajj4 4#g 4#g ,( c\rse ! fraction1 #g ov] decimal #g1 equals #g divid$ by #g_/aj1 :i* is ! same z #gj divid$ by #g1 or #aj4 ,!n #aj multipli$ by #aj is #ajj1 & !re y are6 ,x w be se5 t ?is solution applies equally to any numb] :atev] t y may sub/itute = #g4 #if4--,,! ,,dice ,,numb]s4 ,! sum ( all ! numb]s t c be =m$ ) any giv5 set ( f\r diff]5t figures is always #f1fff multipli$ by ! sum ( ! f\r figures4 ,?us1 #a1 #b1 #c1 #d add up #aj1 & t5 times #f1fff is #ff1ffj4 ,n[1 !re are ?irty-five diff]5t ways ( select+ f\r figures f ! sev5 on ! dice--rememb]+ ! #f & #i trick4 ,! figures ( all ^! ?irty-five gr\ps add up to #fjj4 ,!re=e #f1fff multipli$ by #fjj gives u #c1iii1fjj z ! correct answ]4 ,let u disc>d ! dice & deal ) ! problem g5]ally1 us+ ! n9e digits1 b exclud+ n\ght4 ,n[1 if y 7 giv5 simply ! sum ( ! digits--t is1 if ! condition 7 t y c\ld use any f\r figures s long z !y summ$ to a giv5 am\nt--!n we h to rememb] t sev]al comb9ations ( f\r digits w1 9 _m cases1 make ! same sum4 #aj #aa #ab #ac #ad #ae #af #ag #ah #ai #bj #a #a #b #c #e #f #h #i #aa #aa #ab #ba #bb #bc #bd #be #bf #bg #bh #bi #cj #aa #aa #i #h #f #e #c #b #a #a ,"h ! top r[ ( numb]s gives all ! possible sums ( f\r diff]5t figures1 & ! bottom r[ ! numb] ( diff]5t ways 9 :i* ea* sum may be made4 ,= example #ac may be made 9 ?ree ways3 #abcg1 #abdf1 & #acde4 ,x w be f\nd t ! numb]s 9 ! bottom r[ add up to #abf1 :i* is ! numb] ( comb9ations ( n9e figures tak5 f\r at a "t4 ,f ?is table we may at once calculate ! answ] to su* a "q z ?is3 ,:at is ! sum ( all ! numb]s compos$ ( our diff]5t digits 7n\ght exclud$7 t add up to #ad8 ,multiply #ad by ! numb] b5ea? t 9 ! table1 #e1 & multiply ! result by #f1fff1 & y w h ! answ]4 ,x foll[s t1 to "k ! sum ( all ! numb]s compos$ ( f\r diff]5t digits1 if y multiply all ! pairs 9 ! two r[s & !n add ! results toge!r1 y w get #b1ebj1 :i*1 multipli$ by #f1fff1 gives ! answ] #af1gih1cbj4 ,! foll[+ g5]al solution = any numb] ( digits w d\btless int]e/ read]s4 ,let n repres5t numb] ( digits1 !n #e 7#ajn - #a7 #h6 divid$ by 7#i - n76 equals ! requir$ sum4 ,note t #j6 equals #a4 ,?is may be r$uc$ to ! foll[+ practical rule3 ,multiply toge!r #d #g #f #e 444 to 7n - #a7 factors2 n[ add 7n #a7 ciph]s to ! "r1 & f ?is result subtract ! same set ( figures ) a s+le ciph] to ! "r4 ,?us = n #d 7z 9 ! case la/ m5tion$71 #d #g #f #afh4 ,!re=e #af1hjj1jjj less #a1fhj gives u #af1gih1cbj 9 ano!r way4 #ig4--,,! ,,spot ,,on ,,! ,,table4 ,! ord9>y s*oolboy w\ld correctly treat ?is z a quadratic equation4 ,"h is ! actual ari?metic4 ,d\ble ! product ( ! two di/ances f ! walls4 ,?is gives u #add1 :i* is ! squ>e ( #ab4 ,! sum ( ! two di/ances is #ag4 ,if we add ^! two numb]s1 #ab & #ag1 toge!r1 & also subtract "o f ! o!r1 we get ! two answ]s t #bi or #e 0 ! radius1 or half-diamet]1 ( ! table4 ,consequ5tly1 ! full diamet] 0 #eh 94 or #aj 94 ,b a table ( ! latt] dim5sions w\ld be absurd1 & n at all 9 accordance ) ! illu/ration4 ,!re=e ! table mu/ h be5 #eh 94 9 diamet]4 ,9 ?is case ! spot 0 on ! edge ne>e/ to ! corn] ( ! room--to :i* ! boy 0 po9t+4 ,if ! o!r answ] 7 admissible1 ! spot w\ld be on ! edge f>!/ f ! corn] ( ! room4 #ih4--;,,academic ,,c\rtesies4 ,!re mu/ h be5 t5 boys & tw5ty girls4 ,! numb] ( b[s girl to girl 0 !re=e #chj1 ( boy to boy #ij1 ( girl ) boy #djj1 & ( boys & girls to tea*] #cj1 mak+ toge!r #ijj1 z /at$4 ,x w be rememb]$ t x 0 n said t ! tea*] himself return$ ! b[s ( any *ild4 #ii4--,,! ,,?irty-,,?ree ,,pe>ls4 ,! value ( ! l>ge c5tral pe>l mu/ h be5 #c1jjj4 ,! pe>l at "o end 7f :i* !y increas$ 9 value by #ajj7 0 #a1djj2 ! pe>l at ! o!r end1 #fjj4 #ajj4--,,! ,,lab\r]',s ,,puzzle4 ,! man said1 ,i am go+ twice z deep1 n z deep aga94 ,t is to say1 he 0 /ill go+ twice z deep z he _h gone already1 s t :5 f9i%$ ! hole w\ld be ?ree times its pres5t dep?4 ,!n ! answ] is t at pres5t ! hole is #c ft4 #f 94 deep & ! man #b ft4 #d 94 above gr\nd4 ,:5 complet$ ! hole w be #aj ft4 #f 94 deep1 & !re=e ! man w !n be #d ft4 #h 94 bel[ ! surface1 or twice ! di/ance t he is n[ above gr\nd4 #aja4--,,! ,,trusses ,,( ,,hay4 ,add toge!r ! t5 weights & divide by #d1 & we get #bhi lbs4 z ! weight ( ! five trusses toge!r4 ,if we call ! five trusses 9 ! ord] ( weight ,a1 ,b1 ,c1 ,d1 & ,e1 ! lighte/ be+ ,a & ! heavie/ ,e1 !n ! lighte/1 no lbs41 mu/ be ! weight ( ,a & ,b2 & ! next lighte/1 #aab lbs41 mu/ be ! weight ( ,a & ,c4 ,!n ! two heavie/1 ,d & ,e1 mu/ weigh #aba lbs41 & ,c & ,e mu/ weigh #abj lbs4 ,we ?us "k t ,a1 ,b1 ,d1 & ,e weigh toge!r #bca lbs41 :i*1 d$uct$ f #bhi lbs4 7! weight ( ! five trusses71 gives u ! weight ( ,c z #eh lbs4 ,n[1 by m]e subtraction1 we f9d ! weight ( ea* ( ! five trusses--#ed lbs41 #ef lbs41 #eh lbs41 #ei lbs41 & #fb lbs4 respectively4 #ajb4--,,mr4 ,,gubb9s ,,9 ,a ,,fog4 ,! c&les mu/ h burnt = ?ree h\rs & ?ree-qu>t]s4 ,"o c&le _h "o-sixte5? ( its total l5g? left & ! o!r f\r-sixte5?s4 #ajc4--,,pa9t+ ,,! ,,lamp-,,po/s4 ,pat mu/ h pa9t$ six m po/s ?an ,tim1 no matt] h[ _m lamp-po/s !re 74 ,= example1 suppose twelve on ea* side2 !n ,pat pa9t$ fifte5 & ,tim n9e4 ,if a hundr$ on ea* side1 ,pat pa9t$ "o hundr$ & ?ree1 & ,tim only n9ety-sev5 #ajd4--;,,cat*+ ,,! ,,?ief4 ,! con/able took ?irty /eps4 ,9 ! same "t ! ?ief w\ld take =ty-eight1 :i*1 add$ to 8 />t ( tw5ty-sev51 c>ri$ him sev5ty-five /eps4 ,?is di/ance w\ld be exactly equal to ?irty /eps ( ! con/able4 #aje4--,,! ,,p>i% ,,c\ncil ,,election1 ,! vot] c vote = "o c&idate 9 #bc ways1 = two 9 #bec ways1 = ?ree 9 #a1gga1 = f\r 9 #h1hee1 = five 9 #cc1fdi1 = six 9 #ajj1idg1 = sev5 9 #bde1aeg1 = eight 9 #dij1cad1 & = n9e c&idates 9 #hag1aij diff]5t ways4 ,add ^! toge!r1 & we get ! total ( #a1fih1aei ways ( vot+4 #ajf4--,,! ,,muddlet[n ,,election4 ,! numb]s ( votes poll$ respectively by ! ,lib]al1 ! ,cons]vative1 ! ,indep5d5t1 & ! ,sociali/ 7 #a1eec1 #a1ece1 #a1djg1 & #igh ,all t 0 necess>y 0 to add ! sum ( ! ?ree majorities 7#gci7 to ! total poll ( #e1dgc 7mak+ #f1bab7 & divide by #d1 :i* gives u #a1eec z ! poll ( ! ,lib]al4 ,!n ! polls ( ! o!r ?ree c&idates c1 ( c\rse1 be f\nd by d$uct+ ! successive majorities f ! la/-m5tion$ numb]4 #ajg4--,,! ,,suffragi/s' ,,meet+4 ,eighte5 7 pres5t at ! meet+ & elev5 left4 ,if twelve _h gone1 two-?irds w\ld h retir$4 ,if only n9e _h gone1 ! meet+ w\ld h lo/ half its memb]s4 #ajh4--,,! ,,leap-,,ye> ,,ladies4 ,! correct & only answ] is t #aa1faf ladies made proposals ( m>riage4 ,"h are all ! details1 :i* ! read] c *eck = himself ) ! orig9al /atem5ts4 ,( #aj1afd sp9/]s1 #h1jhe m>ri$ ba*elors1 #fbg m>ri$ wid[]s1 #a1bba 7 decl9$ by ba*elors1 & #bca decl9$ by wid[]s4 ,( ! #a1deb wid[s1 #a1aee m>ri$ ba*elors1 & #big m>ri$ wid[]s4 ,no wid[s 7 decl9$4 ,! problem is n difficult1 by algebra1 :5 once we h succe$$ 9 correctly /at+ x4 #aji4--,,! ,,great ,,scramble4 ,! smalle/ numb] ( sug> plums t w fulfil ! conditions is #bf1hhj4 ,! five boys obta9$ respectively3 ,&rew1 #b1hfc2 ,bob1 #f1cce2 ,*>lie1 #b1dch2 ,david1 #aj1bid2 ,edg>1 #d1iej4 ,!re is a little trap conceal$ 9 ! words ne> ! end1 "o-fif? ( ! same1 t seems at fir/ sight to upset ! :ole acc\nt ( ! affair4 ,b a little ?\ght w %[ t ! words c\ld only mean "o-fif? ( five-eigh?s1 ! fraction la/ m5tion$--t is1 "o-eigh? ( ! ?ree-qu>t]s t ,bob & ,&rew _h la/ acquir$4 #aaj4--,,! ,,abbot',s ,,puzzle4 ,! only answ] is t !re 7 #e m51 #be wom51 & #gj *ildr54 ,!re 7 ?us #ajj p]sons 9 all1 #e times z _m wom5 z m51 & z ! m5 w\ld toge!r receive #ae bu%els1 ! wom5 #ej bu%els1 & ! *ildr5 #ce bu%els1 exactly #ajj bu%els w\ld be di/ribut$4 #aaa4--,,reap+ ,,! ,,corn4 ,! :ole field mu/ h conta9$ #df4fbf squ>e rods4 ,! side ( ! c5tral squ>e1 left by ! f>m]1 is #d4hbhd rods1 s x conta9s #bc4cac squ>e rods4 ,! area ( ! field 0 ?us some?+ m ?an a qu>t] ( an acre & less ?an "o-?ird2 to be m precise1 4#biad ( an acre4 #aab4--;,a ,,puzzl+ ,,legacy4 ,z ! %>e ( ,*>les falls 9 ?r\gh 8 dea?1 we h m]ely to divide ! :ole hundr$ acres betwe5 ,alfr$ & ,b5jam9 9 ! proportion ( "o-?ird to "o-f\r?--t is 9 ! proportion ( f\r-twelf?s to ?ree-twelf?s1 :i* is ! same z f\r to ?ree4 ,!re=e ,alfr$ takes f\r-sev5?s ( ! hundr$ acres & ,b5jam9 ?ree-sev5?s4 #aac4--,,! ,,torn ,,numb]4 ,! o!r numb] t answ]s all ! requirem5ts ( ! puzzle is #i1hja4 ,if we divide ?is 9 ! middle into two numb]s & add !m toge!r we get #ii1 :i*1 multipli$ by itself1 produces #i1hja4 ,x is true t #b1jbe may be treat$ 9 ! same way1 only ?is numb] is exclud$ by ! condition :i* requires t no two figures %\ld be alike4 ,! g5]al solution is curi\s4 ,call ! numb] ( figures 9 ea* half ( ! torn label n4 ,!n1 if we add #a to ea* ( ! expon5ts ( ! prime factors 7o!r ?an #c7 ( #ajn - #a 7#a be+ reg>d$ z a factor ) ! con/ant expon5t1 #a71 _! product w be ! numb] ( solutions4 ,?us1 = a label ( six figures1 n #c4 ,! factors ( #ajn - #a are #a #cg 7n consid]+ ! #c71 & ! product ( #b #b #d1 ! numb] ( solutions4 ,?is always includes ! special cases #ih - #ja1 #jj - #ja1 #iih - #ja1 #jjj - #jja1 etc4 ,! solutions are obta9$ z foll[s3--,factorize #aj - #a 9 all possible ways1 always keep+ ! p[]s ( #c toge!r1 ?us1 #cg #bg1 #iii #a4 ,!n solve ! equation #cgx #bgy #a4 ,"h x #ai & y #bf4 ,!re=e1 #ai #cg #gjc1 ! squ>e ( :i* gives "o label1 #did1bji4 ,a complem5t>y solution 7?r\gh #bgx #cgx #a7 c at once be f\nd by #ajn - #gjc #big1 ! squ>e ( :i* gives #jhh1bji = second label4 7,^! non-significant n\ghts to ! left mu/ be includ$1 ?\gh !y lead to peculi> cases l #jjbch - #jdfda #dhgi1 :]e #jbch - #dfda w\ld n "w47 ,! special case #iii #a we c write at once #iih1jja1 accord+ to ! law %[n above1 by add+ n9es on "o half & n\ghts on ! o!r1 & its complem5t>y w be #a prec$$ by five n\ghts1 or #jjjjja4 ,?us we get ! squ>es ( #iii & #a4 ,^! are ! f\r solutions4 #aad4--;,,curi\s ,,numb]s4 ,! ?ree smalle/ numb]s1 9 addition to #dh1 are #a1fhj1 #eg1abj1 & #a1idj1ddh4 ,x w be f\nd t #a1fha & #hda1 #eg1aba & #bh1efa1 #a1idj1ddi & #igj1bbe1 are respectively ! squ>es ( #da & #bi1 #bci & #afi1 #a1cic & #ihe4 #aae4--;,a ,,pr9t]',s ,,error4 ,! answ] is t #be 4#ib is ! same z #beib1 & ?is is ! only possible solution to ! puzzle4 #aaf4--,,! ,,conv]t$ ,,mis]4 ,z we are n told 9 :at ye> ,mr4 ,jasp] ,bullyon made ! g5]\s di/ribution ( 8 accumulat$ weal?1 b are requir$ to f9d ! l[e/ possible am\nt ( money1 x is cle> t we mu/ look = a ye> ( ! mo/ fav\rable =m4 ,!re are f\r cases to be consid]$--an ord9>y ye> ) fifty-two ,sundays & ) fifty-?ree ,sundays1 & a leap-ye> ) fifty-two & fifty-?ree ,sundays respectively4 ,"h are ! l[e/ possible am\nts 9 ea* case3-- #cac weekdays1 #eb ,sundays #aab1jee #cab weekdays1 #ec ,sundays #ai1cde #cad weekdays1 #eb ,sundays ,no solution possible4 #cac weekdays1 #ec ,sundays #fi1agd ,! l[e/ possible am\nt1 & !re=e ! correct answ]1 is #ai1cde1 di/ribut$ 9 an ord9>y ye> t began on a ,sunday4 ,! la/ ye> ( ?is k9d 0 #aiaa4 ,he w\ld h paid #ec on e day ( ! ye>1 or #fb on e weekday1 ) #a left ov]1 z requir$1 9 ! latt] ev5t4 #aag4--;,a ,,f5ce ,,problem4 ,?\gh ?is puzzle pres5ts no great difficulty to any "o possess+ a k ( algebra1 x has p]haps r int]e/+ features4 ,see+1 z "o does 9 ! illu/ration1 j "o corn] ( ! propos$ squ>e1 "o is sc>cely prep>$ = ! fact t ! field1 9 ord] to comply ) ! conditions1 mu/ conta9 exactly #eja1gfj acres1 ! f5ce requir+ ! same numb] ( rails4 ,yet ?is is ! correct answ]1 & ! only answ]1 & if t g5tleman 9 ,i[a c>ries out 8 int5tion1 8 field w be tw5ty-eight miles long on ea* side1 & a little l>g] ?an ! c\nty ( ,we/morl&4 ,i am n aw>e t any limit has ev] be5 fix$ to ! size ( a field1 ?\gh !y d n run s l>ge z ?is 9 ,great ,brita94 ,/ill1 out 9 ,i[a1 :]e my correspond5t resides1 !y d ^! ?+s on a v big scale4 ,i h1 h[ev]1 reason to believe t :5 he f9ds ! sort ( task he has set himself1 he w decide to ab&on x2 = if t c[ decides to roam to fre% woods & pa/ures new1 ! milkmaid may h to />t out a week 9 advance 9 ord] to obta9 ! morn+'s milk4 ,"h is a little rule t w always apply :]e ! l5g? ( ! rail is half a pole4 ,multiply ! numb] ( rails 9 a hurdle by f\r1 & ! result is ! exact numb] ( miles 9 ! side ( a squ>e field conta9+ ! same numb] ( acres z !re are rails 9 ! complete f5ce4 ,?us1 ) a "o-rail f5ce ! field is f\r miles squ>e2 a two-rail f5ce gives eight miles squ>e2 a ?ree-rail f5ce1 twelve miles squ>e2 & s on1 until we f9d t a sev5-rail f5ce multipli$ by f\r gives a field ( tw5ty-eight miles squ>e4 ,9 ! case ( our pres5t problem1 if ! field be made small]1 !n ! numb] ( rails w exce$ ! numb] ( acres2 :ile if ! field be made l>g]1 ! numb] ( rails w be less ?an ! acres ( ! field4 #aah4--;,,circl+ ,,! ,,squ>es4 ,?\gh ?is problem might /rike ! novice z be+ r difficult1 x is1 z a matt] ( fact1 q easy1 & is made /ill easi] by ins]t+ f\r out ( ! t5 numb]s4 ,fir/1 x w be f\nd t squ>es t are diametrically opposite h a common diff]5ce4 ,= example1 ! diff]5ce betwe5 ! squ>e ( #ad & ! squ>e ( #b1 9 ! diagram1 is #aib2 & ! diff]5ce betwe5 ! squ>e ( #af & ! squ>e ( #h is also #aib4 ,?is mu/ be s 9 e case4 ,!n x %\ld be rememb]$ t ! diff]5ce betwe5 squ>es ( two consecutive numb]s is always twice ! small] numb] plus #a1 & t ! diff]5ce betwe5 ! squ>es ( any two numb]s c always be express$ z ! diff]5ce ( ! numb]s multipli$ by _! sum4 ,?us ! squ>e ( #e 7#be7 less ! squ>e ( #d 7#af7 equals 7#b #d7 #a1 or #i2 also1 ! squ>e ( #g 7#di7 less ! squ>e ( #c 7#i7 equals 7#g #c7 7#g - #c71 or #dj4 ,n[1 ! numb] #aib1 ref]r$ to above1 may be divid$ into five diff]5t pairs ( ev5 factors3 #b #if1 #d #dh1 #f #cb1 #h #bd1 & #ab #af1 & ^! divid$ by #b give u1 #a #dh1 #b #bd1 #c #af1 #d #ab1 & #f #h4 ,! diff]5ce & sum respectively ( ea* ( ^! pairs 9 turn produce #dg1 #di2 #bb1 #bf2 #ac1 #ai2 #h1 #af2 & #b1 #ad4 ,^! are ! requir$ numb]s1 f\r ( :i* are already plac$4 ,! six numb]s t h to be add$ may be plac$ 9 j six diff]5t ways1 "o ( :i* is z foll[s1 read+ r\nd ! circle clockwise3 #af1 #b1 #di1 #bb1 #ai1 #h1 #ad1 #dg1 #bf1 #ac4 ,i w j draw ! read]'s att5tion to "o o!r little po9t4 ,9 all circles ( ?is k9d1 ! diff]5ce betwe5 diametrically opposite numb]s increases by a c]ta9 ratio1 ! fir/ numb]s 7) ! exception ( a circle ( #f7 be+ #d & #f1 & ! o!rs =m$ by d\bl+ ! next prec$+ b "o4 ,?us1 9 ! above case1 ! fir/ diff]5ce is #b1 & !n ! numb]s increase by #d1 #f1 #h1 & #ab4 ,( c\rse1 an inf9ite numb] ( solutions may be f\nd if we admit fractions4 ,! numb] ( squ>es 9 a circle ( ?is k9d mu/1 h[ev]1 be ( ! =m #dn #f2 t is1 x mu/ be a numb] compos$ ( #f plus a multiple ( #d4 #aai4--,,rackbrane',s ,,little ,,loss4 ,! pr(essor mu/ h />t$ ! game ) ?irte5 %ill+s1 ,mr4 ,potts ) f\r %ill+s1 & ,mrs4 ,potts ) sev5 %ill+s4 #abj4--,,! ,,f>m] ,,& ,,8 ,,%eep4 ,! f>m] _h "o %eep only6 ,if he divid$ ?is %eep 7:i* is be/ done by weight7 into two p>ts1 mak+ "o "p two-?irds & ! o!r "p "o-?ird1 !n ! diff]5ce betwe5 ^! two numb]s is ! same z ! diff]5ce betwe5 _! squ>es--t is1 "o-?ird4 ,any two fractions w d if ! d5om9ator equals ! sum ( ! two num]ators4 #aba4--;,,heads ,,or ,,tails4 ,crooks mu/ h lo/1 & ! long] he w5t on ! m he w\ld lose4 ,9 two tosses he w\ld be left ) ?ree-qu>t]s ( 8 money1 9 f\r tosses ) n9e-sixte5?s ( 8 money1 9 six tosses ) tw5ty-sev5 sixty-f\r?s ( 8 money1 & s on4 ,! ord] ( ! w9s & losses makes no diff]5ce1 s long z _! numb] is 9 ! end equal4 #abb4--,,! ,,see-,,saw ,,puzzle4 ,! boy's weight mu/ h be5 ab\t #ci4gi lbs4 ,a brick weigh$ #c lbs4 ,!re=e #af bricks weigh$ #dh lbs4 & #aa bricks #cc lbs4 ,multiply #dh by #cc & take ! squ>e root4 #abc4--;,a ,,legal ,,difficulty4 ,x 0 cle>ly ! int5tion ( ! deceas$ to give ! son twice z mu* z ! "m1 or ! daught] half z mu* z ! "m4 ,!re=e ! mo/ equitable division w\ld be t ! "m %\ld take two-sev5?s1 ! son f\r-sev5?s1 & ! daught] "o-sev5?4 #abd4--;,a ,,"q ,,( ,,def9ition4 ,!re is1 ( c\rse1 no diff]5ce 9 area betwe5 a mile squ>e & a squ>e mile4 ,b !re may be consid]able diff]5ce 9 %ape4 ,a mile squ>e c be no o!r %ape ?an squ>e2 ! expression describes a surface ( a c]ta9 specific size & %ape4 ,a squ>e mile may be ( any %ape2 ! expression names a unit ( area1 b does n prescribe any p>ticul> %ape4 #abe4--,,! ,,m9]s' ,,holiday4 ,bill ,h>ris mu/ h sp5t ?irte5 %ill+s & sixp5ce1 :i* w\ld be ?ree %ill+s m ?an ! av]age = ! sev5 m5--half a gu9ea4 #abf4--,,simple ,,multiplication4 ,! numb] requir$ is #c1ebi1daa1gfd1gje1hhb1 :i* may be multipli$ by #c & divid$ by #b1 by ! simple exp$i5t ( remov+ ! #c f "o end ( ! r[ to ! o!r4 ,if y want a long] numb]1 y c increase ?is "o to any ext5t by repeat+ ! sixte5 figures 9 ! same ord]4 #abg4--,,simple ,,division4 ,subtract e numb] 9 turn f e o!r numb]1 & we get #ceh 7twice71 #gaf1 #a1faa1 #a1bec1 & #hie4 ,n[1 we see at a glance t1 z #ceh equals #b #agi1 ! only numb] t c divide 9 e case )\t a rema9d] w be #agi4 ,on trial we f9d t ?is is su* a divisor4 ,!re=e1 #agi is ! divisor we want1 :i* always leaves a rema9d] #afd 9 ! case ( ! orig9al numb]s giv54 #abh4--;,a ,,problem ,,9 ,,squ>es4 ,! sides ( ! ?ree bo>ds measure #ca 941 #da 941 & #di 94 ,! common diff]5ce ( area is exactly five squ>e feet4 ,?ree numb]s ^: squ>es are 9 ,a4,p41 ) a common diff]5ce ( #g1 are #aac_/abj1 #ccg_/abj1 #dfc_/abj2 & ) a common diff]5ce ( #ac are #hjibi_/aichj1 #ajfiba_/aichj1 & #abggbi_/aichj4 ,9 ! case ( :ole squ>e numb]s ! common diff]5ce w always be divisible by #bd1 s x is obvi\s t our squ>es mu/ be fractional4 ,read]s %\ld n[ try to solve ! case :]e ! common diff]5ce is #bc4 ,x is r a h>d nut4 #abi4--,,! ,,battle ,,( ,,ha/+s4 ,any numb] 7n itself a squ>e numb]7 may be multipli$ by a squ>e t w give a product #a less ?an ano!r squ>e4 ,! giv5 numb] mu/ n itself be a squ>e1 because a squ>e multipli$ by a squ>e produces a squ>e1 & no squ>e plus #a c be a squ>e4 ,my rem>ks ?r\gh\t mu/ be und]/ood to apply to :ole numb]s1 because fractional soldi]s are n ( mu* use 9 w>4 ,n[1 ( all ! numb]s f #b to #ii inclusive1 #fa happ5s to be ! mo/ awkw>d "o to "w1 & ! l[e/ possible answ] to our puzzle is t ,h>old's army consi/$ ( #c1aai1hhb1ihb1hfj1bfd1djj m54 ,t is1 !re w\ld be #ea1ade1fbb1ffi1hdj1djj m5 7! squ>e ( #bbf1aec1ihj7 9 ea* ( ! sixty-"o squ>es4 ,add "o man 7,h>old71 & !y c\ld !n =m "o l>ge squ>e ) #a1gff1cai1jdi m5 on e side4 ,! g5]al problem1 ( :i* ?is is a p>ticul> case1 is kn[n z ! ,pellian ,equation--app>5tly because ,pell nei!r fir/ prop\nd$ ! "q nor fir/ solv$ x6 ,x 0 issu$ z a *all5ge by ,f]mat to ! ,engli% ma!maticians ( 8 day4 ,x is readily solv$ by ! use ( cont9u$ fractions4 ,next to #fa1 ! mo/ difficult numb] "u #ajj is #ig1 :]e #ig #f1cgg1ceb #a a squ>e4 ,! reason :y ,i assum$ t !re mu/ be some?+ wrong ) ! figures 9 ! *ronicle is t we c confid5tly say t ,h>old's army did n conta9 ov] ?ree trillion m56 ,if ?is army 7n to m5tion ! ,normans7 _h _h ! :ole surface ( ! e>? 7sea includ$7 on :i* to encamp1 ea* man w\ld h _h slightly m ?an a qu>t] ( a squ>e in* ( space 9 :i* to move ab\t6 ,put ano!r way3 ,all[+ "o squ>e foot ( /&+-room p] man1 ea* small squ>e w\ld h requir$ all ! space all[$ by a globe ?ree times ! diamet] ( ! e>?4 #acj4--,,! ,,sculptor',s ,,problem4 ,a little ?\ght w make x cle> t ! answ] mu/ be fractional1 & t 9 "o case ! num]ator w be great] & 9 ! o!r case less ?an ! d5om9ator4 ,z a matt] ( fact1 ! height ( ! l>g] cube mu/ be #h_/g ft41 & ( ! small] #c_/g ft41 if we are to h ! answ] 9 ! smalle/ possible figures4 ,"h ! l9eal measurem5t is #aa_/g ft4--t is1 #ad_/g ft4 ,:at are ! cubic cont5ts ( ! two cubes8 ,fir/ #h_/g #c_/g #h_/g #eab_/cdc1 & secondly #c_/g #c_/g #c_/g #bg_/cdc4 ,add ^! toge!r & ! result is #eci_/cdc1 :i* r$uces to #aa_/g or #ad_/g ft4 ,we ?us see t ! answ]s 9 cubic feet & l9eal feet are precisely ! same4 ,! g]m ( ! idea is to be f\nd 9 ! works ( ,diophantus ( ,alex&ria1 :o wrote ab\t ! beg9n+ ( ! f\r? c5tury4 ,^! fractional numb]s appe> 9 triads1 & are obta9$ f ?ree g5]ators1 a1 b1 c1 :]e a is ! l>ge/ & c ! smalle/4 ,!n ab c d5om9ator1 & a - c1 b - c1 & a - b w be ! ?ree num]ators4 ,?us1 us+ ! g5]ators #c1 #b1 #a1 we get #h_/g1 #c_/g1 #e_/g & we c pair ! fir/ & second1 z 9 ! above solution1 or ! fir/ & ?ird = a second solution4 ,! d5om9ator mu/ always be a prime numb] ( ! =m #fn #a1 or compos$ ( su* primes4 ,?us y c h #ac1 #ai1 etc41 z d5om9ators1 b n #be1 #ee1 #ahg1 etc4 ,:5 ! pr9ciple is und]/ood !re is no difficulty 9 writ+ d[n ! dim5sions ( z _m sets ( cubes z ! mo/ exact+ collector may require4 ,if ! read] w\ld l "o1 = example1 ) pl5ty ( n9es1 p]haps ! foll[+ w\ld satisfy him3 #iiiiiiii_/iiiijjja & #aiiii_/iiiijjja4 #aca4--,,! ,,spani% ,,mis]4 ,!re mu/ h be5 #chf d\bloons 9 "o box1 #h1dej 9 ano!r1 & #af1ead 9 ! ?ird1 because #chf is ! smalle/ numb] t c occur4 ,if ,i _h ask$ = ! smalle/ aggregate numb] ( co9s1 ! answ] w\ld h be5 #dhb1 #c1cfb1 & #f1bdb4 ,x w be f\nd 9 ei!r case t if ! cont5ts ( any two ( ! ?ree boxes be comb9$1 !y =m a squ>e numb] ( co9s4 ,x is a curi\s co9cid5ce 7no?+ m1 = x w n always happ57 t 9 ! fir/ solution ! digits ( ! ?ree numb]s add to #ag 9 e case1 & 9 ! second solution to #ad4 ,x %\ld be not$ t ! middle "o ( ! ?ree numb]s w always be half a squ>e4 #acb4--,,! ,,n9e ,,treasure ,,boxes4 ,"h is ! answ] t fulfils ! conditions3-- ,a #d ,b #c1cfd ,c #f1gbd ,d #b1aaf ,e #e1dgf ,f #h1hcf ,g #i1dji ,h #ab1gfi ,i #af1abi ,ea* ( ^! is a squ>e numb]1 ! roots1 tak5 9 alphabetical ord]1 be+ #b1 #eh1 #hb1 #df1 #gd1 #id1 #ig1 #aac1 & #abg1 :ile ! requir$ diff]5ce betwe5 ,a & ,b1 ,b & ,c1 ,d & ,e4 etc41 is 9 e case #c1cfj4 #acc4--,,! ,,five ,,brig&s4 ,! sum ( #bjj d\bloons might h be5 held by ! five brig&s 9 any "o ( #f1fbg diff]5t ways4 ,alfonso may h held any numb] f #a to #aa4 ,if he held #a d\bloon1 !re are #a1jje diff]5t ways ( di/ribut+ ! rema9d]2 if he held #b1 !re are #ihe ways2 if #c1 !re are #igg ways2 if #d1 !re are #ijc ways2 if #e d\bloons1 #hcb ways2 if #f d\bloons1 #gjd ways2 if #g d\bloons1 #egj ways2 if #h d\bloons1 #chh ways2 if #i d\bloons1 #bjj ways2 if #aj d\bloons1 #fj ways2 & if ,alfonso held #aa d\bloons1 ! rema9d] c\ld be di/ribut$ 9 #c diff]5t ways4 ,m ?an #aa d\bloons he c\ld n possibly h _h4 ,x w sc>cely be expect$ t ,i %all give all ^! #f1fbg ways at l5g?4 ,:at ,i propose to d is to enable ! read]1 if he %\ld feel s dispos$1 to write out all ! answ]s :]e ,alfonso has "o & ! same am\nt4 ,let u take ! cases :]e ,alfonso has #f d\bloons1 & see h[ we may obta9 all ! #gjd diff]5t ways indicat$ above4 ,"h are two tables t w s]ve z keys to all ^! answ]s3-- ,table ,i4 ,table ,,ii4 ,a #f4 ,a #f4 ,b n4 ,b n4 ,c 7#fc - #en7 m4 ,c #a m4 ,d 7#abh #dn7 - #dm4 ,d 7#cgf - #afn7 - #dm4 ,e #c #cm4 ,e 7#aen - #ahc7 #cm4 ,9 ! fir/ table we may sub/itute = n any :ole numb] f #a to #ab inclusive1 & m may be n\ght or any :ole numb] f #a to 7#ca n7 inclusive4 ,9 ! second table n may h ! value ( any :ole numb] f #ac to #bc inclusive1 & m may be n\ght or any :ole numb] f #a to 7#ic - #dn7 inclusive4 ,! fir/ table ?us gives 7#cb n7 answ]s = e value ( n2 & ! second table gives 7#id - #dn7 answ]s = e value ( n4 ,! =m]1 !re=e1 produces #dfb & ! latt] #bdb answ]s1 :i* toge!r make #gjd1 z already /at$4 ,let u take ,table ,i41 & say n #e & m #b2 also 9 ,table ,,ii4 take n #ac & m #j4 ,!n we at once get ^! two answ]s3-- ,a #f ,a #f ,b #e ,b #ac ,c #dj ,c #a ,d #adj ,d #afh ,e #i ,e #ab --- --- #bjj d\bloons #bjj d\bloons4 ,^! w be f\nd to "w correctly4 ,all ! re/ ( ! #gjd answ]s1 :]e ,alfonso always holds six d\bloons1 may be obta9$ 9 ?is way f ! two tables by sub/itut+ ! diff]5t numb]s = ! lett]s m & n4 ,put 9 ano!r way1 = e hold+ ( ,alfonso ! numb] ( answ]s is ! sum ( two ari?metical progressions1 ! common diff]5ce 9 "o case be+ #a & 9 ! o!r -#d4 ,?us 9 ! case :]e ,alfonso holds #f d\bloons "o progression is #cc #cd #ce #cf 444 #dc #dd1 & ! o!r #db #ch #cd #cj 444 #f #b4 ,! sum ( ! fir/ s]ies is #dfb1 & ( ! second #bdb--results :i* aga9 agree ) ! figures already giv54 ,! problem may be said to consi/ 9 f9d+ ! fir/ & la/ t]ms ( ^! progressions4 ,i %\ld rem>k t :]e ,alfonso holds #i1 #aj1 or #aa !re is only "o progression1 ( ! second =m4 #acd4--,,! ,,bank]',s ,,puzzle4 ,9 ord] t a numb] ( sixp5ces may n be divisible into a numb] ( equal piles1 x is necess>y t ! numb] %\ld be a prime4 ,if ! bank] c br+ ab\t a prime numb]1 he w w92 & ,i w %[ h[ he c always d ?is1 :atev] ! cu/om] may put 9 ! box1 & t !re=e ! bank] w w9 to a c]ta9ty4 ,! bank] mu/ fir/ deposit =ty sixp5ces1 & !n1 no matt] h[ _m ! cu/om] may add1 he w desire ! latt] to transf] f ! c\nt] ! squ>e ( ! numb] next bel[ :at ! cu/om] put 94 ,?us1 bank] puts #dj1 cu/om]1 we w say1 adds #f1 !n transf]s f ! c\nt] #be 7! squ>e ( #e71 :i* leaves #ga 9 all1 a prime numb]4 ,try aga94 ,bank] puts #dj1 cu/om] adds #ab1 !n transf]s #aba 7! squ>e ( #aa71 z desir$1 :i* leaves #agc1 a prime numb]4 ,! key to ! puzzle is ! curi\s fact t any numb] up to #ci1 if add$ to its squ>e & ! sum increas$ by #da1 makes a prime numb]4 ,?is 0 fir/ discov]$ by ,eul]1 ! great ma!matician4 ,x has be5 sugge/$ t ! bank] might desire ! cu/om] to transf] suffici5t to raise ! cont5ts ( ! box to a giv5 numb]2 b ?is w\ld n only make ! ?+ an absurdity1 b breaks ! rule t nei!r kn[s :at ! o!r puts 94 #ace4--,,! ,,/onemason',s ,,problem4 ,! puzzle am\nts to ?is4 ,f9d ! smalle/ squ>e numb] t may be express$ z ! sum ( m ?an ?ree consecutive cubes1 ! cube #a be+ b>r$4 ,z m ?an ?ree heaps 7 to be suppli$1 ?is condition %uts out ! o!rwise smalle/ answ]1 #bc #bd #be #bjd4 ,b x admits ! answ]1 #be #bf #bg #bh #bi #cae4 ,! correct answ]1 h[ev]1 requires m heaps1 b a small] aggregate numb] ( blocks4 ,"h x is3 #ad #ae 444 up to #be inclusive1 or twelve heaps 9 all1 :i*1 add$ toge!r1 make #ig1cdd blocks ( /one t may be laid out to =m a squ>e #cab #cab4 ,i w j rem>k t "o key to ! solution lies 9 :at are call$ triangul> numb]s4 7,see pp4 #ac1 #be1 & #aff47 #acf4--,,! ,,sultan',s ,,army4 ,! smalle/ primes ( ! =m #dn #a are #e1 #ac1 #ag1 #bi1 & #cg1 & ! smalle/ ( ! =m #dn - #a are #c1 #g1 #aa1 #ai1 & #bc4 ,n[1 primes ( ! fir/ =m c always be express$ z ! sum ( two squ>es1 & 9 only "o way4 ,?us1 #e #d #a2 #ac #i #d2 #ag #af #a2 #bi #be #d2 #cg #cf #a4 ,b primes ( ! second =m c nev] be express$ z ! sum ( two squ>es 9 any way :atev]4 ,9 ord] t a numb] may be express$ z ! sum ( two squ>es 9 sev]al diff]5t ways1 x is necess>y t x %all be a composite numb] conta9+ a c]ta9 numb] ( primes ( our fir/ =m4 ,?us1 #e or #ac alone c only be s express$ 9 "o way2 b #fe1 7#e #ac71 c be express$ 9 two ways1 #a1aje1 7#e #ac #ag71 9 f\r ways1 #cb1jde1 7#e #ac #ag #bi71 9 eight ways4 ,we ?us get d\ble z _m ways = e new factor ( ?is =m t we introduce4 ,note1 h[ev]1 t ,i say new factor1 = ! repetition ( factors is subject to ano!r law4 ,we _c express #be1 7#e #e71 9 two ways1 b only 9 "o2 yet #abe1 7#e #e #e71 c be giv5 9 two ways1 & s c #fbe1 7#e #e #e #e72 :ile if we take 9 yet ano!r #e we c express ! numb] z ! sum ( two squ>es 9 ?ree diff]5t ways4 ,if a prime ( ! second =m gets into y\r composite numb]1 !n t numb] _c be ! sum ( two squ>es4 ,?us #ae1 7#c #e71 w n "w1 nor w #ace1 7#c #c #c #e72 b if we take 9 an ev5 numb] ( #c's x w "w1 because ^! #c's w !mselves =m a squ>e numb]1 b y w only get "o solution4 ,?us1 #de1 7#c #c #e1 or #i #e7 #cf #i4 ,simil>ly1 ! factor #b may always occur1 or any p[] ( #b1 su* z #d1 #h1 #af1 #cb2 b its introduction or omission w nev] affect ! numb] ( y\r solutions1 except 9 su* a case z #ej1 :]e x d\bles a squ>e & !re=e gives y ! two answ]s1 #di #a & #be #be4 ,n[1 directly a numb] is decompos$ into its prime factors1 x is possible to tell at a glance :e!r or n x c be split into two squ>es2 & if x c be1 ! process ( discov]y 9 h[ _m ways is s simple t x c be done 9 ! head )\t any ef=t4 ,! numb] ,i gave 0 #acj4 ,i at once saw t ?is 0 #b #e #ac1 & consequ5tly t1 z #fe c be express$ 9 two ways 7#fd #a & #di #af71 #acj c also be express$ 9 two ways1 ! factor #b n affect+ ! "q4 ,! smalle/ numb] t c be express$ z ! sum ( two squ>es 9 twelve diff]5t ways is #afj1bbe1 & ?is is !re=e ! smalle/ army t w\ld answ] ! ,sultan's purpose4 ,! numb] is compos$ ( ! factors #e #e #ac #ag #bi1 ea* ( :i* is ( ! requir$ =m4 ,if !y 7 all diff]5t factors1 !re w\ld be sixte5 ways2 b z "o ( ! factors is repeat$1 !re are j twelve ways4 ,"h are ! sides ( ! twelve pairs ( squ>es3 7#djj & #ae71 7#cii & #cb71 7#cic & #gf71 7#cib & #ha71 7#chd & #aac71 7#cge & #adj71 7#cfj & #age71 7#cef & #ahc71 7#ccg & #baf71 7#cbi & #bbh71 7#caa & #beb71 7#bfe & #cjj74 ,squ>e ! two numb]s 9 ea* pair1 add !m toge!r1 & _! sum w 9 e case be #afj1bbe4 #acg4--;,a ,,/udy ,,9 ,,?rift4 ,mrs4 ,s&y ,mc,allister w h to save a trem5d\s sum out ( h] h\sekeep+ all[ance if %e is to w9 t six? pres5t t h] canny husb& promis$ h]4 ,& ! all[ance mu/ be a v lib]al "o if x is to admit ( su* sav+s4 ,! problem requir$ t we %\ld f9d five numb]s high] ?an #cf ! units ( :i* may be display$ s z to =m a squ>e1 a triangle1 two triangles1 & ?ree triangles1 us+ ! complete numb] 9 e "o ( ! f\r cases4 ,e triangul> numb] is su* t if we multiply x by #h & add #a ! result is an odd squ>e numb]4 ,= example1 multiply #a1 #c1 #f1 #aj1 #ae respectively by #h & add #a1 & we get #i1 #be1 #di1 #ha1 #aba1 :i* are ! squ>es ( ! odd numb]s #c1 #e1 #g1 #i1 #aa4 ,!re=e 9 e case :]e #hx #a a squ>e numb]1 x is also a triangul>4 ,?is po9t is dealt ) 9 our puzzle1 ,! ,battle ( ,ha/+s4 ,i w n[ m]ely %[ aga9 h[1 :5 ! fir/ solution is f\nd1 ! o!rs may be discov]$ )\t any difficulty4 ,fir/ ( all1 "h are ! figures3-- #h #a #a #c #h #f #a #ag #h #ce #a #ii #h #bjd #a #egg #h #aahi #a #ccfc #h #ficj #a #aifja #h #djcia #a #aadbdc ,! successive pairs ( numb]s are f\nd 9 ?is way3-- 7#a #c7 7#c #a7 #f 7#h #a7 7#c #c7 #ag 7#a #ag7 7#c #f7 #ce 7#h #f7 7#c #ag7 #ii 7#a #ii7 7#c #ce7 #bjd 7#h #ce7 7#c #ii7 #egg & s on4 ,look = ! numb]s 9 ! table above1 & ! me?od w expla9 itself4 ,?us we f9d t ! numb]s #cf1 #abbe1 #dafaf1 #adacgba1 #dhjbdijj1 & #afcadcbhha w =m squ>es ) sides ( #f1 #ce1 #bjd1 #aahi1 #ficj1 & #djcia2 & !y w also =m s+le triangles ) sides ( #h1 #di1 #bhh1 #afha1 #ihjj1 & #egaba4 ,^! numb]s may be obta9$ f ! la/ column 9 ! fir/ table above 9 ?is way3 simply divide ! numb]s by #b & reject ! rema9d]4 ,?us ! integral halves ( #ag1 #ii1 & #egg are #h1 #di1 & #bhh4 ,all ! numb]s we h f\nd w =m ei!r two or ?ree triangles at w4 ,! foll[+ little diagram w %[ y graphically at a glance t e squ>e numb] mu/ necess>ily be ! sum ( two triangul>s1 & t ! side ( "o triangle w be ! same z ! side ( ! correspond+ squ>e1 :ile ! o!r w be j #a less4 ,illu/ration ----------- --------- 4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4_/4 4 4 4 4 _/4 4 4 4 4 _/4 4 4 4 4 4 --------- ----------- ,?us a squ>e may always be divid$ easily into two triangles1 & ! sum ( two consecutive triangul>s w always make a squ>e4 ,9 numb]s x is equally cle>1 = if we exam9e ! fir/ triangul>s--#a1 #c1 #f1 #aj1 #ae1 #ba1 #bh--we f9d t by add+ all ! consecutive pairs 9 turn we get ! s]ies ( squ>e numb]s--#i1 #af1 #be1 #cf1 #di1 etc4 ,! me?od ( =m+ ?ree triangles f our numb]s is equally direct1 & n at all a matt] ( trial4 ,b ,i mu/ cont5t myself ) giv+ actual figures1 & j /at+ t e triangul> high] ?an #f w =m ?ree triangul>s4 ,i give ! sides ( ! triangles1 & read]s w "k f my rem>ks :5 /at+ ! puzzle h[ to f9d f ^! sides ! numb] ( c\nt]s or co9s 9 ea*1 & s *eck ! results if !y s wi%4 ---------------------------------------- ------------------------------- ,numb] ,side ( ,side ( ,sides ( ,two ,sides ( ,?ree ,squ>e4 ,triangle4 ,triangles4 ,triangles4 ---------------------------------------- ------------------------------ #cf #f #h #f #e #e #e #c #abbe #ce #di #cf #cd #cc #cb #af #dafaf #bjd #bhh #bjd #bjc #aib #aib #ie #adacgba #aahi #afha #aahi #aahh #aaba #aabj #efj #dhjbdijj #ficj #ihjj #ficj #fibi #fecc #fecc #cbfg #afcadcbhha #djcia #egaba #djcia #djcij #chjha #chjhj #aijdj ---------------------------------------- ------------------------------ ,i %\ld p]haps expla9 t ! arrangem5ts giv5 9 ! la/ two columns are n ! only ways ( =m+ two & ?ree triangles4 ,!re are o!rs1 b "o set ( figures w fully s]ve our purpose4 ,we ?us see t be=e ,mrs4 ,mc,allister c claim h] six? #e pres5t %e mu/ save ! respectable sum ( #a1fca1dcb1hha4 #ach4--,,! ,,artill]ym5',s ,,dilemma4 ,we 7 requir$ to f9d ! smalle/ numb] ( cannon balls t we c\ld lay on ! gr\nd to =m a p]fect squ>e1 & c\ld pile into a squ>e pyramid4 ,i w try to make ! matt] cle> to ! m]e/ novice4 #a #b #c #d #e #f #g #a #c #f #aj #ae #ba #bh #a #d #aj #bj #ce #ef #hd #a #e #ad #cj #ee #ia #adj ,"h 9 ! fir/ r[ we place 9 regul> ord] ! natural numb]s4 ,ea* numb] 9 ! second r[ repres5ts ! sum ( ! numb]s 9 ! r[ above1 f ! beg9n+ to ! numb] j ov] x4 ,?us #a1 #b1 #c1 #d1 add$ toge!r1 make #aj4 ,! ?ird r[ is =m$ 9 exactly ! same way z ! second4 ,9 ! f\r? r[ e numb] is =m$ by add+ toge!r ! numb] j above x & ! prec$+ numb]4 ,?us #d & #aj make #ad1 #bj & #ce make #ee4 ,n[1 all ! numb]s 9 ! second r[ are triangul> numb]s1 :i* means t ^! numb]s ( cannon balls may be laid out on ! gr\nd s z to =m equilat]al triangles4 ,! numb]s 9 ! ?ird r[ w all =m our triangul> pyramids1 :ile ! numb]s 9 ! f\r? r[ w all =m squ>e pyramids4 ,?us ! v process ( =m+ ! above numb]s %[s u t e squ>e pyramid is ! sum ( two triangul> pyramids1 "o ( :i* has ! same numb] ( balls 9 ! side at ! base1 & ! o!r "o ball few]4 ,if we cont9ue ! above table to tw5ty-f\r places1 we %all rea* ! numb] #d1ijj 9 ! f\r? r[4 ,z ?is numb] is ! squ>e ( #gj1 we c lay out ! balls 9 a squ>e1 & c =m a squ>e pyramid ) !m4 ,?is mann] ( writ+ out ! s]ies until we come to a squ>e numb] does n appeal to ! ma!matical m9d1 b x s]ves to %[ h[ ! answ] to ! p>ticul> puzzle may be easily arriv$ at by anybody4 ,z a matt] ( fact1 ,i confess my failure to discov] any numb] o!r ?an #d1ijj t fulfils ! conditions1 nor h ,i f\nd any rigid pro( t ?is is ! only answ]4 ,! problem is a difficult "o1 & ! second answ]1 if x exi/s 7:i* ,i d n believe71 c]ta9ly runs into big figures4 ,= ! b5efit ( m advanc$ ma!maticians ,i w add t ! g5]al expression = squ>e pyramid numb]s is 7#bn #cn n7_/#f4 ,= ?is expression to be also a squ>e numb] 7! special case ( #a except$7 x is necess>y t n p - #a #ft1 :]e #bp - #a q 7! ,pellian ,equation74 ,9 ! case ( our solution above1 n #bd1 p #e1 t #b1 q #g4 #aci4--,,! ,,dut*m5',s ,,wives4 ,! money paid 9 e case 0 a squ>e numb] ( %ill+s1 because !y b\ght #a at #as41 #b at #bs41 #c at #cs41 & s on4 ,b e husb& pays altoge!r #fcs4 m ?an 8 wife1 s we h to f9d 9 h[ _m ways #fc may be ! diff]5ce betwe5 two squ>e numb]s4 ,^! are ! ?ree only possible ways3 ! squ>e ( #h less ! squ>e ( #a1 ! squ>e ( #ab less ! squ>e ( #i1 & ! squ>e ( #cb less ! squ>e ( #ca4 ,"h #a1 #i1 & #ca repres5t ! numb] ( pigs b\ght & ! numb] ( %ill+s p] pig paid by ea* woman1 & #h1 #ab1 & #cb ! same 9 ! case ( _! respective husb&s4 ,f ! fur!r in=mation giv5 z to _! pur*ases1 we c n[ pair !m (f z foll[s3 ,cornelius & ,gurtrn b\ght #h & #a2 ,elas & ,katrn b\ght #ab & #i2 ,h5drick & ,anna b\ght #cb & #ca4 ,& ^! pairs repres5t correctly ! ?ree m>ri$ c\ples4 ,! read] may "h desire to "k h[ we may det]m9e ! maximum numb] ( ways 9 :i* a numb] may be express$ z ! diff]5ce betwe5 two squ>es1 & h[ we are to f9d ! actual squ>es4 ,any integ] except #a1 #d1 & twice any odd numb]1 may be express$ z ! diff]5ce ( two integral squ>es 9 z _m ways z x c be split up into pairs ( factors1 c\nt+ #a z a factor4 ,suppose ! numb] to be #e1idj4 ,! factors are #b4c4e4aa4 ,"h ! expon5ts are #b1 #c1 #a1 #a4 ,always d$uct #a f ! expon5ts ( #b & add #a to all ! o!r expon5ts2 !n we get #a1 #d1 #b1 #b1 & half ! product ( ^! f\r numb]s w be ! requir$ numb] ( ways 9 :i* #e1idj may be ! diff]5ce ( two squ>es--t is1 #h4 ,to f9d ^! eight squ>es1 z x is an ev5 numb]1 we fir/ divide by #d & get #adhe1 ! eight pairs ( factors ( :i* are #a #adhe1 #c #die1 #e #big1 #i #afe1 #aa #ace1 #ae #ii1 #bg #ee1 & #cc #de4 ,! sum & diff]5ce ( any "o ( ^! pairs w give ! requir$ numb]s4 ,?us1 ! squ>e ( #a1dhf less ! squ>e ( #a1dhd is #e1idj1 ! squ>e ( #dih less ! squ>e ( #dib is ! same1 & s on4 ,9 ! case ( #fc above1 ! numb] is odd2 s we factorize at once1 #a #fc1 #c #ba1 #g #i4 ,!n we f9d t half ! sum & diff]5ce w give u ! numb]s #cb & #ca1 #ab & #i1 & #h & #a1 z %[n 9 ! solution to ! puzzle4 ,! rev]se problem1 to f9d ! factors ( a numb] :5 y h express$ x z ! diff]5ce ( two squ>es1 is obvi\s4 ,= example1 ! sum & diff]5ce ( any pair ( numb]s 9 ! la/ s5t5ce w give u ! factors ( #fc4 ,e prime numb] 7except #a & #b7 may be express$ z ! diff]5ce ( two squ>es 9 "o way1 & 9 "o way only4 ,if a numb] c be express$ z ! diff]5ce ( two squ>es 9 m ?an "o way1 x is composite2 & hav+ s express$ x1 we may at once obta9 ! factors1 z we h se54 ,f]mat %[$ 9 a lett] to ,m]s5ne or ,frnicle1 9 #afdc1 h[ we may discov] :e!r a numb] may be express$ z ! diff]5ce ( two squ>es 9 m ?an "o way1 or prov$ to be a prime4 ,b ! me?od1 :5 deal+ ) l>ge numb]s1 is necess>ily t$i\s1 ?\gh 9 practice x may be consid]ably %ort5$4 ,9 _m cases x is ! %orte/ me?od kn[n = factoriz+ l>ge numb]s1 & ,i h always held ! op9ion t ,f]mat us$ x 9 p]=m+ a c]ta9 feat 9 factoriz+ t is hi/orical & wrapp$ 9 my/]y4 #adj4--;,,f9d ,,ada',s ,,surname4 ,! girls' names 7 ,ada ,smi?1 ,annie ,br[n1 ,emily ,jones1 ,m>y ,rob9son1 & ,bessie ,evans4 #ada4--,,saturday ,,m>ket+4 ,z e p]son's pur*ase 0 ( ! value ( an exact numb] ( %ill+s1 & z ! p>ty possess$ :5 !y />t$ out =ty %ill+ co9s altoge!r1 !re 0 no necessity = any lady to h any small] *ange1 or any evid5ce t !y actually _h su* *ange4 ,?is be+ s1 ! only answ] possible is t ! wom5 7 nam$ respectively ,anne ,jones1 ,m>y ,rob9son1 ,jane ,smi?1 & ,kate ,br[n4 ,x w n[ be f\nd t !re w\ld be exactly eight %ill+s left1 :i* may be divid$ equally among ! eight p]sons 9 co9 )\t any *ange be+ requir$4 #adb4--,,! ,,silk ,,pat*work4 ,illu/ration ,our illu/ration w %[ h[ to cut ! /it*es ( ! pat*work s z to get ! squ>e ,f entire1 & f\r equal pieces1 ,g1 ,h1 ,i1 ,k1 t w =m a p]fect ,greek cross4 ,! read] w "k h[ to assemble ^! f\r pieces f ,fig4 #ac 9 ! article4 ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 #adc4--,,two ,,crosses ,,f ,,"o4 ,x w be se5 t "o cross is cut out entire1 z ,a 9 ,fig4 #a1 :ile ! f\r pieces m>k$ ,b1 ,c1 ,d & ,e =m ! second cross1 z 9 ,fig4 #b1 :i* w be ( exactly ! same size z ! o!r4 ,i w leave ! read] ! pleasant task ( discov]+ = himself ! be/ way ( f9d+ ! direction ( ! cuts4 ,note t ! ,swa/ika aga9 appe>s4 ,! difficult "q n[ pres5ts itself3 ,h[ are we to cut ?ree ,greek crosses f "o 9 ! fewe/ possible pieces8 ,z a matt] ( fact1 ?is problem may be solv$ 9 z few z ?irte5 pieces2 b z ,i "k _m ( my read]s1 advanc$ geometricians1 w be glad to h some?+ to "w on ( :i* !y are n %[n ! solution1 ,i leave ! my/]y = ! pres5t undisclos$4 #add4--,,! ,,cross ,,& ,,! ,,triangle4 ,! l9e ,a ,b 9 ! foll[+ diagram repres5ts ! side ( a squ>e hav+ ! same area z ! cross4 ,i h %[n else:]e1 z /at$1 h[ to make a squ>e & equilat]al triangle ( equal area4 ,i ne$ n g1 !re=e1 into ! prelim9>y "q ( f9d+ ! dim5sions ( ! triangle t is to equal our cross4 ,we w assume t we h already f\nd ?is1 & ! "q !n becomes1 ,h[ are we to cut up "o ( ^! into pieces t w =m ! o!r8 ,fir/ draw ! l9e ,a ,b :]e ,a & ,b are midway betwe5 ! extremities ( ! two side arms4 ,next make ! l9es ,d ,c & ,e ,f equal 9 l5g? to half ! side ( ! triangle4 ,n[ f ,e & ,f describe ) ! same radius ! int]sect+ arcs at ,g & draw ,f ,g4 ,f9ally make ,i ,k equal to ,h ,c & ,l ,b equal to ,a ,d4 ,if we n[ draw ,i ,l1 x %\ld be p>allel to ,f ,g1 & all ! six pieces are m>k$ out4 ,^! fit toge!r & =m a p]fect equilat]al triangle1 z %[n 9 ! second diagram4 ,or we might h fir/ f\nd ! direction ( ! l9e ,m ,n 9 our triangle1 !n plac$ ! po9t ,o ov] ! po9t ,e 9 ! cross & turn$ r\nd ! triangle ov] ! cross until ! l9e ,m ,n 0 p>allel to ,a ,b4 ,! piece #e c !n be m>k$ (f & ! o!r pieces 9 succession4 ,illu/ration ,i h se5 _m attempts at a solution involv+ ! assumption t ! height ( ! triangle is exactly ! same z ! height ( ! cross4 ,?is is a fallacy3 ! cross w always be high] ?an ! triangle ( equal area4 #ade4--,,! ,,fold$ ,,cross4 ,illu/ration3 ,,fig4 #a41 ,,fig #b4 ,fir/ fold ! cross along ! dott$ l9e ,a ,b 9 ,fig4 #a4 ,y !n h x 9 ! =m %[n 9 ,fig4 #b4 ,next fold x along ! dott$ l9e ,c ,d 7:]e ,d is1 ( c\rse1 ! c5tre ( ! cross71 & y get ! =m %[n 9 ,fig4 #c4 ,n[ take y\r scissors & cut f ,g to ,f1 & ! f\r pieces1 all ( ! same size & %ape1 w fit toge!r & =m a squ>e1 z %[n 9 ,fig4 #d4 ,illu/ration3 ,,fig4 #c41 ,,fig4 #d4 #adf4--;,,an ,,easy ,,dissection ,,puzzle4 ,illu/ration - 3 3 3 3 3 ----------- _/ 3 3 _/ 3 3 _/ 3 3 _/ 3 3 _/ 3 3 ,! solution to ?is puzzle is %[n 9 ! illu/ration4 ,divide ! figure up into twelve equal triangles1 & x is easy to discov] ! directions ( ! cuts1 z indicat$ by ! d>k l9es4 #adg4--;,,an ,,easy ,,squ>e ,,puzzle4 ,illu/ration ---------------------------------------- - 4 _/ 4 _/ 4 _/ _/ _/ _/ 4 _/ _/ 4 _/ _/ 4 _/ _/ 4_/ -------------------- _/ _/ _/ _/ _/ _/ _/ 4 _/ _/ 4 _/ _/ 4 _/ _/ 4 _/ _/ 4 _/ 4 _/ 4 _/ 4 ---------------------------------------- - ,! diagram expla9s itself1 "o ( ! five pieces hav+ be5 cut 9 two to =m a squ>e4 #adh4--,,! ,,bun ,,puzzle4 ,illu/ration 4 4 4 4 4 4 4 ,a 4 4 ,c 4 _/ 4 _/ 4 4 4 ,b 4 4 4 4 4 - 4 4 4 4 4 4 ,d ,e 4 4 4 4 4 4 4 4 4 -- 4 4 4 4 4 - - 4 ,g ,f - - 4 4 4 4 4 - - 4 4 4 - -- 4 4 - - 4 ,h - - 4 4 - - ,! secret ( ! bun puzzle lies 9 ! fact t1 ) ! relative dim5sions ( ! circles z giv51 ! ?ree diamet]s w =m a "r-angl$ triangle1 z %[n by ,a1 ,b1 ,c4 ,x foll[s t ! two small] buns are exactly equal to ! l>ge bun4 ,!re=e1 if we give ,david & ,edg> ! two halves m>k$ ,d & ,e1 !y w h _! fair %>es--"o qu>t] ( ! confection]y ea*4 ,!n if we place ! small bun1 ,h1 on ! top ( ! rema9+ "o & trace its circumf]5ce 9 ! mann] %[n1 ,fr$'s piece1 ,f1 w exactly equal ,h>ry's small bun1 ,h1 ) ! addition ( ! piece m>k$ ,g--half ! rim ( ! o!r4 ,?us ea* boy gets an exactly equal %>e1 & !re are only five pieces necess>y4 #adi4--,,! ,,*ocolate ,,squ>es4 ,illu/ration ,squ>e ,a is left entire2 ! two pieces m>k$ ,b fit toge!r & make a second squ>e2 ! two pieces ,c make a ?ird squ>e2 & ! f\r pieces m>k$ ,d w =m ! f\r? squ>e4 #aej4--;,,dissect+ ,a ,,mitre4 ,! diagram on ! next page %[s h[ to cut into five pieces to =m a squ>e4 ,! dott$ l9es are int5d$ to %[ h[ to f9d ! po9ts ,c & ,f--! only difficulty4 ,a ,b is half ,b ,d1 & ,a ,e is p>allel to ,b ,h4 ,) ! po9t ( ! compasses at ,b describe ! arc ,h ,e1 & ,a ,e w be ! di/ance ( ,c f ,b4 ,!n ,f ,g equals ,b ,c less ,a ,b4 ,?is puzzle--) ! add$ condition t x %all be cut into f\r p>ts ( ! same size & %ape--,i h n be5 able to trace to an e>li] date ?an #ahce4 ,/rictly speak+1 x is1 9 t =m1 impossible ( solution2 b ,i give ! answ] t is always pres5t$1 & t seems to satisfy mo/ p4 ,illu/ration ,we are ask$ to assume t ! two portions conta9+ ! same lett]--,,aa1 ,,bb1 ,,cc1 ,,dd--are jo9$ by a m]e hair1 & are1 !re=e1 only "o piece4 ,to ! geometrician ?is is absurd1 & ! f\r %>es are n equal 9 area unless !y consi/ ( two pieces ea*4 ,if y make !m equal 9 area1 !y w n be exactly alike 9 %ape4 ,illu/ration #aea4--,,! ,,jo9]',s ,,problem4 ,illu/ration ,no?+ c\ld be easi] ?an ! solution ( ?is puzzle--:5 y "k h[ to d x4 ,& yet x is apt to p]plex ! novice a good deal if he wants to d x 9 ! fewe/ possible pieces--?ree4 ,all y h to d is to f9d ! po9t ,a1 midway betwe5 ,b & ,c1 & !n cut f ,a to ,d & f ,a to ,e4 ,! ?ree pieces !n =m a squ>e 9 ! mann] %[n4 ,( c\rse1 ! proportions ( ! orig9al figure mu/ be correct2 ?us ! triangle ,,bef is j a qu>t] ( ! squ>e ,,bcdf4 ,draw l9es f ,b to ,d & f ,c to ,f & ?is w be cle>4 #aeb4--;,,ano!r ,,jo9]',s ,,problem4 ,illu/ration ,,! po9t 0 to f9d a g5]al rule = =m+ a p]fect squ>e out ( ano!r squ>e comb9$ ) a "r-angl$ isosceles triangle4 ,! triangle to :i* geometricians give ?is high-s\nd+ "n is1 ( c\rse1 no?+ m or less ?an half a squ>e t has be5 divid$ f corn] to corn]4 ,! precise relative proportions ( ! squ>e & triangle are ( no consequ5ce :atev]4 ,x is only necess>y to cut ! wood or mat]ial into five pieces4 ,suppose our orig9al squ>e to be ,,aclf 9 ! above diagram & our triangle to be ! %ad$ portion ,,c$4 ,n[1 we fir/ f9d half ! l5g? ( ! long side ( ! triangle 7,,cd7 & measure (f ?is l5g? at ,,ab4 ,!n we place ! triangle 9 its pres5t position aga9/ ! squ>e & make two cuts--"o f ,b to ,f1 & ! o!r f ,b to ,e4 ,/range z x may seem1 t is all t is necess>y6 ,if we n[ remove ! pieces ,g1 ,h1 & ,m to _! new places1 z %[n 9 ! diagram1 we get ! p]fect squ>e ,,bekf4 ,take any two squ>e pieces ( pap]1 ( diff]5t sizes b p]fect squ>es1 & cut ! small] "o 9 half f corn] to corn]4 ,n[ proce$ 9 ! mann] %[n1 & y w f9d t ! two pieces may be comb9$ to =m a l>g] squ>e by mak+ ^! two simple cuts1 & t no piece w be requir$ to be turn$ ov]4 ,! rem>k t ! triangle might be a little l>g] or a good deal small] 9 proportion 0 int5d$ to b> cases :]e area ( triangle is great] ?an area ( squ>e4 ,9 su* cases six pieces are necess>y1 & if triangle & squ>e are ( equal area !re is an obvi\s solution 9 ?ree pieces1 by simply cutt+ ! squ>e 9 half diagonally4 #aec4--;,a ,,cutt+-,,out ,,puzzle4 ,illu/ration ,! illu/ration %[s h[ to cut ! f\r pieces & =m ) !m a squ>e4 ,fir/ f9d ! side ( ! squ>e 7! mean proportional betwe5 ! l5g? & height ( ! rectangle71 & ! me?od is obvi\s4 ,if our /rip is exactly 9 ! proportions #i #a1 or #af #a1 or #be #a1 we c cle>ly cut x 9 #c1 #d1 or #e rectangul> pieces respectively to =m a squ>e4 ,exclud+ ^! special cases1 ! g5]al law is t = a /rip 9 l5g? m ?an n times ! bread?1 & n m ?an 7n#a7 times ! bread?1 x may be cut 9 n#b pieces to =m a squ>e1 & !re w be n-#a rectangul> pieces l piece #d 9 ! diagram4 ,?us1 = example1 ) a /rip #bd #a1 ! l5g? is m ?an #af & less ?an #be times ! bread?4 ,!re=e x c be done 9 #f pieces 7n "h be+ #d71 #c ( :i* w be rectangul>4 ,9 ! case :]e n equals #a1 ! rectangle disappe>s & we get a solution 9 ?ree pieces4 ,)9 ^! limits1 ( c\rse1 ! sides ne$ n be rational3 ! solution is purely geometrical4 #aed4--,,mrs4 ,,hobson',s ,,he>?rug4 ,illu/ration ,z ,i gave full measurem5ts ( ! mutilat$ rug1 x 0 q an easy matt] to f9d ! precise dim5sions = ! squ>e4 ,! two pieces cut (f w\ld1 if plac$ toge!r1 make an oblong piece #ab #f1 giv+ an area ( #gb 7in*es or y>ds1 z we please71 & z ! orig9al complete rug measur$ #cf #bg1 x _h an area ( #igb4 ,if1 !re=e1 we d$uct ! pieces t h be5 cut away1 we f9d t our new rug w conta9 #igb less #gb1 or #ijj2 & z #ijj is ! squ>e ( #cj1 we "k t ! new rug mu/ measure #cj #cj to be a p]fect squ>e4 ,?is is a great help t[>ds ! solution1 because we may safely conclude t ! two horizontal sides measur+ #cj ea* may be left intact4 ,!re is a v easy way ( solv+ ! puzzle 9 f\r pieces1 & also a way 9 ?ree pieces t c sc>cely be call$ difficult1 b ! correct answ] is 9 only two pieces4 ,x w be se5 t if1 aft] ! cuts are made1 we ins]t ! tee? ( ! piece ,b "o too? l[] d[n1 ! two portions w fit toge!r & =m a squ>e4 #aee4--,,! ,,p5tagon ,,& ,,squ>e4 ,a regul> p5tagon may be cut into z few z six pieces t w fit toge!r )\t any turn+ ov] & =m a squ>e1 z ,i %all %[ bel[4 ,hi!rto ! be/ answ] has be5 9 sev5 pieces--! solution produc$ "s ye>s ago by a =eign ma!matician1 ,paul ,buss*op4 ,we fir/ =m a p>allelogram1 & f t ! squ>e4 ,! process w be se5 9 ! diagram on ! next page4 ,! p5tagon is ,,abcde4 ,by ! cut ,,ac & ! cut ,,fm 7,f be+ ! middle po9t betwe5 ,a & ,c1 & ,m be+ ! same di/ance f ,a z ,f7 we get two pieces t may be plac$ 9 position at ,,ghea & =m ! p>allelogram ,,ghdc4 ,we !n f9d ! mean proportional betwe5 ! l5g? ,,hd & ! height ( ! p>allelogram4 ,?is di/ance we m>k (f f ,c at ,k1 !n draw ,,ck1 & f ,g drop ! l9e ,,gl1 p]p5dicul> to ,,kc4 ,! re/ is easy & r obvi\s4 ,x w be se5 t ! six pieces w =m ei!r ! p5tagon or ! squ>e4 ,i h receiv$ :at purport$ to be a solution 9 five pieces1 b ! me?od 0 bas$ on ! r subtle fallacy t half ! diagonal plus half ! side ( a p5tagon equals ! side ( a squ>e ( ! same area4 ,i say subtle1 because x is an extremely close approximation t w deceive ! eye1 & is q difficult to prove inexact4 ,i am n aw>e t att5tion has be=e be5 drawn to ?is curi\s approximation4 ,illu/ration ,ano!r correspond5t made ! side ( 8 squ>e #a ( ! side ( ! p5tagon4 ,z a matt] ( fact1 ! ratio is irrational4 ,i calculate t if ! side ( ! p5tagon is #a--;in*1 foot1 or any?+ else--! side ( ! squ>e ( equal area is #a4caag ne>ly1 or say r\ghly #ac_/aj4 ,s we c only hope to solve ! puzzle by geometrical me?ods4 #aef4--,,! ,,dissect$ ,,triangle4 ,diagram ,a is our orig9al triangle4 ,we w say x measures #e in*es 7or #e feet7 on ea* side4 ,if we take (f a slice at ! bottom ( any equilat]al triangle by a cut p>allel ) ! base1 ! portion t rema9s w always be an equilat]al triangle2 s we fir/ cut (f piece #a & get a triangle #c in*es on e side4 ,! mann] ( f9d+ directions ( ! o!r cuts 9 ,a is obvi\s f ! diagram4 ,n[1 if we want two triangles1 #a w be "o ( !m1 & #b1 #c1 #d1 & #e w fit toge!r1 z 9 ,b1 to =m ! o!r4 ,if we want ?ree equilat]al triangles1 #a w be "o1 #d & #e w =m ! second1 z 9 ,c1 & #b & #c w =m ! ?ird1 z 9 ,d4 ,9 ,b & ,c ! piece #e is turn$ ov]2 b !re c be no objection to ?is1 z x is n =bidd51 & is 9 no way oppos$ to ! nature ( ! puzzle4 ,illu/ration #aeg4--,,! ,,table-,,top ,,& ,,/ools4 ,illu/ration ,"o object t ,i _h 9 view :5 pres5t+ ?is little puzzle 0 to po9t out ! unc]ta9ty ( ! mean+ convey$ by ! ^w oval4 ,?\gh orig9ally d]iv$ f ! ,lat9 ^w ovum1 an egg1 yet :at we und]/& z ! egg-%ape 7) "o end small] ?an ! o!r7 is only "o ( _m =ms ( ! oval2 :ile "s eggs are sph]ical 9 %ape1 & a sph]e or circle is mo/ c]ta9ly n an oval4 ,if we speak ( an ellipse--a conical ellipse--we are on saf] gr\nd1 b "h we mu/ be c>eful ( error4 ,i recollect a ,liv]pool t[n c\ncillor1 _m ye>s ago1 ^: ignorance ( ! p\ltry-y>d l$ him to sub/itute ! ^w h5 = f[l1 rem>k+1 ,we mu/ rememb]1 g5tlem51 t al?\gh e cock is a h51 e h5 is n a cock6 ,simil>ly1 we mu/ always note t al?\gh e ellipse is an oval1 e oval is n an ellipse4 ,x is correct to say t an oval is an oblong curvil9e> figure1 hav+ two unequal diamet]s1 & b\nd$ by a curve l9e return+ into itself2 & ?is includes ! ellipse1 b all o!r figures :i* 9 any way approa* t[>ds ! =m ( an oval )\t necess>ily hav+ ! prop]ties above describ$ are includ$ 9 ! t]m oval4 ,?us ! foll[+ solution t ,i give to our puzzle involves ! po9t$ oval1 kn[n among ar*itects z ! vesica piscis4 ,illu/ration3 ,,! ,,two ,,/ools4 ,! dott$ l9es 9 ! table are giv5 = great] cle>ness1 ! cuts be+ made along ! o!r l9es4 ,x w be se5 t ! eight pieces =m two /ools ( exactly ! same size & %ape ) simil> h&-holes4 ,^! holes are a trifle long] ?an ^? 9 ! s*oolma/]'s /ools1 b !y are mu* n>r[] & ( consid]ably small] area4 ,( c\rse #e & #f c be cut out 9 "o piece--also #g & #h--mak+ only six pieces 9 all4 ,b ,i wi%$ to keep ! same numb] z 9 ! orig9al /ory4 ,:5 ,i fir/ gave ! above puzzle 9 a ,london newspap]1 9 competition1 no correct solution 0 receiv$1 b an +5i\s & neatly execut$ attempt by a man ly+ 9 a ,london infirm>y 0 accompani$ by ! foll[+ note3 ,hav+ no compasses "h1 ,i 0 compell$ to improvise a pair ) ! aid ( a small p5knife1 a bit ( firewood f a bundle1 a piece ( t9 f a toy eng9e1 a t9 tack1 & two portions ( a hairp91 = po9ts4 ,!y are a fairly s]viceable pair ( compasses1 & ,i %all keep !m z a mem5to ( y\r puzzle4 #aeh4--,,! ,,great ,,monad4 ,! areas ( circles are to ea* o!r z ! squ>es ( _! diamet]s4 ,if y h a circle #b 94 9 diamet] & ano!r #d 94 9 diamet]1 !n "o circle w be f\r times z great 9 area z ! o!r1 because ! squ>e ( #d is f\r times z great z ! squ>e ( #b4 ,n[1 if we ref] to ,diagram #a1 we see h[ two equal squ>es may be cut into f\r pieces t w =m "o l>g] squ>e2 f :i* x is self-evid5t t any squ>e has j half ! area ( ! squ>e ( its diagonal4 ,9 ,diagram #b ,i h introduc$ a squ>e z x (t5 occurs 9 anci5t draw+s ( ! ,monad2 :i* 0 my reason = believ+ t ! symbol _h ma!matical mean+s1 s9ce x w be f\nd to demon/rate ! fact t ! area ( ! out] r+ or annulus is exactly equal to ! area ( ! inn] circle4 ,comp>e ,diagram #b ) ,diagram #a1 & y w see t z ! squ>e ( ! diamet] ,,cd is d\ble ! squ>e ( ! diamet] ( ! inn] circle1 or ,,ce1 !re=e ! area ( ! l>g] circle is d\ble ! area ( ! small] "o1 & consequ5tly ! area ( ! annulus is exactly equal to t ( ! inn] circle4 ,?is answ]s our fir/ "q4 ,illu/ration3 #a #b #c #d ,9 ,diagram #c ,i %[ ! simple solution to ! second "q4 ,x is obvi\sly correct1 & may be prov$ by ! cutt+ & sup]position ( p>ts4 ,! dott$ l9es w also s]ve to make x evid5t4 ,! ?ird "q is solv$ by ! cut ,,cd 9 ,diagram #b1 b x rema9s to be prov$ t ! piece ,f is really "o-half ( ! ,y9 or ! ,yan4 ,?is we w d 9 ,diagram #d4 ,! circle ,k has "o-qu>t] ! area ( ! circle conta9+ ,y9 & ,yan1 because its diamet] is j "o-half ! l5g?4 ,also ,l 9 ,diagram #c is1 we "k1 "o-qu>t] ! area4 ,x is !re=e evid5t t ,g is exactly equal to ,h1 & !re=e half ,g is equal to half ,h4 ,s t :at ,f loses f ,l x ga9s f ,k1 & ,f mu/ be half ( ,y9 or ,yan4 #aei4--,,! ,,squ>e ,,( ,,v5e]4 ,illu/ration3 --------------------------------------- 3 3 3 3 3 3 3 3 --------------------------------------- 3 3 3 3 3 3 3 3 --------------------------------------- 3 3 3 3 3333 --------------------------------------- 3 3 3 3 3 3 3 3 --------------------------------- 3 3 3 3 3 3 3 3 --------------------------------------- 3333 3 3 3 3 --------------------------------------- 3 3 3 3 3 3 ,b 3 3 ,c --------------------------------------- 3 3 3 3 3333 ------------------------ 3 3 3 3 3 3 3 3 --------------------------------------- 3 3 3 3 3 3 3 3 ,a --------------------------------------- 3333 3 3 3 3 --------------------------------------- 3 3 3 3 3 3 3 3 --------------- 3 3 3 3 3 ,d 3 3 3 ------------------------ ,any squ>e numb] may be express$ z ! sum ( two squ>es 9 an inf9ite numb] ( diff]5t ways4 ,! solution ( ! pres5t puzzle =ms a simple demon/ration ( ?is rule4 ,x is a condition t we give actual dim5sions4 ,9 ?is puzzle ,i ignore ! kn[n dim5sions ( our squ>e & "w on ! assumption t x is #acn by #acn4 ,! value ( n we c aft]w>ds det]m9e4 ,divide ! squ>e z %[n 7:]e ! dott$ l9es indicate ! orig9al m>k+s7 into #afi squ>es4 ,z #afi is ! sum ( ! two squ>es #add & #be1 we w proce$ to divide ! v5e] into two squ>es1 measur+ respectively #ab #ab & #e #e2 & z we "k t two squ>es may be =m$ f "o squ>e by dissection 9 f\r pieces1 we seek a solution 9 ?is numb]4 ,! d>k l9es 9 ! diagram %[ :]e ! cuts are to be made4 ,! squ>e #e #e is cut out :ole1 & ! l>g] squ>e is =m$ f ! rema9+ ?ree pieces1 ,b1 ,c1 & ,d1 :i* ! read] c easily fit toge!r4 ,n[1 n is cle>ly #e_/ac ( an in*4 ,consequ5tly our l>g] squ>e mu/ be #fj_/ac 94 #fj_/ac 941 & our small] squ>e #be_/ac 94 #be_/ac 94 ,! squ>e ( #fj_/ac add$ to ! squ>e ( #be_/ac is #be4 ,! squ>e is ?us divid$ into z few z f\r pieces t =m two squ>es ( kn[n dim5sions1 & all ! sixte5 nails are avoid$4 ,"h is a g5]al =mula = f9d+ two squ>es ^: sum %all equal a giv5 squ>e1 say a4 ,9 ! case ( ! solution ( our puzzle p #c1 q #b1 & a #e4 #bpqa _/ a7 p q7 - 7#bpqa7 --------- x2 --------------------------- y p q p q ,"h x y a4 #afj4--,,! ,,two ,,horse%oes4 ,! puzzle 0 to cut ! two %oes 7includ+ ! ho( conta9$ )9 ! outl9es7 into f\r pieces1 two pieces ea*1 t w\ld fit toge!r & =m a p]fect circle4 ,x 0 also /ipulat$ t all f\r pieces %\ld be diff]5t 9 %ape4 ,z a matt] ( fact1 x is a puzzle bas$ on ! pr9ciple conta9$ 9 t curi\s ,*9ese symbol ! ,monad4 7,see ,no4 #aeh47 ,illu/ration ,! above diagrams give ! correct solution to ! problem4 ,x w be notic$ t #a & #b are cut into ! requir$ f\r pieces1 all diff]5t 9 %ape1 t fit toge!r & =m ! p]fect circle %[n 9 ,diagram #c4 ,x w fur!r be obs]v$ t ! two pieces ,a & ,b ( "o %oe & ! two pieces ,c & ,d ( ! o!r =m two exactly simil> halves ( ! circle--! ,y9 & ! ,yan ( ! great ,monad4 ,x w be se5 t ! %ape ( ! horse%oe is m easily det]m9$ f ! circle ?an ! dim5sions ( ! circle f ! horse%oe1 ?\gh ! latt] pres5ts no difficulty :5 y "k t ! curve ( ! long side ( ! %oe is "p ( ! circumf]5ce ( y\r circle4 ,! diff]5ce betwe5 ,b & ,d is in/ructive1 & ! idea is useful 9 all su* cases :]e x is a condition t ! pieces mu/ be diff]5t 9 %ape4 ,9 =m+ ,d we simply add on a symmetrical piece1 a curvil9e> squ>e1 to ! piece ,b4 ,!re=e1 9 giv+ ei!r ,b or ,d a qu>t] turn be=e plac+ 9 ! new position1 a precisely simil> effect mu/ be produc$4 #afa4--,,! ,,betsy ,,ross ,,puzzle4 ,fold ! circul> piece ( pap] 9 half along ! dott$ l9e %[n 9 ,fig4 #a1 & divide ! upp] half into five equal p>ts z indicat$4 ,n[ fold ! pap] along ! l9es1 & x w h ! appe>ance %[n 9 ,fig4 #b4 ,if y want a /> l ,fig4 #c1 cut f ,a to ,b2 if y wi% "o l ,fig4 #d1 cut f ,a to ,c4 ,?us1 ! ne>] y cut to ! po9t at ! bottom ! long] w be ! po9ts ( ! />1 & ! f>!r (f f ! po9t t y cut ! %ort] w be ! po9ts ( ! />4 ,illu/ration #afb4--,,! ,,c>dbo>d ,,*a94 ,! read] w probably feel rew>d$ = any c>e & pati5ce t he may be/[ on cutt+ out ! c>dbo>d *a94 ,we w suppose t he has a piece ( c>dbo>d measur+ #h 94 by #b 941 ?\gh ! dim5sions are ( no importance4 ,yet if y want a long *a9 y mu/1 ( c\rse1 take a long /rip ( c>dbo>d4 ,fir/ rule p5cil l9es ,b ,b & ,c ,c1 half an in* f ! edges1 & also ! %ort p]p5dicul> l9es half an in* ap>t4 7,see next page47 ,rule l9es on ! o!r side 9 j ! same way1 & 9 ord] t !y %all co9cide x is well to prick ?r\gh ! c>d ) a ne$le ! po9ts :]e ! %ort l9es end4 ,n[ take y\r p5knife & split ! c>d f ,a ,a d[n to ,b ,b1 & f ,d ,d up to ,c ,c4 ,!n cut "r ?r\gh ! c>d along all ! %ort p]p5dicul> l9es1 & half ?r\gh ! c>d along ! %ort portions ( ,b ,b & ,c ,c t are n dott$4 ,next turn ! c>d ov] & cut half ?r\gh along ! %ort l9es on ,b ,b & ,c ,c at ! places t are imm$iately b5ea? ! dott$ l9es on ! upp] side4 ,) a little c>eful sep>ation ( ! p>ts ) ! p5knife1 ! c>dbo>d may n[ be divid$ into two int]lac+ ladd]-l portions1 z %[n 9 ,fig4 #b2 & if y cut away all ! %ad$ p>ts y w get ! *a91 cut solidly out ( ! c>dbo>d1 )\t any jo91 z %[n 9 ! illu/rations on page #dj4 ,x is an int]e/+ v>iant ( ! puzzle to cut out two keys on a r+--9 ! same mann] )\t jo94 ,illu/ration #afd4--,,! ,,potato ,,puzzle4 ,z _m z tw5ty-two pieces may be obta9$ by ! six cuts4 ,! illu/ration %[s a pretty symmetrical solution4 ,! rule 9 su* cases is t e cut %all int]sect e o!r cut & no two int]sections co9cide2 t is to say1 e l9e passes ?r\gh e o!r l9e1 b m ?an two l9es d n cross at ! same po9t any:]e4 ,!re are o!r ways ( mak+ ! cuts1 b ?is rule mu/ always be obs]v$ if we are to get ! full numb] ( pieces4 ,! g5]al =mula is t ) n cuts we c always produce 77n7n#a77_/b7a 4 ,"o ( ! problems propos$ by ! late ,sam ,loyd 0 to produce ! maximum numb] ( pieces by n /raight cuts ?r\gh a solid *eese4 ,( c\rse1 aga91 ! pieces cut (f may n be mov$ or pil$4 ,"h we h to deal ) ! int]section ( planes 7in/ead ( l9es71 & ! g5]al =mula is t ) n cuts we may produce 77n - #a7n7n #a77_/f n #a pieces4 ,x is extremely difficult to see ! direction & effects ( ! successive cuts = m ?an a few ( ! l[e/ values ( n4 #afe4--,,! ,,sev5 ,,pigs4 ,! illu/ration %[s ! direction = plac+ ! ?ree f5ces s z to enclose e pig 9 a sep>ate /y4 ,! greate/ numb] ( spaces t c be enclos$ ) ?ree /raight l9es 9 a squ>e is sev51 z %[n 9 ! la/ puzzle4 ,be>+ ?is fact 9 m9d1 ! puzzle mu/ be solv$ by trial4 ,illu/ration3 ,,! ,,sev5 ,,pigs4 #aff4--,,! ,,l&[n]',s ,,f5ces4 ,f\r f5ces only are necess>y1 z foll[s3-- ,illu/ration #afg4--,,! ,,wiz>d',s ,,cats4 ,! illu/ration requires no explanation4 ,x %[s cle>ly h[ ! ?ree circles may be drawn s t e cat has a sep>ate enclosure1 & _c approa* ano!r cat )\t cross+ a l9e4 ,illu/ration3 ,,! ,,wiz>ds' ,,cats4 #afh4--,,! ,,*ri/mas ,,pudd+4 ,! illu/ration %[s h[ ! pudd+ may be cut into two p>ts ( exactly ! same size & %ape4 ,! l9es mu/ necess>ily pass ?r\gh ! po9ts ,a1 ,b1 ,c1 ,d1 & ,e4 ,b1 subject to ?is condition1 !y may be v>i$ 9 an inf9ite numb] ( ways4 ,= example1 at a po9t midway betwe5 ,a & ! edge1 ! l9e may be complet$ 9 an unlimit$ numb] ( ways 7/raight or crook$71 provid$ x be exactly reflect$ f ,e to ! opposite edge4 ,& simil> v>iations may be introduc$ at o!r places4 ,illu/ration #afi4--;,a ,,tangram ,,p>adox4 ,! diagrams w %[ h[ ! figures are con/ruct$--ea* ) ! sev5 ,tangrams4 ,x w be notic$ t 9 bo? cases ! head1 hat1 & arm are precisely alike1 & ! wid? at ! base ( ! body ! same4 ,b ?is body conta9s f\r pieces 9 ! fir/ case1 & 9 ! second design only ?ree4 ,! fir/ is l>g] ?an ! second by exactly t n>r[ /rip indicat$ by ! dott$ l9e betwe5 ,a & ,b4 ,?is /rip is !re=e exactly equal 9 area to ! piece =m+ ! foot 9 ! o!r design1 ?\gh :5 ?us di/ribut$ along ! side ( ! body ! increas$ dim5sion is n easily app>5t to ! eye4 ,illu/ration #agj4--,,! ,,cu%ion ,,cov]s4 ,illu/ration ,! two pieces ( brocade m>k$ ,a w fit toge!r & =m "o p]fect squ>e cu%ion top1 & ! two pieces m>k$ ,b w =m ! o!r4 #aga4--,,! ,,bann] ,,puzzle4 ,! illu/ration expla9s itself4 ,divide ! bunt+ into #be squ>es 7because ?is numb] is ! sum ( two o!r squ>es--#af & #i71 & !n cut along ! ?ick l9es4 ,! two pieces m>k$ ,a =m "o squ>e1 & ! two pieces m>k$ ,b =m ! o!r4 ,illu/ration #agb4--,,mrs4 ,,smiley',s ,,*ri/mas ,,pres5t4 ,illu/ration ,illu/ration ,! fir/ /ep is to f9d six diff]5t squ>e numb]s t sum to #aif4 ,= example1 #a #d #be #cf #di #ha #aif2 #a #d #i #be #cf #aba #aif2 #a #i #af #be #fd #ha #aif4 ,! re/ calls = individual judgm5t & +5uity1 & no def9ite rules c be giv5 = proc$ure4 ,! annex$ diagrams w %[ solutions = ! fir/ two cases /at$4 ,( c\rse ! ?ree pieces m>k$ ,a & ^? m>k$ ,b w fit toge!r & =m a squ>e 9 ea* case4 ,! assembl+ ( ! p>ts may be slightly v>i$1 & ! read] may be int]e/$ 9 f9d+ a solution = ! ?ird set ( squ>es ,i h giv54 #agc4--,,mrs4 ,,p]k9s',s ,,quilt4 ,! foll[+ diagram %[s h[ ! quilt %\ld be con/ruct$4 ,illu/ration ,!re is1 ,i believe1 practically only "o solution to ?is puzzle4 ,! fewe/ sep>ate squ>es mu/ be elev54 ,! portions mu/ be ( ! sizes giv51 ! ?ree l>ge/ pieces mu/ be arrang$ z %[n1 & ! rema9+ gr\p ( eight squ>es may be reflect$1 b _c be diff]5tly arrang$4 #agd4--,,! ,,squ>es ,,( ,,brocade4 ,illu/ration3 ,diagram #a ,s f> z ,i h be5 able to discov]1 !re is only "o possible solution to fulfil ! conditions4 ,! pieces fit toge!r z 9 ,diagram #a1 ,diagrams #b & #c %[+ h[ ! two orig9al squ>es are to be cut4 ,x w be se5 t ! pieces ,a & ,c h ea* tw5ty *equ]s1 & are !re=e ( equal area4 ,diagram #d 7built up ) ! dissect$ squ>e ,no4 #e7 solves ! puzzle1 except = ! small condition conta9$ 9 ! words1 ,i cut ! two squ>es 9 ! mann] desir$4 ,9 ?is case ! small] squ>e is pres]v$ intact4 ,/ill ,i give x z an illu/ration ( a feature ( ! puzzle4 ,x is impossible 9 a problem ( ?is k9d to give a qu>t]-turn to any ( ! pieces if ! patt]n is to prop]ly mat*1 b 7z 9 ! case ( ,f1 9 ,diagram #d7 we may give a symmetrical piece a half-turn--t is1 turn x upside d[n4 ,:e!r or n a piece may be giv5 a qu>t]-turn1 a half-turn1 or no turn at all 9 ^! *equ]$ problems1 dep5ds on ! "* ( ! design1 on ! mat]ial employ$1 & also on ! =m ( ! piece itself4 ,illu/ration3 ,diagram #b ,illu/ration3 ,diagram #c ,illu/ration3 ,diagram #d ,illu/ration3 ,diagram #e #age4--;,,ano!r ,,pat*work ,,puzzle4 ,! lady ne$ only unpick ! /it*es along ! d>k l9es 9 ! l>g] portion ( pat*work1 :5 ! f\r pieces w fit toge!r & =m a squ>e1 z %[n 9 our illu/ration4 ,illu/ration #agf4--,,l9oleum ,,cutt+4 ,!re is only "o solution t w enable u to reta9 ! l>g] ( ! two pieces ) z little z possible cut f x4 ,fig4 #a 9 ! foll[+ diagram %[s h[ ! small] piece is to be cut1 & ,fig4 #b h[ we %\ld dissect ! l>g] piece1 :ile 9 ,fig4 #c we h ! new squ>e #aj #aj =m$ by ! f\r pieces ) all ! *equ]s prop]ly mat*$4 ,x w be se5 t ! piece ,d conta9s fifty-two *equ]s1 & ?is is ! l>ge/ piece t x is possible to pres]ve "u ! conditions4 ,illu/ration #agg4--;,,ano!r ,,l9oleum ,,puzzle4 ,cut along ! ?ick l9es1 & ! f\r pieces w fit toge!r & =m a p]fect squ>e 9 ! mann] %[n 9 ! small] diagram4 ,illu/ration3 ,,ano!r ,,l9oleum ,,puzzle4 #agh4--,,! ,,c>dbo>d ,,box4 ,! areas ( ! top & side multipli$ toge!r & divid$ by ! area ( ! end give ! squ>e ( ! l5g?4 ,simil>ly1 ! product ( top & end divid$ by side gives ! squ>e ( ! bread?2 & ! product ( side & end divid$ by ! top gives ! squ>e ( ! dep?4 ,b we only ne$ "o ( ^! op]ations4 ,let u take ! fir/4 ,?us1 #abj #if divid$ by #hj equals #add1 ! squ>e ( #ab4 ,!re=e ! l5g? is #ab in*es1 f :i* we c1 ( c\rse1 at once get ! bread? & dep?--#aj 94 & #h 94 respectively4 #agi4--,,/eal+ ,,! ,,bell-,,ropes4 ,:5ev] we h "o side 7a7 ( a "r-angl$ triangle1 & "k ! diff]5ce betwe5 ! second side & ! hypot5use 7:i* diff]5ce we w call b71 !n ! l5g? ( ! hypot5use w be a b --- -4 #b;b #b ,9 ! case ( our puzzle ?is w be #dh #dh ------- #a 94 #cb ft4 #a 941 #f :i* is ! l5g? ( ! rope4 #ahj-- ,,! ,,f\r ,,sons4 ,illu/ration ,! diagram %[s ! mo/ equitable division ( ! l& possible1 s t ea* son %all receive l& ( exactly ! same area & exactly simil> 9 %ape1 & s t ea* %all h access to ! well 9 ! c5tre )\t trespass on ano!r's l&4 ,! conditions d n require t ea* son's l& %all be 9 "o piece1 b x is necess>y t ! two portions assign$ to an individual %\ld be kept ap>t1 or two adjo9+ portions might be held to be "o piece1 9 :i* case ! condition z to %ape w\ld h to be brok54 ,at pres5t !re is only "o %ape = ea* piece ( l&--half a squ>e divid$ diagonally4 ,& ,a1 ,b1 ,c1 & ,d c ea* rea* _! l& f ! outside1 & h ea* equal access to ! well 9 ! c5tre4 #aha4--,,! ,,?ree ,,railway ,,/ations4 ,! ?ree /ations =m a triangle1 ) sides #ac1 #ad1 & #ae miles4 ,make ! #ad side ! base2 !n ! height ( ! triangle is #ab & ! area #hd4 ,multiply ! ?ree sides toge!r & divide by f\r times ! area4 ,! result is eight miles & "o-eigh?1 ! di/ance requir$4 #ahb4--,,! ,,g>d5 ,,puzzle4 ,half ! sum ( ! f\r sides is #add4 ,f ?is d$uct 9 turn ! f\r sides1 & we get #fd1 #ii1 #dd1 & #ha4 ,multiply ^! toge!r1 & we h z ! result ! squ>e ( #d1geb4 ,!re=e ! g>d5 conta9$ #d1geb squ>e y>ds4 ,( c\rse ! tree be+ equidi/ant f ! f\r corn]s %[s t ! g>d5 is a quadrilat]al t may be inscrib$ 9 a circle4 #ahc4--;,,draw+ ,a ,,spiral4 ,make a fold 9 ! pap]1 z %[n by ! dott$ l9e 9 ! illu/ration4 ,!n1 tak+ any two po9ts1 z ,a & ,b1 describe semicircles on ! l9e alt]nately f ! c5tres ,b & ,a1 be+ c>eful to make ! ends jo91 & ! ?+ is done4 ,( c\rse ?is is n a true spiral1 b ! puzzle 0 to produce ! p>ticul> spiral t 0 %[n1 & t 0 drawn 9 ?is simple mann]4 ,illu/ration #ahd4--;,,h[ ,,to ,,draw ,,an ,,oval4 ,if y place y\r %eet ( pap] r\nd ! surface ( a cyl9drical bottle or cani/]1 ! oval c be drawn ) "o sweep ( ! compasses4 #ahe4--,,/4 ,,george',s ,,bann]4 ,z ! flag measures #d ft4 by #c ft41 ! l5g? ( ! diagonal 7f corn] to corn]7 is #e ft4 ,all y ne$ d is to d$uct half ! l5g? ( ?is diagonal 7#b ft47 f a qu>t] ( ! di/ance all r\nd ! edge ( ! flag 7#c ft47--a qu>t] ( #ad ft4 ,! diff]5ce 7#a ft47 is ! requir$ wid? ( ! arm ( ! r$ cross4 ,! area ( ! cross w !n be ! same z t ( ! :ite gr\nd4 #ahf4--,,! ,,clo!s ,,l9e ,,puzzle4 ,multiply toge!r1 & also add toge!r1 ! heights ( ! two poles & divide "o result by ! o!r4 ,t is1 if ! two heights are a & b respectively1 !n ab_/7a b7 w give ! height ( ! int]section4 ,9 ! p>ticul> case ( our puzzle1 ! int]section 0 !re=e #b ft4 #aa 94 f ! gr\nd4 ,! di/ance t ! poles are ap>t does n affect ! answ]4 ,! read] :o may h imag9$ t ?is 0 an accid5tal omission w p]haps be int]e/$ 9 discov]+ ! reason :y ! di/ance betwe5 ! poles may be ignor$4 #ahg4--,,! ,,milkmaid ,,puzzle4 ,illu/ration3 ,a ,b ,,riv] ------------------ _/ _/ _/ _/ ,,door ,,/ool ,draw a /raight l9e1 z %[n 9 ! diagram1 f ! milk+-/ool p]p5dicul> to ! ne> bank ( ! riv]1 & cont9ue x to ! po9t ,a1 :i* is ! same di/ance f t bank z ! /ool4 ,if y n[ draw ! /raight l9e f ,a to ! door ( ! dairy1 x w cut ! riv] at ,b4 ,!n ! %orte/ r\te w be f ! /ool to ,b & !nce to ! door4 ,obvi\sly ! %orte/ di/ance f ,a to ! door is ! /raight l9e1 & z ! di/ance f ! /ool to any po9t ( ! riv] is ! same z f ,a to t po9t1 ! correctness ( ! solution w probably appeal to e read] )\t any acqua9tance ) geometry4 #ahh4--,,! ,,ball ,,problem4 ,if a r\nd ball is plac$ on ! level gr\nd1 six simil> balls may be plac$ r\nd x 7all on ! gr\nd71 s t !y %all all t\* ! c5tral ball4 ,z = ! second "q1 ! ratio ( ! diamet] ( a circle to its circumf]5ce we call pi2 & ?\gh we _c express ?is ratio 9 exact numb]s1 we c get suffici5tly ne> to x = all practical purposes4 ,h[ev]1 9 ?is case x is n necess>y to "k ! value ( pi at all4 ,because1 to f9d ! area ( ! surface ( a sph]e we multiply ! squ>e ( ! diamet] by pi2 to f9d ! volume ( a sph]e we multiply ! cube ( ! diamet] by "o-six? ( pi4 ,!re=e we may ignore pi1 & h m]ely to seek a numb] ^: squ>e %all equal "o-six? ( its cube4 ,?is numb] is obvi\sly #f4 ,!re=e ! ball 0 #f ft4 9 diamet]1 = ! area ( its surface w be #cf times pi 9 squ>e feet1 & its volume also #cf times pi 9 cubic feet4 #ahi4--,,! ,,york%ire ,,e/ates4 ,! triangul> piece ( l& t 0 n = sale conta9s exactly elev5 acres4 ,( c\rse x is n difficult to f9d ! answ] if we foll[ ! ecc5tric & tricky tracks ( intricate trigonometry2 or ,i might say t ! application ( a well-kn[n =mula r$uces ! problem to f9d+ "o-qu>t] ( ! squ>e root ( 7#d #cgj #aaf7 -7#cgj #aaf - #gd7--t is a qu>t] ( ! squ>e root ( #aicf1 :i* is "o-qu>t] ( #dd1 or #aa acres4 ,b all t ! read] really requires to "k is ! ,py?agorean law on :i* _m puzzles h be5 built1 t 9 any "r-angl$ triangle ! squ>e ( ! hypot5use is equal to ! sum ( ! squ>es ( ! o!r two sides4 ,i %all disp5se ) all surds & simil> absurdities1 not)/&+ ! fact t ! sides ( our triangle are cle>ly incomm5surate1 s9ce we _c exactly extract ! squ>e roots ( ! ?ree squ>e areas4 ,illu/ration3 ,a 4 4 #e 4 #g 4 ,e --------- ,c 4 4 4 4 4 #d #d 4 4 #g 44 ,d--------------------------- ,b ,f ,9 ! above diagram ,,abc repres5ts our triangle4 ,,adb is a "r-angl$ triangle1 ,,ad measur+ #i & ,,bd measur+ #ag1 because ! squ>e ( #i add$ to ! squ>e ( #ag equals #cgj1 ! kn[n area ( ! squ>e on ,,ab4 ,also ,,aec is a "r-angl$ triangle1 & ! squ>e ( #e add$ to ! squ>e ( #g equals #gd1 ! squ>e e/ate on ,a ,c4 ,simil>ly1 ,,cfb is a "r-angl$ triangle1 = ! squ>e ( #d add$ to ! squ>e ( #aj equals #aaf1 ! squ>e e/ate on ,,bc4 ,n[1 al?\gh ! sides ( our triangul> e/ate are incomm5surate1 we h 9 ?is diagram all ! exact figures t we ne$ to discov] ! area ) precision4 ,! area ( our triangle ,,adb is cle>ly half ( #i #ag1 or #gf acres4 ,! area ( ,,aec is half ( #e #g1 or #ag acres2 ! area ( ,,cfb is half ( #d #aj1 or #bj acres2 & ! area ( ! oblong ,,edfc is obvi\sly #d #g1 or #bh acres4 ,n[1 if we add toge!r #ag1 #bj1 & #bh #fe1 & d$uct ?is sum f ! area ( ! l>ge triangle ,,adb 7:i* we h f\nd to be #gf acres71 :at rema9s mu/ cle>ly be ! area ( ,,abc4 ,t is to say1 ! area we want mu/ be #gf - #fe #aa acres exactly4 #aij4--;,,f>m] ,,wurzel',s ,,e/ate4 ,! area ( ! complete e/ate is exactly "o hundr$ acres4 ,to f9d ?is answ] ,i use ! foll[+ little =mula1 _/#d;ab - 7a b - c7 -------------------- #d :]e a1 b1 c repres5t ! ?ree squ>e areas1 9 any ord]4 ,! expression gives ! area ( ! triangle ,a4 ,?is w be f\nd to be #i acres4 ,x c be easily prov$ t ,a1 ,b1 ,c1 & ,d are all equal 9 area2 s ! answ] is #bf #bj #ah #i #i #i #i #ajj acres4 ,illu/ration ,"h is ! pro(4 ,if e little dott$ squ>e 9 ! diagram repres5ts an acre1 ?is mu/ be a correct plan ( ! e/ate1 = ! squ>es ( #e & #a toge!r equal #bf2 ! squ>es ( #d & #b equal #bj2 & ! squ>es ( #c & #c add$ toge!r equal #ah4 ,n[ we see at once t ! area ( ! triangle ,e is #b1 ,f is #d1 & ,g is #d4 ,^! add$ toge!r make #aa acres1 :i* we d$uct f ! area ( ! rectangle1 #bj acres1 & we f9d t ! field ,a conta9s exactly #i acres4 ,if y want to prove t ,b1 ,c1 & ,d are equal 9 size to ,a1 divide !m 9 two by a l9e f ! middle ( ! longe/ side to ! opposite angle1 & y w f9d t ! two pieces 9 e case1 if cut out1 w exactly fit toge!r & =m ,a4 ,or we c get our pro( 9 a /ill easi] way4 ,! complete area ( ! squ>$ diagram is #ab #ab #add acres1 & ! portions #a1 #b1 #c1 #d1 n includ$ 9 ! e/ate1 h ! respective areas ( #ab1 #ag1 #i1 & #d4 ,^! add$ toge!r make #dd1 :i*1 d$uct$ f #add1 leaves #ajj z ! requir$ area ( ! complete e/ate4 #aia4--,,! ,,cresc5t ,,puzzle4 ,ref]r+ to ! orig9al diagram1 let ,,ac be x1 let ,,cd be x - #i1 & let ,,ec be x - #e4 ,!n x - #e is a mean proportional betwe5 x - #i & x1 f :i* we f9d t x equals #be4 ,!re=e ! diamet]s are #ej 94 & #da 94 respectively4 #aib4--,,! ,,puzzle ,,wall4 ,illu/ration ,! answ] giv5 9 all ! old books is t %[n 9 ,fig4 #a1 :]e ! curv$ wall %uts out ! cottages f access to ! lake4 ,b 9 seek+ ! direction = ! %orte/ possible wall mo/ read]s to-day1 rememb]+ t ! %orte/ di/ance betwe5 two po9ts is a /raight l9e1 w adopt ! me?od %[n 9 ,fig4 #b4 ,?is is c]ta9ly an improvem5t1 yet ! correct answ] is really t indicat$ 9 ,fig4 #c4 ,a measurem5t ( ! l9es w %[ t !re is a consid]able sav+ ( l5g? 9 ?is wall4 #aic4--,,! ,,%eep-,,fold4 ,?is is ! answ] t is always giv5 & accept$ z correct3 ,two m hurdles w\ld be necess>y1 = ! p5 0 tw5ty-f\r by "o 7z 9 ,fig4 ,a on next page71 & by mov+ "o ( ! sides & plac+ an extra hurdle at ea* end 7z 9 ,fig4 ,b7 ! area w\ld be d\bl$4 ,! diagrams are n to scale4 ,n[ !re is no condition 9 ! puzzle t requires ! %eep-fold to be ( any p>ticul> =m4 ,b ev5 if we accept ! po9t t ! p5 0 tw5ty-f\r by "o1 ! answ] utt]ly fails1 = two extra hurdles are c]ta9ly n at all necess>y4 ,= example1 ,i arrange ! fifty hurdles z 9 ,fig4 ,c1 & z ! area is increas$ f tw5ty-f\r squ>e hurdles to #aef1 !re is n[ accommodation = #fej %eep4 ,if x be held t ! area mu/ be exactly d\ble t ( ! orig9al p51 !n ,i con/ruct x 7z 9 ,fig4 ,d7 ) tw5ty-eight hurdles only1 & h tw5ty-two 9 h& = o!r purposes on ! f>m4 ,ev5 if x 7 insi/$ t all ! orig9al hurdles mu/ be us$1 !n ,i %\ld con/ruct x z 9 ,fig4 ,e1 :]e ,i c get ! area z exact z any f>m] c\ld possibly require1 ev5 if we h to all[ = ! fact t ! %eep might n be able to graze at ! extreme ends4 ,?us we see t1 f any po9t ( view1 ! accept$ answ] to ?is anci5t little puzzle breaks d[n4 ,& yet att5tion has nev] be=e be5 drawn to ! absurdity4 ,illu/ration ,a #bd -------------------------------- #bd #a -------------------------------- ,b -------------------------------- #dh #b -------------------------------- #bd ,c -------------------- ,d ---------- #ab #dh #f #aef ---------- #h -------------------- #ac #ab 4 ,e #ac 4 ' ' 4 4 ' ' 4 ' 4 4 ' #ab ' 4 ' #ac #aid4--,,! ,,g>d5 ,,walls4 ,! puzzle 0 to divide ! circul> field into f\r equal p>ts by ?ree walls1 ea* wall be+ ( exactly ! same l5g?4 ,!re are two ess5tial difficulties 9 ?is problem4 ,^! are3 7#a7 ! ?ickness ( ! walls1 & 7#b7 ! condition t ^! walls are ?ree 9 numb]4 ,z to ! fir/ po9t1 s9ce we are told t ! walls are brick walls1 we cle>ly _c ignore _! ?ickness1 :ile we h to f9d a solution t w equally "w1 :e!r ! walls be ( a ?ickness ( "o1 two1 ?ree1 or m bricks4 ,illu/ration ,! second po9t requires a little m consid]ation4 ,h[ are we to di/+ui% betwe5 a wall & walls8 ,a /raight wall )\t any b5d 9 x1 no matt] h[ long1 _c ev] become walls1 if x is nei!r brok5 nor int]sect$ 9 any way4 ,also our circul> field is cle>ly enclos$ by "o wall4 ,b if x _h happ5$ to be a squ>e or a triangul> enclosure1 w\ld !re be respectively f\r & ?ree walls or only "o enclos+ wall 9 ea* case8 ,x is true t we speak ( ! f\r walls ( a squ>e build+ or g>d51 b ?is is only a conv5tional way ( say+ ! f\r sides4 ,if y 7 speak+ ( ! actual brickwork1 y w\ld say1 ,i am go+ to enclose ?is squ>e g>d5 ) a wall4 ,angles cle>ly d n affect ! "q1 = we may h a zigzag wall j z well z a /raight "o1 & ! ,great ,wall ( ,*9a is a good example ( a wall ) pl5ty ( angles4 ,n[1 if y look at ,diagrams #a1 #b1 & #c1 y may be puzzl$ to decl>e :e!r !re are 9 ea* case two or f\r new walls2 b y _c call !m ?ree1 z requir$ 9 our puzzle4 ,! int]section ei!r affects ! "q or x does n affect x4 ,if y tie two pieces ( /r+ firmly toge!r1 or splice !m 9 a nautical mann]1 !y become "o piece ( /r+4 ,if y simply let !m lie across "o ano!r or ov]lap1 !y rema9 two pieces ( /r+4 ,x is all a "q ( jo9+ & weld+4 ,x may simil>ly be held t if two walls be built into "o ano!r--,i might almo/ say1 if !y be made homog5e\s--!y become "o wall1 9 :i* case ,diagrams #a1 #b1 & #c might ea* be said to %[ "o wall or two1 if x be indicat$ t ! f\r ends only t\*1 & are n really built into1 ! out] circul> wall4 ,! objection to ,diagram #d is t al?\gh x %[s ! ?ree requir$ walls 7assum+ ! ends are n built into ! out] circul> wall71 yet x is only absolutely correct :5 we assume ! walls to h no ?ickness4 ,a brick has ?ickness1 & !re=e ! fact ?r[s ! :ole me?od out & r5d]s x only approximately correct4 ,diagram #e %[s1 p]haps1 ! only correct & p]fectly satisfactory solution4 ,x w be notic$ t1 9 addition to ! circul> wall1 !re are ?ree new walls1 :i* t\* 7& s enclose7 b are n built into "o ano!r4 ,?is solution may be adapt$ to any desir$ ?ickness ( wall1 & its correctness z to area & l5g? ( wall space is s obvi\s t x is unnecess>y to expla9 x4 ,i w1 h[ev]1 j say t ! semicircul> piece ( gr\nd t ea* t5ant gives to 8 neighb\r is exactly equal to ! semicircul> piece t 8 neighb\r gives to him1 :ile any section ( wall space f\nd 9 "o g>d5 is precisely repeat$ 9 all ! o!rs4 ,( c\rse !re is an inf9ite numb] ( ways 9 :i* ?is solution may be correctly v>i$4 #aie4--,,lady ,,bel9da',s ,,g>d54 ,all t ,lady ,bel9da ne$ d 0 ?is3 ,%e %\ld measure f ,a to ,b1 fold h] tape 9 f\r & m>k (f ! po9t ,e1 :i* is ?us "o qu>t] ( ! side4 ,!n1 9 ! same way1 m>k (f ! po9t ,f1 "o-f\r? ( ! side ,,ad ,n[1 if %e makes ,,eg equal to ,,af1 & ,,gh equal to ,,ef1 !n ,,ah is ! requir$ wid? = ! pa? 9 ord] t ! b$ %all be exactly half ! area ( ! g>d54 ,an exact num]ical measurem5t c only be obta9$ :5 ! sum ( ! squ>es ( ! two sides is a squ>e numb]4 ,?us1 if ! g>d5 measur$ #ab poles by #e poles 7:]e ! squ>es ( #ab & #e1 #add & #be1 sum to #afi1 ! squ>e ( #ac71 !n #ab add$ to #e1 less #ac1 w\ld equal f\r1 & a qu>t] ( ?is1 #a pole1 w\ld be ! wid? ( ! pa?4 #aif4--,,! ,,te!r$ ,,goat4 ,illu/ration ,?is problem is q simple if prop]ly attack$4 ,let u suppose ! triangle ,,abc to repres5t our half-acre field1 & ! %ad$ portion to be ! qu>t]-acre ov] :i* ! goat w graze :5 te!r$ to ! corn] ,c4 ,n[1 z six equal equilat]al triangles plac$ toge!r w =m a regul> hexagon1 z %[n1 x is evid5t t ! %ad$ pa/ure is j "o-six? ( ! complete area ( a circle4 ,!re=e all we require is ! radius 7,,cd7 ( a circle conta9+ six qu>t]-acres or #a acres1 :i* is equal to #i1djh1ifj squ>e in*es4 ,z we only want our answ] to ! ne>e/ in*1 x is suffici5tly exact = our purpose if we assume t z #a is to #c4adaf1 s is ! diamet] ( a circle to its circumf]5ce4 ,if1 !re=e1 we divide ! la/ numb] ,i gave by #c4adaf1 & extract ! squ>e root1 we f9d t #a1gca in*es1 or #dh y>ds #c in*es1 is ! requir$ l5g? ( ! te!r to ! ne>e/ in*4 #aig4--,,! ,,compasses ,,puzzle4 ,let ,,ab 9 ! foll[+ diagram be ! giv5 /raight l9e4 ,) ! c5tres ,a & ,b & radius ,,ab describe ! two circles4 ,m>k (f ,,de & ,,ef equal to ,,ad4 ,) ! c5tres ,a & ,f & radius ,,df describe arcs int]sect+ at ,g4 ,) ! c5tres ,a & ,b & di/ance ,,bg describe arcs ,,ghk & ,n4 ,make ,,hk equal to ,,ab & ,,hl equal to ,,hb4 ,!n ) c5tres ,k & ,l & radius ,,ab describe arcs int]sect+ at ,i4 ,make ,,bm equal to ,,bi4 ,f9ally1 ) ! c5tre ,m & radius ,,mb cut ! l9e 9 ,c1 & ! po9t ,c is ! requir$ middle ( ! l9e ,,ab4 ,= great] exactitude y c m>k (f ,r f ,a 7z y did ,m f ,b71 & f ,r describe ano!r arc at ,c4 ,?is also solves ! problem1 to f9d a po9t midway betwe5 two giv5 po9ts )\t ! /raight l9e4 ,illu/ration ,i w put ! "y geomet] 9 ! way ( a rigid pro(4 ,fir/ prove t twice ! squ>e ( ! l9e ,,ab equals ! squ>e ( ! di/ance ,,bg1 f :i* x foll[s t ,,habn are ! f\r corn]s ( a squ>e4 ,to prove t ,i is ! c5tre ( ?is squ>e1 draw a l9e f ,h to ,p ?r\gh ,,qib & cont9ue ! arc ,,hk to ,p4 ,!n1 conceiv+ ! necess>y l9es to be drawn1 ! angle ,,hkp1 be+ 9 a semicircle1 is a "r angle4 ,let fall ! p]p5dicul> ,,kq1 & by simil> triangles1 & f ! fact t ,,hki is an isosceles triangle by ! con/ruction1 x c be prov$ t ,,hi is half ( ,,hb4 ,we c simil>ly prove t ,c is ! c5tre ( ! squ>e ( :i* ,,aib are ?ree corn]s4 ,i am aw>e t ?is is n ! simple/ possible solution4 #aih4--,,! ,,eight ,,/icks4 ,! fir/ diagram is ! answ] t ne>ly e "o w give to ?is puzzle1 & at fir/ sight x seems q satisfactory4 ,b consid] ! conditions4 ,we h to lay e "o ( ! /icks on ! table4 ,n[1 if a ladd] be plac$ aga9/ a wall ) only "o end on ! gr\nd1 x c h>dly be said t x is laid on ! gr\nd4 ,& if we place ! /icks 9 ! above mann]1 x is only possible to make "o end ( two ( !m t\* ! table3 to say t e "o lies on ! table w\ld n be correct4 ,to obta9 a solution x is only necess>y to h our /icks ( prop] dim5sions4 ,say ! long /icks are ea* #b ft4 9 l5g? & ! %ort ones #a ft4 ,!n ! /icks mu/ be #c 94 ?ick1 :5 ! ?ree equal squ>es may be enclos$1 z %[n 9 ! second diagram4 ,if ,i _h said mat*es in/ead ( /icks1 ! puzzle w\ld be impossible1 because an ord9>y mat* is ab\t tw5ty-"o times z long z x is broad1 & ! enclos$ rectangles w\ld n be squ>es4 ,illu/ration #aii4--,,papa',s ,,puzzle4 ,i h f\nd t a l>ge numb] ( p imag9e t ! foll[+ is a correct solution ( ! problem4 ,us+ ! lett]s 9 ! diagram bel[1 !y argue t if y make ! di/ance ,,ba "o-?ird ( ,,bc1 & !re=e ! area ( ! rectangle ,,abe equal to t ( ! triangul> rema9d]1 ! c>d mu/ hang ) ! long side horizontal4 ,read]s w rememb] ! je/ ( ,*>les ,,ii41 :o induc$ ! ,royal ,society to meet & discuss ! reason :y ! wat] 9 a vessel w n rise if y put a live fi% 9 x2 b 9 ! middle ( ! proce$+s "o ( ! lea/ di/+ui%$ among !m quietly slipp$ out & made ! exp]im5t1 :5 he f\nd t ! wat] did rise6 ,if my correspond5ts _h simil>ly made ! exp]im5t ) a piece ( c>dbo>d1 !y w\ld h f\nd at once _! error4 ,area is "o ?+1 b gravitation is q ano!r4 ,! fact ( t triangle /ick+ its leg out to ,d has to be comp5sat$ = by additional area 9 ! rectangle4 ,z a matt] ( fact1 ! ratio ( ,,ba to ,,ac is z #a is to ! squ>e root ( #c1 :i* latt] _c be giv5 9 an exact num]ical measure1 b is approximately #a4gcb4 ,n[ let u look at ! correct g5]al solution4 ,!re are _m ways ( arriv+ at ! desir$ result1 b ! "o ,i give is1 ,i ?9k1 ! simple/ = beg9n]s4 ,illu/ration ,fix y\r c>d on a piece ( pap] & draw ! equilat]al triangle ,,bcf1 ,,bf & ,,cf be+ equal to ,,bc4 ,also m>k (f ! po9t ,g s t ,,dg %all equal ,,dc4 ,draw ! l9e ,,cg & produce x until x cuts ! l9e ,,bf 9 ,h4 ,if we n[ make ,,ha p>allel to ,,be1 !n ,a is ! po9t f :i* our cut mu/ be made to ! corn] ,d1 z indicat$ by ! dott$ l9e4 ,a curi\s po9t 9 connection ) ?is problem is ! fact t ! position ( ! po9t ,a is indep5d5t ( ! side ,,cd4 ,! reason = ?is is m obvi\s 9 ! solution ,i h giv5 ?an 9 any o!r me?od t ,i h se51 & 7al?\gh ! problem may be solv$ ) all ! work+ on ! c>dbo>d7 t is p>tly :y ,i h pref]r$ x4 ,x w be se5 at once t h[ev] mu* y may r$uce ! wid? ( ! c>d by br++ ,e ne>] to ,b & ,d ne>] to ,c1 ! l9e ,,cg1 be+ ! diagonal ( a squ>e1 w always lie 9 ! same direction1 & w cut ,,bf 9 ,h4 ,f9ally1 if y wi% to get an approximate measure = ! di/ance ,,ba1 all y h to d is to multiply ! l5g? ( ! c>d by ! decimal 4#cff4 ,?us1 if ! c>d 7 #g in*es long1 we get #g 4#cff #b4efb1 or a little m ?an #b in*es1 = ! di/ance f ,b to ,a4 ,b ! real joke ( ! puzzle is ?is3 ,we h se5 t ! position ( ! po9t ,a is indep5d5t ( ! wid? ( ! c>d1 & dep5ds entirely on ! l5g?4 ,n[1 9 ! illu/ration x w be f\nd t bo? c>ds h ! same l5g?2 consequ5tly all ! little maid _h to d 0 to lay ! clipp$ c>d on top ( ! o!r "o & m>k (f ! po9t ,a at precisely ! same di/ance f ! top left-h& corn]6 ,s1 aft] all1 ,pappus' puzzle1 z he pres5t$ x to 8 little maid1 0 q an infantile problem1 :5 he 0 able to %[ h] h[ to p]=m ! feat )\t fir/ introduc+ h] to ! elem5ts ( /atics & geometry4 #bjj4--;,a ,,kite-,,fly+ ,,puzzle4 ,solv]s ( ?is little puzzle1 ,i h g5]ally f\nd1 may be r\ghly divid$ into two classes3 ^? :o get )9 a mile ( ! correct answ] by means ( m or less complex calculations1 involv+ pi1 & ^? ^: ari?metical kites fly hundr$s & ?\s&s ( miles away f ! tru?4 ,! comp>atively easy me?od t ,i %all %[ does n involve any consid]ation ( ! ratio t ! diamet] ( a circle be>s to its circumf]5ce4 ,i call x ! hat-box me?od4 ,illu/ration ,suppos+ we place our ball ( wire1 ,a1 9 a cyl9drical hat-box1 ,b1 t exactly fits x1 s t x t\*es ! side all r\nd & exactly t\*es ! top & bottom1 z %[n 9 ! illu/ration4 ,!n1 by an inv>iable law t %\ld be kn[n by ev]ybody1 t box conta9s exactly half z mu* aga9 z ! ball4 ,!re=e1 z ! ball is #bd 94 9 diamet]1 a hat-box ( ! same circumf]5ce b two-?irds ( ! height 7t is1 #af 94 high7 w h exactly ! same cont5ts z ! ball4 ,n[ let u consid] t ?is r$uc$ hat-box is a cyl9d] ( metal made up ( an imm5se numb] ( little wire cyl9d]s close toge!r l ! hairs 9 a pa9t]'s bru%4 ,by ! conditions ( ! puzzle we are all[$ to consid] t !re are no spaces betwe5 ! wires4 ,h[ _m ( ^! cyl9d]s "o "o-hundr$? ( an in* ?ick are equal to ! l>ge cyl9d]1 :i* is #bd 94 ?ick8 ,circles are to "o ano!r z ! squ>es ( _! diamet]s4 ,! squ>e ( #a_/ajj is #a_/ajjjjj1 & ! squ>e ( #bd is #egf2 !re=e ! l>ge cyl9d] conta9s #e1gfj1jjj ( ! little wire cyl9d]s4 ,b we h se5 t ea* ( ^! wires is #af 94 long2 h5ce #af #e1gfj1jjj #ib1afj1jjj in*es z ! complete l5g? ( ! wire4 ,r$uce ?is to miles1 & we get #a1ded miles #b1hhj ft4 z ! l5g? ( ! wire atta*$ to ! pr(essor's kite4 ,:e!r a kite w\ld fly at su* a height1 or support su* a weight1 are que/ions t d n ent] into ! problem4 #bja4--;,,h[ ,,to ,,make ,,ci/]ns4 ,"h is a g5]al =mula = solv+ ?is problem4 ,call ! two sides ( ! rectangle a & b4 ,!n a b - 7a b - ab7 --------------------------- #f equals ! side ( ! little squ>e pieces to cut away4 ,! measurem5ts giv5 7 #h ft4 by #c ft41 & ! above rule gives #h 94 z ! side ( ! squ>e pieces t h to be cut away4 ,( c\rse x w n always come out exact1 z 9 ?is case 7on acc\nt ( t squ>e root71 b y c get z ne> z y l ) decimals4 #bjb4--,,! ,,cone ,,puzzle4 ,! simple rule is t ! cone mu/ be cut at "o-?ird ( its altitude4 #bjc4--;,,conc]n+ ,,:eels4 ,if y m>k a po9t ,a on ! circumf]5ce ( a :eel t runs on ! surface ( a level road1 l an ord9>y c>t-:eel1 ! curve describ$ by t po9t w be a common cycloid1 z 9 ,fig4 #a4 ,b if y m>k a po9t ,b on ! circumf]5ce ( ! flange ( a locomotive-:eel1 ! curve w be a curtate cycloid1 z 9 ,fig4 #b1 t]m9at+ 9 nodes4 ,n[1 if we consid] "o ( ^! nodes or loops1 we %all see t at any giv5 mom5t c]ta9 po9ts at ! bottom ( ! loop mu/ be mov+ 9 ! opposite direction to ! tra94 ,z !re is an inf9ite numb] ( su* po9ts on ! flange's circumf]5ce1 !re mu/ be an inf9ite numb] ( ^! loops be+ describ$ :ile ! tra9 is 9 motion4 ,9 fact1 at any giv5 mom5t c]ta9 po9ts on ! flanges are always mov+ 9 a direction opposite to t 9 :i* ! tra9 is go+4 ,illu/ration3 #a ,illu/ration3 #b ,9 ! case ( ! two :eels1 ! :eel t runs r\nd ! /ation>y "o makes two revolutions r\nd its own c5tre4 ,z bo? :eels are ( ! same size1 x is obvi\s t if at ! />t we m>k a po9t on ! circumf]5ce ( ! upp] :eel1 at ! v top1 ?is po9t w be 9 contact ) ! l[] :eel at its l[e/ "p :5 half ! j\rney has be5 made4 ,!re=e ?is po9t is aga9 at ! top ( ! mov+ :eel1 & "o revolution has be5 made4 ,consequ5tly !re are two su* revolutions 9 ! complete j\rney4 #bjd4--;,a ,,new ,,mat* ,,puzzle4 #a4 ,! easie/ way is to arrange ! eighte5 mat*es z 9 ,diagrams #a & #b1 mak+ ! l5g? ( ! p]p5dicul> ,,ab equal to a mat* & a half4 ,!n1 if ! mat*es are an in* 9 l5g?1 ,fig4 #a conta9s two squ>e in*es & ,fig4 #b conta9s six squ>e in*es--#d #a4 ,! second case 7#b7 is a little m difficult to solve4 ,! solution is giv5 9 ,figs4 #c & #d4 ,= ! purpose ( con/ruction1 place mat*es tempor>ily on ! dott$ l9es4 ,!n x w be se5 t z #c conta9s five equal equilat]al triangles & #d conta9s fifte5 simil> triangles1 "o figure is ?ree times z l>ge z ! o!r1 & exactly eighte5 mat*es are us$4 ,illu/ration3 ,figures #a1 #b1 #c1 #d4 #bje4--,,! ,,six ,,%eep-,,p5s4 ,illu/ration ,place ! twelve mat*es 9 ! mann] %[n 9 ! illu/ration1 & y w h six p5s ( equal size4 #bjf4--,,! ,,k+ ,,& ,,! ,,ca/les4 ,!re are v>i\s ways ( build+ ! t5 ca/les s t !y %all =m five r[s ) f\r ca/les 9 e r[1 b ! arrangem5t 9 ! next column is ! only "o t also provides t two ca/les 7! greate/ numb] possible7 %all n be approa*able f ! outside4 ,x w be se5 t y mu/ cross ! walls to rea* ^! two4 ,illu/ration3 ,! ,k+ & ! ,ca/les #bjg4--;,,*]ries ,,& ,,plums4 ,!re are sev]al ways 9 :i* ?is problem might be solv$ 7 x n = ! condition t z few *]ries & plums z possible %all be plant$ on ! nor? & ea/ sides ( ! or*>d4 ,! be/ possible arrangem5t is t %[n 9 ! diagram1 :]e ! *]ries1 plums1 & apples are indicat$ respectively by ! lett]s ,c1 ,p1 & ,a4 ,! dott$ l9es connect ! *]ries1 & ! o!r l9es ! plums4 ,x w be se5 t ! t5 *]ry trees & ! t5 plum trees are s plant$ t ea* fruit =ms five l9es ) f\r trees ( its k9d 9 l9e4 ,?is is ! only arrangem5t t all[s ( s few z two *]ries or plums be+ plant$ on ! nor? & ea/ outside r[s4 ,illu/ration #bjh4--;,a ,,plantation ,,puzzle4 ,! illu/ration %[s ! t5 trees t mu/ be left to =m five r[s ) f\r trees 9 e r[4 ,! dots repres5t ! positions ( ! trees t h be5 cut d[n4 ,illu/ration #bji4--,,! ,,tw5ty-,,"o ,,trees4 ,i give two pleas+ arrangem5ts ( ! trees4 ,9 ea* case !re are twelve /raight r[s ) five trees 9 e r[4 ,illu/ration3 ,figure #a1 ,figure #b4 #baj4--,,! ,,t5 ,,co9s4 ,! answ] is t !re are j #b1djj diff]5t ways4 ,any ?ree co9s may be tak5 f "o side to comb9e ) "o co9 tak5 f ! o!r side4 ,i give f\r examples on ?is & ! next page4 ,we may ?us select ?ree f ! top 9 t5 ways & "o f ! bottom 9 five ways1 mak+ fifty4 ,b we may also select ?ree f ! bottom & "o f ! top 9 fifty ways4 ,we may ?us select ! f\r co9s 9 "o hundr$ ways1 & ! f\r remov$ may be arrang$ by p]mutation 9 tw5ty-f\r ways4 ,?us !re are #bd #ajj #b1djj diff]5t solutions4 ,illu/ration ,z all ! po9ts & l9es puzzles t ,i h giv5 s f>1 except+ ! la/1 are v>iations ( ! case ( t5 po9ts arrang$ to =m five l9es ( f\r1 x w be well to consid] ?is p>ticul> case g5]ally4 ,!re are six fundam5tal solutions1 & no m1 z %[n 9 ! six diagrams4 ,^!1 = ! sake ( conv5i5ce1 ,i nam$ "s ye>s ago ! ,/>1 ! ,d>t1 ! ,compasses1 ! ,funnel1 ! ,scissors1 & ! ,nail4 7,see next page47 ,read]s w und]/& t any "o ( ^! =ms may be di/ort$ 9 an inf9ite numb] ( diff]5t ways )\t de/roy+ its real "*4 ,9 ,! ,k+ & ! ,ca/les we h ! ,/>1 & its solution gives ! ,compasses4 ,9 ! ,*]ries & ,plums solution we f9d t ! ,*]ries repres5t ! ,funnel & ! ,plums ! ,d>t4 ,! solution ( ! ,plantation ,puzzle is an example ( ! ,d>t di/ort$4 ,any solution to ! ,t5 ,co9s w repres5t ! ,scissors4 ,?us examples ( all h be5 giv5 except ! ,nail4 ,on a r$uc$ *essbo>d1 #g by #g1 we may place ! t5 pawns 9 j ?ree diff]5t ways1 b !y mu/ all repres5t ! ,d>t4 ,! ,plantation %[s "o way1 ! ,plums %[ a second way1 & ! read] may l to f9d ! ?ird way = himself4 ,on an ord9>y *essbo>d1 #h by #h1 we c also get 9 a beautiful example ( ! ,funnel--symmetrical 9 relation to ! diagonal ( ! bo>d4 ,! smalle/ bo>d t w take a ,/> is "o #i by #g4 ,! ,nail requires a bo>d #aa by #g1 ! ,scissors ,illu/ration #aa by #i1 & ! ,compasses #ag by #ab4 ,at lea/ ^! are ! be/ results record$ 9 my note-book4 ,!y may be beat51 b ,i d n ?9k s4 ,if y divide a *essbo>d into two p>ts by a diagonal zigzag l9e1 s t ! l>g] "p conta9s #cf squ>es & ! small] "p #bh squ>es1 y c place ?ree sep>ate s*emes on ! l>g] "p & "o on ! small] "p 7all ,d>ts7 )\t _! conflict+--t is1 !y occupy =ty diff]5t squ>es4 ,!y c be plac$ 9 o!r ways )\t a division ( ! bo>d4 ,! smalle/ squ>e bo>d t w conta9 six diff]5t s*emes 7n fundam5tally diff]5t71 )\t any l9e ( "o s*eme cross+ ! l9e ( ano!r1 is #ad by #ad2 & ! smalle/ bo>d t w conta9 "o s*eme entirely enclos$ )9 ! l9es ( a second s*eme1 )\t any ( ! l9es ( ! "o1 :5 drawn f po9t to po9t1 cross+ a l9e ( ! o!r1 is #ad by #ab4 ,illu/ration3 ,,/> ,,d>t ,,compasses ,,funnel ,,scissors ,,nail #baa4--,,! ,,twelve ,,m9ce-,,pies4 ,if y ignore ! f\r black pies 9 our illu/ration1 ! rema9+ twelve are 9 _! orig9al positions4 ,n[ remove ! f\r deta*$ pies to ! places occupi$ by ! black ones1 & y w h y\r sev5 /raight r[s ( f\r1 z %[n by ! dott$ l9es4 ,illu/ration3 ,! ,twelve ,m9ce ,pies4 #bab4--,,! ,,burmese ,,plantation4 ,! arrangem5t on ! next page is ! mo/ symmetrical answ] t c probably be f\nd = tw5ty-"o r[s1 :i* is1 ,i believe1 ! greate/ numb] ( r[s possible4 ,!re are sev]al ways ( do+ x4 #bac4--,,turks ,,& ,,russians4 ,! ma9 po9t is to discov] ! smalle/ possible numb] ( ,russians t !re c\ld h be54 ,z ! enemy op5$ fire f all directions1 x is cle>ly necess>y to f9d :at is ! smalle/ numb] ( heads t c\ld =m sixte5 l9es ) ?ree heads 9 e l9e4 ,note t ,i say sixte51 & n ?irty-two1 because e l9e tak5 by a bullet may be also tak5 by ano!r bullet fir$ 9 exactly ! opposite direction4 ,n[1 z few z elev5 po9ts1 or heads1 may be arrang$ to =m ! requir$ sixte5 l9es ( ?ree1 b ! discov]y ( ?is arrangem5t is a h>d nut4 ,! diagram at ! foot ( ?is page w %[ exactly h[ ! ?+ is to be done4 ,illu/ration ,if1 !re=e1 elev5 ,russians 7 9 ! positions %[n by ! />s1 & ! ?irty-two ,turks 9 ! positions indicat$ by ! black dots1 x w be se51 by ! l9es %[n1 t ea* ,turk may fire exactly ov] ! heads ( ?ree ,russians4 ,b z ea* bullet kills a man1 x is ess5tial t e ,turk %all %oot "o ( 8 comrades & be %ot by him 9 turn2 o!rwise we %\ld h to provide extra ,russians to be %ot1 :i* w\ld be de/ructive ( ! correct solution ( our problem4 ,z ! fir+ 0 simultane\s1 ?is po9t pres5ts no difficulties4 ,! answ] we ?us see is t !re 7 at lea/ elev5 ,russians among/ :om !re 0 no casualty1 & t all ! ?irty-two ,turks 7 %ot by "o ano!r4 ,x 0 n /at$ :e!r ! ,russians fir$ any %ots1 b x w be evid5t t ev5 if !y did _! fir+ c\ld n h be5 effective3 = if "o ( _! bullets kill$ a ,turk1 !n we h imm$iately to provide ano!r man = "o ( ! ,turki% bullets to kill2 & z ! ,turks 7 kn[n to be ?irty-two 9 numb]1 ?is w\ld necessitate our introduc+ ano!r ,russian soldi] &1 ( c\rse1 de/roy+ ! solution4 ,i repeat t ! difficulty ( ! puzzle consi/s 9 f9d+ h[ to arrange elev5 po9ts s t !y %all =m sixte5 l9es ( ?ree4 ,i am told t ! possibility ( do+ ?is 0 fir/ discov]$ by ! ,rev4 ,mr4 ,wilk9son "s tw5ty ye>s ago4 #bad4--,,! ,,six ,,frogs4 ,move ! frogs 9 ! foll[+ ord]3 #b1 #d1 #f1 #e1 #c1 #a 7repeat ^! moves 9 ! same ord] twice m71 #b1 #d1 #f4 ,?is is a solution 9 tw5ty-"o moves--! fewe/ possible4 ,if n1 ! numb] ( frogs1 be ev51 we require 7n n7_/#b moves1 ( :i* 7n - n7_/#b w be leaps & n simple moves4 ,if n be odd1 we %all ne$ 77n #cn7_/#b7 - #d moves1 ( :i* 7n - n7_/#b w be leaps & #bn - #d simple moves4 ,9 ! ev5 cases write1 = ! moves1 all ! ev5 numb]s 9 asc5d+ ord] & ! odd numb]s 9 desc5d+ ord]4 ,?is s]ies mu/ be repeat$ n times & foll[$ by ! ev5 numb]s 9 asc5d+ ord] once only4 ,?us ! solution = #ad frogs w be 7#b1 #d1 #f1 #h1 #aj1 #ab1 #ad1 #ac1 #aa1 #i1 #g1 #e1 #c1 #a7 repeat$ #g times & foll[$ by #b1 #d1 #f1 #h1 #aj1 #ab1 #ad #aje moves4 ,9 ! odd cases1 write ! ev5 numb]s 9 asc5d+ ord] & ! odd numb]s 9 desc5d+ ord]1 repeat ?is s]ies 7n - #a7 times1 foll[ ) ! ev5 numb]s 9 asc5d+ ord] 7omitt+ n - #a71 ! odd numb]s 9 desc5d+ ord] 7omitt+ #a71 & conclude ) all ! numb]s 7odd & ev57 9 _! natural ord] 7omitt+ #a & n74 ,?us = #aa frogs3 7#b1 #d1 #f1 #h1 #aj1 #aa1 #i1 #g1 #e1 #c1 #a7 repeat$ #e times1 #b1 #d1 #f1 #h1 #aa1 #i1 #g1 #e1 #c1 & #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #aj #gc moves4 ,?is complete g5]al solution is publi%$ "h = ! fir/ "t4 #bae4--,,! ,,gras%opp] ,,puzzle4 ,move ! c\nt]s 9 ! foll[+ ord]4 ,! moves 9 brackets are to be made f\r times 9 succession4 #ab1 #a1 #c1 #b1 #ab1 #aa1 #a1 #c1 #b 7#e1 #g1 #i1 #aj1 #h1 #f1 #d71 #c1 #b1 #ab1 #aa1 #b1 #a1 #b4 ,! gras%opp]s w !n be rev]s$ 9 =ty-f\r moves4 ,! g5]al solution ( ?is problem is v difficult4 ,( c\rse x c always be solv$ by ! me?od giv5 9 ! solution ( ! la/ puzzle1 if we h no desire to use ! fewe/ possible moves4 ,b to employ a full economy ( moves we h two ma9 po9ts to consid]4 ,!re are always :at ,i call a l[] movem5t 7,l7 & an upp] movem5t 7,u74 ,l consi/s 9 ex*ang+ c]ta9 ( ! highe/ numb]s1 su* z #ab1 #aa1 #aj 9 our ,gras%opp] ,puzzle1 ) c]ta9 ( ! l[] numb]s1 #a1 #b1 #c2 ! =m] mov+ 9 a clockwise direction1 ! latt] 9 a non-clockwise direction4 ,u consi/s 9 rev]s+ ! int]m$iate c\nt]s4 ,9 ! above solution = #ab1 x w be se5 t #ab1 #aa1 & #a1 #b1 #c are engag$ 9 ! ,l movem5t1 & #d1 #e1 #f1 #g1 #h1 #i1 #aj 9 ! ,u movem5t4 ,! ,l movem5t ne$s #af moves & ,u #bh1 mak+ toge!r #dd4 ,we might also involve #aj 9 ! ,l movem5t1 :i* w\ld result 9 ,l #bc1 ,u #ba1 mak+ also toge!r #dd moves4 ,^! ,i call ! fir/ & second me?ods4 ,b any o!r s*eme w entail an increase ( moves4 ,y always get ^! two me?ods 7( equal economy7 = odd or ev5 c\nt]s1 b ! po9t is to det]m9e j h[ _m to involve 9 ,l & h[ _m 9 ,u4 ,"h is ! solution 9 table =m4 ,b fir/ note1 9 giv+ values to n1 t #b1 #c1 & #d c\nt]s are special cases1 requir+ respectively #c1 #c1 & #f moves1 & t #e & #f c\nt]s d n give a m9imum solution by ! second me?od--only by ! fir/4 ,,fir/ ,,me?od4 ---------------------------------------- ------------------------------- ,total ,no4 ,l ,,movem5t4 ,u ,,movem5t4 ( ---------------------------------------- -------- ,total ,no4 ,c\nt]s4 ,no4 ( ,no4 ( ,no4 ( ,no4 ( ( ,moves4 ,c\nt]s4 ,moves4 ,c\nt]s4 ,moves4 ---------------------------------------- ----------------------------- #dn n-#a & n #b7n-#a7en-#g #bn#a #bn#cn#a #d7nn-#a7 #dn-#b n-#a n #b7n-#a7en-#g #bn-#a #b7n-#a7cn-#bdn-#e #dn#a n n#a #bn#en-#b #bn #bn#cn-#d #b7bn#dn-#c7 #dn-#a n-#a n #b7n-#a7en-#g #bn #bn#cn-#d #dn#dn-#i ---------------------------------------- ----------------------------- ,,second ,,me?od4 ---------------------------------------- ------------------------------- ,total ,no4 ,l ,,movem5t4 ,u ,,movem5t4 ( ---------------------------------------- --------- ,total ,no4 ,c\nt]s4 ,no4 ( ,no4 ( ,no4 ( ,no4 ( ( ,moves4 ,c\nt]s4 ,moves4 ,c\nt]s4 ,moves4 ---------------------------------------- ----------------------------- #dn n & n #bn#cn-#d #bn #b7n-#a7en-#b #d7nn-#a7 #dn-#b n-#a n-#a #b7n-#a7cn-#g #bn #b7n-#a7en-#b #dn-#e #dn#a n n #bn#cn-#d #bn#a #bn#en-#b #b7bn#dn-#c7 #dn-#a n n #bn#cn-#d #bn-#a #b7n-#a7en-#g #dn#dn-#i ---------------------------------------- ----------------------------- ,m g5]ally we may say t ) m c\nt]s1 :]e m is ev5 & great] ?an #d1 we require 7m #dm - #af7_/d moves2 & :]e m is odd & great] ?an #c1 7m #fm - #ca7_/d moves4 ,i h ?us %[n ! read] h[ to f9d ! m9imum numb] ( moves = any case1 & ! "* & direction ( ! moves4 ,i w leave him to discov] = himself h[ ! actual ord] ( moves is to be det]m9$4 ,?is is a h>d nut1 & requires c>eful adju/m5t ( ! ,l & ! ,u movem5ts1 s t !y may be mutually accommodat+4 #baf4--,,! ,,educat$ ,,frogs4 ,! foll[+ leaps solve ! puzzle 9 t5 moves3 #b to #a1 #e to #b1 #c to #e1 #f to #c1 #g to #f1 #d to #g1 #a to #d1 #c to #a1 #f to #c1 #g to #f4 #bag4--,,! ,,twick5ham ,,puzzle4 ,play ! c\nt]s 9 ! foll[+ ord]3 ,k ,c ,e ,k ,w ,t ,c ,e ,h ,m ,k ,w ,t ,a ,n ,c ,e ,h ,m ,i ,k ,c ,e ,h ,m ,t1 & !re y are1 at ,twick5ham4 ,! position itself w always det]m9e :e!r y are to make a leap or a simple move4 #bah4--,,! ,,victoria ,,cross ,,puzzle4 ,9 solv+ ?is puzzle !re 7 two ?+s to be a*iev$3 fir/1 s to manipulate ! c\nt]s t ! ^w ,,victoria %\ld read r\nd ! cross 9 ! same direction1 only ) ! ,v on "o ( ! d>k arms2 & secondly1 to p]=m ! feat 9 ! fewe/ possible moves4 ,n[1 z a matt] ( fact1 x w\ld be impossible to p]=m ! fir/ "p 9 any way :atev] if all ! lett]s ( ! ^w 7 diff]5t2 b z !re are two ,i's1 x c be done by mak+ ^! lett]s *ange places--t is1 ! fir/ ,i *anges f ! #bnd place to ! #g?1 & ! second ,i f ! #g? place to ! #bnd4 ,b ! po9t ,i ref]r$ to1 :5 introduc+ ! puzzle1 z a little rem>kable is ?is3 t a solution 9 tw5ty-two moves is obta9able by mov+ ! lett]s 9 ! ord] ( ! foll[+ words3 ,a ,,victor6 ,a ,,victor6 ,a ,,victor ,i6 ,!re are1 h[ev]1 j six solutions 9 eighte5 moves1 & ! foll[+ is "o ( !m3 ,i 7#a71 ,v1 ,a1 ,i 7#b71 ,r1 ,o1 ,t1 ,i 7#a71 ,i 7#b71 ,a1 ,v1 ,i 7#b71 ,i 7#a71 ,c1 ,i 7#b71 ,v1 ,a1 ,i 7#a74 ,! fir/ & second ,i 9 ! ^w are di/+ui%$ by ! numb]s #a & #b4 ,x w be notic$ t 9 ! fir/ solution giv5 above "o ( ! ,i's nev] moves1 ?\gh ! movem5ts ( ! o!r lett]s cause x to *ange its relative position4 ,!re is ano!r peculi>ity ,i may po9t out--t !re is a solution 9 tw5ty-eight moves requir+ no lett] to move to ! c5tral division except ! ,i's4 ,i may also m5tion t1 9 ea* ( ! solutions 9 eighte5 moves1 ! lett]s ,c1 ,t1 ,o1 ,r move once only1 :ile ! second ,i always moves f\r times1 ! ,v always be+ transf]r$ to ! "r arm ( ! cross4 #bai4--,,! ,,lett] ,,block ,,puzzle4 ,?is puzzle c be solv$ 9 #bc moves--! fewe/ possible4 ,move ! blocks 9 ! foll[+ ord]3 ,a1 ,b1 ,f1 ,e1 ,c1 ,a1 ,b1 ,f1 ,e1 ,c1 ,a1 ,b1 ,d1 ,h1 ,g1 ,a1 ,b1 ,d1 ,h1 ,g1 ,d1 ,e1 ,f4 #bbj4--;,a ,,lodg+-,,h\se ,,difficulty4 ,! %orte/ possible way is to move ! articles 9 ! foll[+ ord]3 ,piano1 bookcase1 w>drobe1 piano1 cab9et1 *e/ ( draw]s1 piano1 w>drobe1 bookcase1 cab9et1 w>drobe1 piano1 *e/ ( draw]s1 w>drobe1 cab9et1 bookcase1 piano4 ,?us sev5te5 removals are necess>y4 ,! l&lady c\ld !n move *e/ ( draw]s1 w>drobe1 & cab9et4 ,mr4 ,dobson did n m9d ! w>drobe & *e/ ( draw]s *ang+ rooms s long z he secur$ ! piano4 #bba4--,,! ,,eight ,,eng9es4 ,! solution to ! ,eight ,eng9es ,puzzle is z foll[s3 ,! eng9e t has _h its fire drawn & !re=e _c move is ,no4 #e4 ,move ! o!r eng9es 9 ! foll[+ ord]3 #g1 #f1 #c1 #g1 #f1 #a1 #b1 #d1 #a1 #c1 #h1 #a1 #c1 #b1 #d1 #c1 #b1 sev5te5 moves 9 all1 leav+ ! eight eng9es 9 ! requir$ ord]4 ,!re are two o!r slightly diff]5t solutions4 #bbb4--;,a ,,railway ,,puzzle4 ,?is little puzzle may be solv$ 9 z few z n9e moves4 ,play ! eng9es z foll[s3 ,f #i to #aj1 f #f to #i1 f #e to #f1 f #b to #e1 f #a to #b1 f #g to #a1 f #h to #g1 f #i to #h1 & f #aj to #i4 ,y w !n h eng9es ,a1 ,b1 & ,c on ea* ( ! ?ree circles & on ea* ( ! ?ree /raight l9es4 ,?is is ! %orte/ solution t is possible4 #bbc4--;,a ,,railway ,,muddle4 ,illu/ration3 #a ,illu/ration3 #b ,illu/ration3 #c ,illu/ration3 #d ,illu/ration3 #e ,illu/ration3 #f ,only six rev]sals are necess>y4 ,! :ite tra9 7f ,a to ,d7 is divid$ into ?ree sections1 eng9e & #g wagons1 #h wagons1 & #a wagon4 ,! black tra9 7,d to ,a7 nev] unc\ples any?+ ?r\gh\t4 ,fig4 #a is orig9al position ) #h & #a unc\pl$4 ,! black tra9 proce$s to position 9 ,fig4 #b 7no rev]sal74 ,! eng9e & #g proce$ t[>ds ,d1 & black tra9 backs1 leaves #h on loop1 & takes up position 9 ,fig4 #c 7fir/ rev]sal74 ,black tra9 goes to position 9 ,fig4 #d to fet* s+le wagon 7second rev]sal74 ,black tra9 pu%es #h (f loop & leaves s+le wagon !re1 proce$+ on its j\rney1 z 9 ,fig4 #e 7?ird & f\r? rev]sals74 ,:ite tra9 n[ backs on to loop to pick up s+le c> & goes "r away to ,d 7fif? & six? rev]sals74 #bbd4--,,! ,,motor-,,g>age ,,puzzle4 ,! ex*ange ( c>s c be made 9 =ty-?ree moves1 z foll[s3 #f-;,g1 #b-;,b1 #a-;,e1 #c-;,h1 #d-;,i1 #c-,l1 #f-,k1 #d-;,g1 #a-;,i1 #b-;,j1 #e-;,h1 #d-;,a1 #g-;,f1 #h-;,e1 #d-;,d1 #h-;,c1 #g-;,a1 #h-;,g1 #e-;,c1 #b-;,b1 #a-;,e1 #h-;,i1 #a-;,g1 #b-;,j1 #g-;,h1 #a-;,a1 #g-;,g1 #b-;,b1 #f-;,e1 #c-;,h1 #h-,l1 #c-;,i1 #g-,k1 #c-;,g1 #f-;,i1 #b-;,j1 #e-;,h1 #c-;,c1 #e-;,g1 #b-;,b1 #f-;,e1 #e-;,i1 #f-;,j4 ,( c\rse1 #f-;,g means t ! c> numb]$ #f moves to ! po9t ,g4 ,!re are o!r ways 9 =ty-?ree moves4 #bbe4--,,! ,,t5 ,,prison]s4 ,illu/ration ,x w be se5 9 ! illu/ration h[ ! prison]s may be arrang$ s z to produce z _m z sixte5 ev5 r[s4 ,!re are #d su* v]tical r[s1 #d horizontal r[s1 #e diagonal r[s 9 "o direction1 & #c diagonal r[s 9 ! o!r direction4 ,! arr[s "h %[ ! movem5ts ( ! f\r prison]s1 & x w be se5 t ! infirm man 9 ! bottom corn] has n be5 mov$4 #bbf4--,,r\nd ,,! ,,coa/4 ,9 ord] to place words r\nd ! circle "u ! conditions1 x is necess>y to select words 9 :i* lett]s are repeat$ 9 c]ta9 relative positions4 ,?us1 ! ^w t solves our puzzle is ,swansea1 9 :i* ! fir/ & fif? lett]s are ! same1 & ! ?ird & sev5? ! same4 ,we make out jumps z foll[s1 tak+ ! lett]s ( ! ^w 9 _! prop] ord]3 #b-e1 #g-b1 #d-g1 #a-d1 #f-a1 #c-f1 #h-c4 ,or we c\ld place a ^w l ,t>apur 79 :i* ! second & f\r? lett]s1 & ! ?ird & sev5?1 are alike7 ) ^! moves3 #f-a1 #g-d1 #b-g1 #e--b1 #h-e1 #c-f1 #h-c4 ,b ,swansea is ! only ^w1 app>5tly1 t w fulfil ! conditions ( ! puzzle4 ,?is puzzle %\ld be comp>$ ) ,%>p's ,puzzle1 ref]r$ to 9 my solution to ,no4 #cda1 ,! ,f\r ,frogs4 ,! condition t\* & jump ov] two is id5tical ) t\* & move along a l9e4 #bbg4--;,,c5tral ,,solitaire4 ,"h is a solution 9 n9ete5 moves2 ! moves enclos$ 9 brackets c\nt z "o move only3 #ai-ag1 #af-ah1 7#bi-ag1 #ag-ai71 #cj-ah1 #bg-be1 7#bb-bd1 #bd-bf71 #ca-bc1 7#d-af1 #af-bh71 #g-i1 #aj-h1 #ab-aj1 #c-aa1 #ah-f1 7#a-c1 #c-aa71 7#ac-bg1 #bg-be71 7#ba-g1 #g-i71 7#cc-ca1 #ca-bc71 7#aj-h1 #h-bb1 #bb-bd1 #bd-bf1 #bf-ab1 #ab-aj71 #e-ag4 ,all ! c\nt]s are n[ remov$ except "o1 :i* is left 9 ! c5tral hole4 ,! solution ne$s judgm5t1 z "o is tempt$ to make sev]al jumps 9 "o move1 :]e x w\ld be ! rev]se ( good play4 ,= example1 aft] play+ ! fir/ #c-aa above1 "o is incl9$ to increase ! l5g? ( ! move by cont9u+ ) #aa-be1 #be-bg1 or ) #aa-i1 #i-g4 ,i d n ?9k ! numb] ( moves c be r$uc$4 #bbh4--,,! ,,t5 ,,apples4 ,numb] ! plates 7#a1 #b1 #c1 #d71 7#e1 #f1 #g1 #h71 7#i1 #aj1 #aa1 #ab71 7#ac1 #ad1 #ae1 #af7 9 successive r[s f ! top to ! bottom4 ,!n transf] ! apple f #h to #aj & play z foll[s1 always remov+ ! apple jump$ ov]3 #i-aa1 #a-i1 #ac-e1 #af-h1 #d-ab1 #ab-aj1 #c-a1 #a-i1 #i-aa4 #bbi4--,,! ,,n9e ,,almonds4 ,?is puzzle may be solv$ 9 z few z f\r moves1 9 ! foll[+ mann]3 ,move #e ov] #h1 #i1 #c1 #a4 ,move #g ov] #d4 ,move #f ov] #b & #g4 ,move #e ov] #f1 & all ! c\nt]s are remov$ except #e1 :i* is left 9 ! c5tral squ>e t x orig9ally occupi$4 #bcj4--,,! ,,twelve ,,p5nies4 ,"h is "o ( sev]al solutions4 ,move #ab to #c1 #g to #d1 #aj to #f1 #h to #a1 #i to #e1 #aa to #b4 #bca4--,,plates ,,& ,,co9s4 ,numb] ! plates f #a to #ab 9 ! ord] t ! boy is se5 to be go+ 9 ! illu/ration4 ,/>t+ f #a1 proce$ z foll[s1 :]e #a to #d means t y take ! co9 f plate ,no4 #a & transf] x to plate ,no4 #d3 #a to #d1 #e to #h1 #i to #ab1 #c to #f1 #g to #aj1 #aa to #b1 & complete ! la/ revolution to #a1 mak+ ?ree revolutions 9 all4 ,or y c proce$ ?is way3 #d to #g1 #h to #aa1 #ab to #c1 #b to #e1 #f to #i1 #aj to #a4 ,x is easy to solve 9 f\r revolutions1 b ! solutions 9 ?ree are m difficult to discov]4 ,?is is ,! ,riddle ( ! ,fi%pond 7,no4 #da1 ,cant]bury ,puzzles7 9 a diff]5t dress4 #bcb4--;,,cat*+ ,,! ,,mice4 ,9 ord] t ! cat %\ld eat e ?irte5? m\se1 & ! :ite m\se la/ ( all1 x is necess>y t ! c\nt %\ld beg9 at ! sev5? m\se 7call+ ! :ite "o ! fir/7--t is1 at ! "o ne>e/ ! tip ( ! cat's tail4 ,9 ?is case x is n at all necess>y to try />t+ at all ! mice 9 turn until y come to ! "r "o1 = y c j />t any:]e & note h[ f> di/ant ! la/ "o eat5 is f ! />t+ po9t4 ,y w f9d x to be ! eigh?1 & !re=e mu/ />t at ! eigh?1 c\nt+ backw>ds f ! :ite m\se4 ,?is is ! "o ,i h indicat$4 ,9 ! case ( ! second puzzle1 :]e y h to f9d ! smalle/ numb] ) :i* ! cat may />t at ! :ite m\se & eat ?is "o la/ ( all1 unless y h ma/]$ ! g5]al solution ( ! problem1 :i* is v difficult1 !re is no bett] c\rse op5 to y ?an to try e numb] 9 succession until y come to "o t works correctly4 ,! smalle/ numb] is tw5ty-"o4 ,if y h to proce$ by trial1 y w %ort5 y\r lab\r a great deal by only c\nt+ out ! rema9d]s :5 ! numb] is divid$ successively by #ac1 #ab1 #aa1 #aj1 etc4 ,?us1 9 ! case ( #ba1 we h ! rema9d]s #h1 #i1 #aj1 #a1 #c1 #e1 #g1 #c1 #a1 #a1 #c1 #a1 #a4 ,note t ,i d n give ! rema9d]s ( #g1 #c1 & #a z n\ght1 b z #g1 #c1 & #a4 ,n[1 c\nt r\nd ea* ( ^! numb]s 9 turn1 & y w f9d t ! :ite m\se is kill$ la/ ( all4 ,( c\rse1 if we want$ simply any numb]1 n ! smalle/1 ! solution is v easy1 = we m]ely take ! lea/ common multiple ( #ac1 #ab1 #aa1 #aj1 etc4 d[n to #b4 ,?is is #cfjcfj1 & y w f9d t ! fir/ c\nt kills ! ?irte5? m\se1 ! next ! twelf?1 ! next ! elev5?1 & s on d[n to ! fir/4 ,b ! mo/ ari?metically incl9$ cat c\ld n be expect$ to take su* a big numb] :5 a small "o l tw5ty-"o w\ld equally s]ve its purpose4 ,9 ! ?ird case1 ! smalle/ numb] is #ajj4 ,! numb] #a1jjj w\ld also d1 & !re are j sev5ty-two o!r numb]s betwe5 ^! t ! cat might employ ) equal success4 #bcc4--,,! ,,ecc5tric ,,*eesemong]4 ,to leave ! ?ree piles at ! extreme ends ( ! r[s1 ! *eeses may be mov$ z foll[s--! numb]s ref] to ! *eeses & n to _! positions 9 ! r[3 #g-b1 #h-g1 #i-h1 #aj-ae1 #f-aj1 #e-f1 #ad-af1 #ac-ad1 #ab-ac1 #c-a1 #d-c1 #aa-d4 ,?is is probably ! easie/ solution ( all to f9d4 ,to get ?ree ( ! piles on *eeses #ac1 #ad1 & #ae1 play ?us3 #i-d1 #aj-i1 #aa-aj1 #f-ad1 #e-f1 #ab-ae1 #h-ab1 #g-h1 #af-e1 #c-ac1 #b-c1 #a-b4 ,to leave ! piles on *eeses #c1 #e1 #ab1 & #ad1 play ?us3 #h-c1 #i-ad1 #af-ab1 #a-e1 #aj-i1 #g-aj1 #aa-h1 #b-a1 #d-af1 #ac-b1 #f-aa1 #ae-d4 #bcd4--,,! ,,ex*ange ,,puzzle4 ,make ! foll[+ ex*anges ( pairs3 ,h-,k1 ,h-,e1 ,h-,c1 ,h-,a1 ,i-,l1 ,i-,f1 ,i-,d1 ,k-,l1 ,g-,j1 ,j-,a1 ,f-,k1 ,l-,e1 ,d-,k1 ,e-,f1 ,e-,d1 ,e-,b1 ,b-,k4 ,x w be f\nd t1 al?\gh ! :ite c\nt]s c be mov$ to _! prop] places 9 #aa moves1 if we omit all consid]ation ( ex*anges1 yet ! black _c be s mov$ 9 few] ?an #ag moves4 ,s we h to introduce wa/e moves ) ! :ite c\nt]s to equal ! m9imum requir$ by ! black4 ,?us few] ?an #ag moves mu/ be impossible4 ,"s ( ! moves are1 ( c\rse1 int]*angeable4 #bce4--,,torp$o ,,practice4 ,illu/ration3 #aj #f #g _/ #d u u #b u _/ #c-u u u u u u u u u u -----#i--- _/ u #h u u _/ #a #e ,if ! enemy's fleet be an*or$ 9 ! =mation %[n 9 ! illu/ration1 x w be se5 t z _m z t5 out ( ! sixte5 %ips may be bl[n up by dis*>g+ ! torp$oes 9 ! ord] indicat$ by ! numb]s & 9 ! directions indicat$ by ! arr[s4 ,z ea* torp$o 9 succession passes "u ?ree %ips & s9ks ! f\r?1 /rike out ea* vessel ) ! p5cil z x is sunk4 #bcf4--,,! ,,hat ,,puzzle4 ,illu/ration3 #a #b #c #d #e #f #g #h #i #aj #aa #ab ------------------------ 99 o 99 ,o 99 ,o 99 ,o 99 ,o ------------------------ 99 ,o 99 ,o 99 ,o 99 ,o ,o 99 ------------------------ 99 99 ,o ,o 99 ,o 99 ,o ,o 99 ------------------------ 99 99 ,o ,o ,o 99 99 ,o ,o 99 ------------------------ 99 99 ,o ,o ,o ,o ,o 99 99 99 ------------------------ ,o ,o ,o ,o ,o 99 99 99 99 99 ------------------------ ,i sugge/$ t ! read] %\ld try ?is puzzle ) c\nt]s1 s ,i give my solution 9 t =m4 ,! silk hats are repres5t$ by black c\nt]s & ! felt hats by :ite c\nt]s4 ,! fir/ r[ %[s ! hats 9 _! orig9al positions1 & !n ea* successive r[ %[s h[ !y appe> aft] "o ( ! five manipulations4 ,x w ?us be se5 t we fir/ move hats #b & #c1 !n #g & #h1 !n #d & #e1 !n #aj & #aa1 &1 f9ally1 #a & #b1 leav+ ! f\r silk hats toge!r1 ! f\r felt hats toge!r1 & ! two vacant pegs at "o end ( ! r[4 ,! fir/ ?ree pairs mov$ are dissimil> hats1 ! la/ two pairs be+ simil>4 ,!re are o!r ways ( solv+ ! puzzle4 #bcg4--;,,boys ,,& ,,girls4 ,!re are a good _m diff]5t solutions to ?is puzzle4 ,any contigu\s pair1 except #g-h1 may be mov$ fir/1 & aft] ! fir/ move !re are v>iations4 ,! foll[+ solution %[s ! position f ! />t "r ?r\gh ea* successive move to ! end3-- 4 4 #a #b #c #d #e #f #g #h #d #c #a #b 4 4 #e #f #g #h #d #c #a #b #g #f #e 4 4 #h #d #c #a #b #g 4 4 #e #f #h #d 4 4 #b #g #a #c #e #f #h #d #h #f #b #g #a #c #e 4 4 #bch4--;,,arrang+ ,,! ,,jam ,,pots4 ,two ( ! pots1 #ac & #ai1 7 9 _! prop] places4 ,z e int]*ange may result 9 a pot be+ put 9 its place1 x is cle> t tw5ty-two int]*anges w get !m all 9 ord]4 ,b ?is numb] ( moves is n ! fewe/ possible1 ! correct answ] be+ sev5te54 ,ex*ange ! foll[+ pairs3 7#c-a1 #b-c71 7#ae-d1 #af-ae71 7#ag-g1 #bj-ag71 7#bd-aj1 #aa-bd1 #ab-aa71 7#h-e1 #f-h1 #ba-f1 #bc-ba1 #bb-bc1 #ad-bb1 #i-ad1 #ah-i74 ,:5 y h made ! int]*anges )9 any pair ( brackets1 all numb]s )9 ^? brackets are 9 _! places4 ,!re are five pairs ( brackets1 & #e f #bb gives ! numb] ( *anges requir$--#ag4 #bci4--;,a ,,juv5ile ,,puzzle4 ,illu/ration3 ----------------- ,c ,e ,d ,f ---------------,b ,g ,a ,h ----------------- ,z ! conditions are g5]ally und]/ood1 ?is puzzle is incapable ( solution4 ,?is c be demon/rat$ q easily4 ,s we h to look = "s cat* or quibble 9 ! /atem5t ( :at we are ask$ to d4 ,n[ if y fold ! pap] & !n pu% ! po9t ( y\r p5cil d[n betwe5 ! fold1 y c ) "o /roke make ! two l9es ,,cd & ,,ef 9 our diagram4 ,!n />t at ,a1 & describe ! l9e end+ at ,b4 ,f9ally put 9 ! la/ l9e ,,gh1 & ! ?+ is done /rictly )9 ! conditions1 s9ce fold+ ! pap] is n actually =bidd54 ,( c\rse ! l9es are "h left unjo9$ = ! purpose ( cle>ness4 ,9 ! rubb+ out =m ( ! puzzle1 fir/ rub out ,a to ,b ) a s+le f+] 9 "o /roke4 ,!n rub out ! l9e ,,gh ) "o f+]4 ,f9ally1 rub out ! rema9+ two v]tical l9es ) two f+]s at once6 ,t is ! old trick4 #bdj4--,,! ,,union ,,jack4 ,illu/ration3 ------- ----- ,a ,b _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ -------_/--_/----- _/ _/ _/ _/ _/ _/ _/ -----_/--_/------- _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ ----- ----- ,!re are j sixte5 po9ts 7all on ! outside7 :]e ?ree roads may be said to jo94 ,^! are call$ by ma!maticians odd nodes4 ,!re is a rule t tells u t 9 ! case ( a draw+ l ! pres5t "o1 :]e !re are sixte5 odd nodes1 x requires eight sep>ate /rokes or r\tes 7t is1 half z _m z !re are odd nodes7 to complete x4 ,z we h to produce z mu* z possible ) only "o ( ^! eight /rokes1 x is cle>ly necess>y to contrive t ! sev5 /rokes f odd node to odd node %all be z %ort z possible4 ,/>t at ,a & end at ,b1 or g ! rev]se way4 #bda4--,,! ,,dissect$ ,,circle4 ,illu/ration3 _/--------------- _/ _/ _/------,b _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ ,a _/ _/ _/ _/ _/ ----------99---------- _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ ,d-------99----------99----,e _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ ----------99---------- _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ _/ ------------_/ _/ _/ ,c-------_/ ,x c be done 9 twelve cont9u\s /rokes1 ?us3 ,/>t at ,a 9 ! illu/ration1 & eight /rokes1 =m+ ! />1 w br+ y back to ,a2 !n "o /roke r\nd ! circle to ,b1 "o /roke to ,c1 "o r\nd ! circle to ,d1 & "o f9al /roke to ,e--twelve 9 all4 ,( c\rse1 9 practice ! second circul> /roke w be ov] ! fir/ "o2 x is sep>at$ 9 ! diagram1 & ! po9ts ( ! /> n jo9$ to ! circle1 to make ! solution cle> to ! eye4 #bdb4--,,! ,,tube ,,inspector',s ,,puzzle4 ,! inspector ne$ only travel n9ete5 miles if he />ts at ,b & takes ! foll[+ r\te3 ,,badgdefifcbehklihgjk4 ,?us ! only portions ( l9e travell$ ov] twice are ! two sections ,d to ,g & ,f to ,i4 ,( c\rse1 ! r\te may be v>i$1 b x _c be %ort5$4 #bdc4--,,visit+ ,,! ,,t[ns4 ,note t !re are six t[ns1 f :i* only two roads issue4 ,?us #a mu/ lie betwe5 #i & #ab 9 ! circul> r\te4 ,m>k ^! two roads z settl$4 ,simil>ly m>k #i1 #e1 #ad1 & #d1 #h1 #ad1 & #aj1 #f1 #ae1 & #aj1 #b1 #ac1 & #c1 #g1 #ac4 ,all ^! roads mu/ be tak54 ,!n y w f9d t he mu/ g f #d to #ae1 z #ac is clos$1 & t he is compell$ to take #c1 #aa1 #af1 & also #af1 #ab4 ,?us1 !re is only "o r\te1 z foll[s3 #a1 #i1 #e1 #ad1 #h1 #d1 #ae1 #f1 #aj1 #b1 #ac1 #g1 #c1 #aa1 #af1 #ab1 #a1 or its rev]se--read+ ! l9e ! o!r way4 ,sev5 roads are n us$4 #bdd4--,,! ,,fifte5 ,,turn+s4 ,illu/ration ,x w be se5 f ! illu/ration 7:]e ! roads n us$ are omitt$7 t ! travell] c g z f> z sev5ty miles 9 fifte5 turn+s4 ,! turn+s are all numb]$ 9 ! ord] 9 :i* !y are tak54 ,x w be se5 t he nev] visits n9ete5 ( ! t[ns4 ,he might visit !m all 9 fifte5 turn+s1 nev] ent]+ any t[n twice1 & end at ! black t[n f :i* he />ts 7see ,! ,rook's ,t\r1 ,no4 #cbj71 b su* a t\r w\ld only take him sixty-f\r miles4 #bde4--,,! ,,fly ,,on ,,! ,,octah$ron4 ,illu/ration ,?\gh we _c really see all ! sides ( ! octah$ron at once1 we c make a projection ( x t suits our purpose j z well4 ,9 ! diagram ! six po9ts repres5t ! six angles ( ! octah$ron1 & f\r l9es proce$ f e po9t "u exactly ! same conditions z ! twelve edges ( ! solid4 ,!re=e if we />t at ! po9t ,a & g ov] all ! l9es once1 we mu/ always end our r\te at ,a4 ,& ! numb] ( diff]5t r\tes is j #a1dhh1 c\nt+ ! rev]se way ( any r\te z diff]5t4 ,x w\ld take too mu* space to %[ h[ ,i make ! c\nt4 ,x c be done 9 ab\t five m9utes1 b an explanation ( ! me?od is difficult4 ,! read] is !re=e ask$ to accept my answ] z correct4 #bdf4--,,! ,,icosah$ron ,,puzzle4 ,illu/ration ,!re are ?irty edges1 ( :i* eighte5 7 visible 9 ! orig9al illu/ration1 repres5t$ 9 ! foll[+ diagram by ! hexagon ,,naesgd4 ,by ?is projection ( ! solid we get an imag9>y view ( ! rema9+ twelve edges1 & are able to see at once _! direction & ! twelve po9ts at :i* all ! edges meet4 ,! diff]5ce 9 ! l5g? ( ! l9es is ( no importance2 all we want is to pres5t _! direction 9 a graphic mann]4 ,b 9 case ! novice %\ld be puzzl$ at only f9d+ n9ete5 triangles in/ead ( ! requir$ tw5ty1 ,i w po9t out t ! app>5tly miss+ triangle is ! outl9e ,,hik4 ,9 ?is case !re are twelve odd nodes2 !re=e six di/9ct & disconnect$ r\tes w be ne$ful if we are n to g ov] any l9es twice4 ,let u !re=e f9d ! greate/ di/ance t we may s travel 9 "o r\te4 ,x w be notic$ t ,i h /ruck out ) little cross /rokes five l9es or edges 9 ! diagram4 ,^! five l9es may be /ruck out any:]e s long z !y d n jo9 "o ano!r1 & s long z "o ( !m does n connect ) ,n1 ! ,nor? ,pole1 f :i* we are to />t4 ,x w be se5 t ! result ( /rik+ out ^! five l9es is t all ! nodes are n[ ev5 except ,n & ,s4 ,consequ5tly if we beg9 at ,n & /op at ,s we may g ov] all ! l9es1 except ! five cross$ out1 )\t trav]s+ any l9e twice4 ,!re are _m ways ( do+ ?is4 ,"h is "o r\te3 ,n to ,h1 ,i1 ,k1 ,s1 ,i1 ,e1 ,s1 ,g1 ,k1 ,d1 ,h1 ,a1 ,n1 ,b1 ,a1 ,e1 ,f1 ,b1 ,c1 ,g1 ,d1 ,n1 ,c1 ,f1 ,s4 ,by ?us mak+ five ( ! r\tes z %ort z is possible--simply f "o node to ! next--we are able to get ! greate/ possible l5g? = our six? l9e4 ,a great] di/ance 9 "o r\te1 )\t go+ ov] ! same gr\nd twice1 x is n possible to get4 ,x is n[ readily se5 t ^? five eras$ l9es mu/ be gone ov] twice1 & !y may be pick$ up1 s to speak1 at any po9ts ( our r\te4 ,?us1 :5ev] ! travell] happ5s to be at ,i he c run up to ,a & back be=e proce$+ on 8 r\te1 or he may wait until he is at ,a & !n run d[n to ,i & back to ,a4 ,& s ) ! o!r l9es t h to be trac$ twice4 ,x is1 !re=e1 cle> t he c g ov] #be ( ! l9es once only 7#be #aj1jjj miles #bej1jjj miles7 & #e ( ! l9es twice 7#e #bj1jjj miles #ajj1jjj miles71 ! total1 #cej1jjj miles1 be+ ! l5g? ( 8 travels & ! %orte/ di/ance t is possible 9 visit+ ! :ole body4 ,x w be notic$ t ,i h made him end 8 travels at ,s1 ! ,s\? ,pole1 b ?is is n imp]ative4 ,i might h made him f9i% at any ( ! o!r nodes1 except ! "o f :i* he />t$4 ,suppose x _h be5 requir$ to br+ him home aga9 to ,n at ! end ( 8 travels4 ,!n in/ead ( suppress+ ! l9e ,,ai we might leave t op5 & close ,,is4 ,?is w\ld enable him to complete 8 #cej1jjj miles t\r at ,a1 & ano!r #aj1jjj miles w\ld take him to 8 own fireside4 ,!re are a great _m diff]5t r\tes1 b z ! l5g?s ( ! edges are all alike1 "o c\rse is z good z ano!r4 ,to make ! complete #cej1jjj miles t\r f ,n to ,s absolutely cle> to ev]ybody1 ,i w give x entire3 ,n to ,h1 ,i1 ,a1 ,i1 ,k1 ,h1 ,k1 ,s1 ,i1 ,e1 ,s1 ,g1 ,f1 ,g1 ,k1 ,d1 ,c1 ,d1 ,h1 ,a1 ,n1 ,b1 ,e1 ,b1 ,a1 ,e1 ,f1 ,b1 ,c1 ,g1 ,d1 ,n1 ,c1 ,f1 ,s--t is1 ?irty-five l9es ( #aj1jjj miles ea*4 #bdg4--;,,inspect+ ,a ,,m9e4 ,/>t+ f ,a1 ! inspector ne$ only travel #cf furlongs if he takes ! foll[+ r\te3 ,a to ,b1 ,g1 ,h1 ,c1 ,d1 ,i1 ,h1 ,m1 ,n1 ,i1 ,j1 ,o1 ,n1 ,s1 ,r1 ,m1 ,l1 ,g1 ,f1 ,k1 ,l1 ,q1 ,r1 ,s1 ,t1 ,o1 ,j1 ,e1 ,d1 ,c1 ,b1 ,a1 ,f1 ,k1 ,p1 ,q4 ,he ?us passes betwe5 ,a & ,b twice1 betwe5 ,c & ,d twice1 betwe5 ,f & ,k twice1 betwe5 ,j & ,o twice1 & betwe5 ,r & ,s twice--five repetitions4 ,!re=e #ca passages plus #e repeat$ equal #cf furlongs4 ,! little pitfall 9 ?is puzzle lies 9 ! fact t we />t f an ev5 node4 ,o!rwise we ne$ only travel #ce furlongs4 #bdh4--,,! ,,cycli/',s ,,t\r4 ,:5 ,mr4 ,maggs repli$1 ,no way1 ,i'm sure1 he 0 n say+ t ! ?+ 0 impossible1 b 0 really giv+ ! actual r\te by :i* ! problem c be solv$4 ,/>t+ f ! />1 if y visit ! t[ns 9 ! ord]1 ,,no ,,way1 ,i',m ,,sure1 y w visit e t[n once1 & only once1 & end at ,e4 ,s bo? m5 7 correct4 ,?is 0 ! little joke ( ! puzzle1 :i* is n by any means difficult4 #bdi4--,,! ,,sailor',s ,,puzzle4 ,illu/ration ,!re are only f\r diff]5t r\tes 7or eight1 if we c\nt ! rev]se ways7 by :i* ! sailor c />t at ! isl& m>k$ ,a1 visit all ! isl&s once1 & once only1 & return aga9 to ,a4 ,"h !y are3-- ,a ,i ,p ,t ,l ,o ,e ,h ,r ,q ,d ,c ,f ,u ,g ,n ,s ,k ,m ,b ,a ,a ,i ,p ,t ,s ,n ,g ,l ,o ,e ,u ,f ,c ,d ,k ,m ,b ,q ,r ,h ,a ,a ,b ,m ,k ,s ,n ,g ,l ,t ,p ,i ,o ,e ,u ,f ,c ,d ,q ,r ,h ,a ,a ,i ,p ,t ,l ,o ,e ,u ,g ,n ,s ,k ,m ,b ,q ,d ,c ,f ,r ,h ,a ,n[1 if ! sailor takes ! fir/ r\te he w make ,c 8 #ab? isl& 7c\nt+ ,a z #a72 by ! second r\te he w make ,c 8 #ac? isl&2 by ! ?ird r\te1 8 #af? isl&2 & by ! f\r? r\te1 8 #ag? isl&4 ,if he goes ! rev]se way1 ,c w be respectively 8 #aj?1 #i?1 #f?1 & #e? isl&4 ,z ^! are ! only possible r\tes1 x is evid5t t if ! sailor puts (f 8 visit to ,c z long z possible1 he mu/ take ! la/ r\te read+ f left to "r4 ,?is r\te ,i %[ by ! d>k l9es 9 ! diagram1 & x is ! correct answ] to ! puzzle4 ,! map may be greatly simplifi$ by ! buttons & /r+ me?od1 expla9$ 9 ! solution to ,no4 #cda1 ,! ,f\r ,frogs4 #bej4--,,! ,,gr& ,,t\r4 ,! fir/ ?+ to d 9 try+ to solve a puzzle l ?is is to attempt to simplify x4 ,if y look at ,fig4 #a1 y w see t x is a simplifi$ v]sion ( ! map4 ,imag9e ! circul> t[ns to be buttons & ! railways to be connect+ /r+s4 7,see solution to ,no4 #cda47 ,!n1 x w be se51 we h simply /raight5$ out ! previ\s diagram )\t affect+ ! conditions4 ,n[ we c fur!r simplify by conv]t+ ,fig4 #a into ,fig4 #b1 :i* is a portion ( a *essbo>d4 ,"h ! directions ( ! railways w resemble ! moves ( a rook 9 *ess--t is1 we may move 9 any direction p>allel to ! sides ( ! diagram1 b n diagonally4 ,!re=e ! fir/ t[n 7or squ>e7 visit$ mu/ be a black "o2 ! second mu/ be a :ite2 ! ?ird mu/ be a black2 & s on4 ,e odd squ>e visit$ w ?us be black & e ev5 "o :ite4 ,n[1 we h #bc squ>es to visit 7an odd numb]71 s ! la/ squ>e visit$ mu/ be black4 ,b ,z happ5s to be :ite1 s ! puzzle w\ld seem to be impossible ( solution4 ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,z we 7 told t ! man succe$$ 9 c>ry+ put 8 plan1 we mu/ try to f9d "s loophole 9 ! conditions4 ,he 0 to ent] e t[n once & only once1 & we f9d no prohibition aga9/ 8 ent]+ once ! t[n ,a aft] leav+ x1 especially z he has nev] left x s9ce he 0 born1 & w\ld ?us be ent]+ x = ! fir/ "t 9 8 life4 ,b he mu/ return at once f ! fir/ t[n he visits1 & !n he w h only #bb t[ns to visit1 & z #bb is an ev5 numb]1 !re is no reason :y he %\ld n end on ! :ite squ>e ,z4 ,a possible r\te = him is indicat$ by ! dott$ l9e f ,a to ,z4 ,?is r\te is repeat$ by ! d>k l9es 9 ,fig4 #a1 & ! read] w n[ h no difficulty 9 apply+2 x to ! orig9al map4 ,we h ?us prov$ t ! puzzle c only be solv$ by a return to ,a imm$iately aft] leav+ x4 #bea4--,,wat]1 ,,gas1 ,,& ,,electricity4 ,illu/ration ,accord+ to ! conditions1 9 ! /rict s5se 9 :i* "o at fir/ und]/&s !m1 !re is no possible solution to ?is puzzle4 ,9 su* a dilemma "o always has to look = "s v]bal quibble or trick4 ,if ! own] ( h\se ,a w all[ ! wat] company to run _! pipe = h\se ,c ?r\gh 8 prop]ty 7& we are n b\nd to assume t he w\ld object71 !n ! difficulty is got ov]1 z %[n 9 our illu/ration4 ,x w be se5 t ! dott$ l9e f ,w to ,c passes ?r\gh h\se ,a1 b no pipe ev] crosses ano!r pipe4 #beb4--;,a ,,puzzle ,,= ,,motori/s4 ,illu/ration ,! r\tes tak5 by ! eight driv]s are %[n 9 ! illu/ration1 :]e ! dott$ l9e roads are omitt$ to make ! pa?s cle>] to ! eye4 #bec4--;,a ,,bank ,,holiday ,,puzzle4 ,! simple/ way is to write 9 ! numb] ( r\tes to all ! t[ns 9 ?is mann]4 ,put a #a on all ! t[ns 9 ! top r[ & 9 ! fir/ column4 ,!n ! numb] ( r\tes to any t[n w be ! sum ( ! r\tes to ! t[n imm$iately above & to ! t[n imm$iately to ! left4 ,?us ! r\tes 9 ! second r[ w be #a1 #b1 #c1 #d1 #e1 #f1 etc41 9 ! ?ird r[1 #a1 #c1 #f1 #aj1 #ae1 #ba1 etc42 & s on ) ! o!r r[s4 ,x w !n be se5 t ! only t[n to :i* !re are exactly #a1cfe diff]5t r\tes is ! twelf? t[n 9 ! fif? r[--! "o imm$iately ov] ! lett] ,e4 ,?is t[n 0 !re=e ! cycli/'s de/9ation4 ,! g5]al =mula = ! numb] ( r\tes f "o corn] to ! corn] diagonally opposite on any su* rectangul> reticulat$ arrangem5t1 "u ! conditions z to direction1 is 7mn76_/m6n61 :]e m is ! numb] ( t[ns on "o side1 less "o1 & n ! numb] on ! o!r side1 less "o4 ,our solution involves ! case :]e !re are #ab t[ns by #e4 ,!re=e m #aa & n #d4 ,!n ! =mula gives u ! answ] #a1cfe z above4 #bed4-- ,,! ,,motor-,,c> ,,t\r4 ,fir/ ( all ,i w ask ! read] to comp>e ! orig9al squ>e diagram ) ! circul> "o %[n 9 ,figs4 #a1 #b1 & #c bel[4 ,if = ! mom5t we ignore ! %ad+ 7! purpose ( :i* ,i %all proce$ to expla971 we f9d t ! circul> diagram 9 ea* case is m]ely a simplification ( ! orig9al squ>e "o--t is1 ! roads f ,a lead to ,b1 ,e1 & ,m 9 bo? cases1 ! roads f ,l 7,london7 lead to ,i1 ,k1 & ,s1 & s on4 ,! =m bel[1 be+ circul> & symmetrical1 answ]s my purpose bett] 9 apply+ a me*anical solution1 & ,i !re=e adopt x )\t alt]+ 9 any way ! conditions ( ! puzzle4 ,if su* a "q z di/ances f t[n to t[n came into ! problem1 ! new diagrams might require ! addition ( numb]s to indicate ^! di/ances1 or !y might conceivably n be at all practicable4 ,illu/ration3 ,figs4 #a1 #b1 & #c ,n[1 ,i draw ! ?ree circul> diagrams1 z %[n1 on a %eet ( pap] & !n cut out ?ree pieces ( c>dbo>d ( ! =ms indicat$ by ! %ad$ p>ts ( ^! diagrams4 ,x c be %[n t e r\te1 if m>k$ out ) a r$ p5cil1 w =m "o or o!r ( ! designs indicat$ by ! edges ( ! c>ds1 or a reflection !re(4 ,let u direct our att5tion to ,fig4 #a4 ,"h ! c>d is s plac$ t ! /> is at ! t[n ,t2 x !re=e gives u 7by foll[+ ! edge ( ! c>d7 "o ( ! circul> r\tes f ,london3 ,l1 ,s1 ,r1 ,t1 ,m1 ,a1 ,e1 ,p1 ,o1 ,j1 ,d1 ,c1 ,b1 ,g1 ,n1 ,q1 ,k1 ,h1 ,f1 ,i1 ,l4 ,if we w5t ! o!r way1 we %\ld get ,l1 ,i1 ,f1 ,h1 ,k1 ,q1 etc41 b ^! rev]se r\tes 7 n to be c\nt$4 ,:5 we h writt5 out ?is fir/ r\te we revolve ! c>d until ! /> is at ,m1 :5 we get ano!r diff]5t r\te1 at ,a a ?ird r\te1 at ,e a f\r? r\te1 & at ,p a fif? r\te4 ,we h ?us obta9$ five diff]5t r\tes by revolv+ ! c>d z x lies4 ,b x is evid5t t if we n[ take up ! c>d & replace x ) ! o!r side upp]mo/1 we %all 9 ! same mann] get five o!r r\tes by revolution4 ,we !re=e see h[1 by us+ ! revolv+ c>d 9 ,fig4 #a1 we may1 )\t any difficulty1 at once write out t5 r\tes4 ,& if we employ ! c>ds 9 ,figs4 #b & #c1 we simil>ly obta9 9 ea* case t5 o!r r\tes4 ,^! ?irty r\tes are all t are possible4 ,i d n give ! actual pro( t ! ?ree c>ds exhau/ all ! possible cases1 b leave ! read] to reason t out = himself4 ,if he works out any r\te at haphaz>d1 he w c]ta9ly f9d t x falls into "o or o!r ( ! ?ree categories4 #bee4--,,! ,,level ,,puzzle4 ,let u conf9e our att5tion to ! ,l 9 ! top left-h& corn]4 ,suppose we g by way ( ! ,e on ! "r3 we mu/ !n g /raight on to ! ,v1 f :i* lett] ! ^w may be complet$ 9 f\r ways1 = !re are f\r ,e's available ?r\gh :i* we may rea* an ,l4 ,!re are !re=e f\r ways ( read+ ?r\gh ! "r-h& ,e4 ,x is also cle> t !re mu/ be ! same numb] ( ways ?r\gh ! ,e t is imm$iately bel[ our />t+ po9t4 ,t makes eight4 ,if1 h[ev]1 we take ! ?ird r\te ?r\gh ! ,e on ! diagonal1 we !n h ! option ( any "o ( ! ?ree ,v's1 by means ( ea* ( :i* we may complete ! ^w 9 f\r ways4 ,we c !re=e spell ,,level 9 twelve ways ?r\gh ! diagonal ,e4 ,twelve add$ to eight gives tw5ty read+s1 all emanat+ f ! ,l 9 ! top left-h& corn]2 & z ! f\r corn]s are equal1 ! answ] mu/ be f\r times tw5ty1 or eighty diff]5t ways4 #bef4--,,! ,,diamond ,,puzzle4 ,!re are #beb diff]5t ways4 ,! g5]al =mula is t1 = words ( n lett]s 7n pal9dromes1 z 9 ! case ( ! next puzzle71 :5 gr\p$ 9 ?is mann]1 !re are always #b7n#a7 - #d diff]5t read+s4 ,?is does n all[ diagonal read+s1 su* z y w\ld get if y us$ in/ead su* a ^w z ,,digg+1 :]e x w\ld be possible to pass f "o ,g to ano!r ,g by a diagonal /ep4 #beg4--,,! ,,deifi$ ,,puzzle4 ,! correct answ] is #a1iib diff]5t ways4 ,e ,f is ei!r a corn] ,f or a side ,f--/&+ next to a corn] 9 its own squ>e ( ,f's4 ,n[1 ,,fi$ may be read f a corn] ,f 9 #af ways2 !re=e ,,deif may be read into a corn] ,f also 9 #af ways2 h5ce ,,deifi$ may be read ?r\gh a corn] ,f 9 #af #af #bef ways4 ,consequ5tly1 ! f\r corn] ,f's give #d #bef #a1jbd ways4 ,!n ,,fi$ may be read f a side ,f 9 #aa ways1 & ,,deifi$ !re=e 9 #aba ways4 ,b !re are eight side ,f's2 consequ5tly ^! give toge!r #h #aba #ifh ways4 ,add #ifh to #a1jbd & we get ! answ]1 #a1iib4 ,9 ?is =m ! solution w dep5d on :e!r ! numb] ( lett]s 9 ! pal9drome be odd or ev54 ,= example1 if y apply ! ^w ,,nun 9 precisely ! same mann]1 y w get #fd diff]5t read+s2 b if y use ! ^w ,,noon1 y w only get #ef1 because y _c use ! same lett] twice 9 imm$iate succession 7s9ce y mu/ always pass f "o lett] to ano!r7 or diagonal read+s1 & e read+ mu/ involve ! use ( ! c5tral ,n4 ,! read] may l to f9d = himself ! g5]al =mula 9 ?is case1 :i* is complex & difficult4 ,i w m]ely add t = su* a case z ,,madam1 dealt ) 9 ! same way z ,,deifi$1 ! numb] ( read+s is #djj4 #beh4-- ,,! ,,vot]s' ,,puzzle4 ,,! numb] ( read+s "h is #fc1ejd1 z 9 ! case ( ,,0 ,,x ,a ,,rat ,i ,,saw 7,no4 #cj1 ,cant]bury ,puzzles74 ,! g5]al =mula is t = pal9dromic s5t5ces conta9+ #bn #a lett]s !re are 7#d7bn -#a77 read+s4 #bei4-- ,,hannah',s ,,puzzle4 ,/>t+ f any "o ( ! ,n's1 !re are #ag diff]5t read+s ( ,,nah1 or #fh 7#d times #ag7 = ! #d ,n's4 ,!re=e !re are also #fh ways ( spell+ ,,han4 ,if we 7 all[$ to use ! same ,n twice 9 a spell+1 ! answ] w\ld be #fh times #fh1 or #d1fbd ways4 ,b ! conditions 71 always pass+ f "o lett] to ano!r4 ,!re=e1 = e "o ( ! #ag ways ( spell+ ,,han ) a p>ticul> ,n1 !re w\ld be #ea ways 7#c times #ag7 ( complet+ ! ,,nah1 or #hfg 7#ag times #ea7 ways = ! complete ^w4 ,h5ce1 z !re are f\r ,n's to use 9 ,,han1 ! correct solution ( ! puzzle is #c1dfh 7#d times #hfg7 diff]5t ways4 #bfj4--,,! ,,honeycomb ,,puzzle4 ,! requir$ prov]b is1 ,!re is _m a slip 'twixt ! cup & ! lip4 ,/>t at ! ,t on ! outside at ! bottom "r-h& corn]1 pass to ! ,h above x1 & ! re/ is easy4 #bfa4-- ,,! ,,monk ,,& ,,! ,,bridges4 ,illu/ration ,! problem ( ! ,bridges may be r$uc$ to ! simple diagram %[n 9 illu/ration4 ,! po9t ,m repres5ts ! ,monk1 ! po9t ,i ! ,isl&1 & ! po9t ,y ! ,mona/]y4 ,n[ ! only direct ways f ,m to ,i are by ! bridges a & b2 ! only direct ways f ,i to ,y are by ! bridges c & d2 & !re is a direct way f ,m to ,y by ! bridge e4 ,n[1 :at we h to d is to c\nt all ! r\tes t w lead f ,m to ,y1 pass+ ov] all ! bridges1 a1 b1 c1 d1 & e once & once only4 ,) ! simple diagram "u ! eye x is q easy1 )\t any elaborate rule1 to c\nt ^! r\tes me?odically4 ,?us1 />t+ f a1 b1 we f9d !re are only two ways ( complet+ ! r\te2 ) a1 c1 !re are only two r\tes2 ) a1 d1 only two r\tes2 & s on4 ,x w be f\nd t !re are sixte5 su* r\tes 9 all1 z 9 ! foll[+ li/3-- a b e c d b c d a e a b e d c b c e a d a c d b e b d c a e a c e b d b d e a c a d e b c e c a b d a d c b e e c b a d b a e c d e d a b c b a e d c e d b a c ,if ! read] w transf] ! lett]s indicat+ ! bridges f ! diagram to ! correspond+ bridges 9 ! orig9al illu/ration1 ev]y?+ w be q obvi\s4 #bfb4--,,^? ,,fifte5 ,,%eep4 ,if we read ! exact words ( ! writ] 9 ! cyclopdia1 we f9d t we are n told t ! p5s 7 all necess>ily empty6 ,9 fact1 if ! read] w ref] back to ! illu/ration1 he w see t "o %eep is already 9 "o ( ! p5s4 ,x 0 j at ?is po9t t ! wily f>m] said to me1 ,n[ ,i'm go+ to />t plac+ ! fifte5 %eep4 ,he !reupon proce$$ to drive ?ree f 8 flock into ! already occupi$ p51 & !n plac$ f\r %eep 9 ea* ( ! o!r ?ree p5s4 ,!re1 says he1 y h se5 me place fifte5 %eep 9 f\r p5s s t !re %all be ! same numb] ( %eep 9 e p54 ,i 01 ( c\rse1 =c$ to admit t he 0 p]fectly correct1 accord+ to ! exact word+ ( ! "q4 #bfc4--,,k+ ,,ar?ur',s ,,knights4 ,on ! second ev5+ ,k+ ,ar?ur arrang$ ! knights & himself 9 ! foll[+ ord] r\nd ! table3 ,a1 ,f1 ,b1 ,d1 ,g1 ,e1 ,c4 ,on ! ?ird ev5+ !y sat ?us1 ,a1 ,e1 ,b1 ,g1 ,c1 ,f1 ,d4 ,he ?us _h ,b next b "o to him on bo? occasions 7! ne>e/ possible71 & ,g 0 ! ?ird f him at bo? sitt+s 7! fur!/ position possible74 ,no o!r way ( sitt+ ! knights w\ld h be5 s satisfactory4 #bfd4--,,! ,,city ,,lun*eons4 ,! m5 may be gr\p$ z foll[s1 :]e ea* l9e repres5ts a day & ea* column a table3-- ,,ab ,,cd ,,ef ,,gh ,,ij ,,kl ,,ae ,,dl ,,gk ,,fi ,,cb ,,hj ,,ag ,,lj ,,fh ,,kc ,,de ,,ib ,,af ,,jb ,,ki ,,hd ,,lg ,,ce ,,ak ,,be ,,hc ,,il ,,jf ,,dg ,,ah ,,eg ,,id ,,cj ,,bk ,,lf ,,ai ,,gf ,,cl ,,db ,,eh ,,jk ,,ac ,,fk ,,dj ,,le ,,gi ,,bh ,,ad ,,kh ,,lb ,,jg ,,fc ,,ei ,,al ,,hi ,,je ,,bf ,,kd ,,gc ,,aj ,,ic ,,bg ,,ek ,,hl ,,fd ,note t 9 e column 7except 9 ! case ( ! ,a's7 all ! lett]s desc5d cyclically 9 ! same ord]1 ,b1 ,e1 ,g1 ,f1 up to ,j1 :i* is foll[$ by ,b4 #bfe4--;,a ,,puzzle ,,= ,,c>d-,,play]s4 ,9 ! foll[+ solution ea* ( ! elev5 l9es repres5ts a sitt+1 ea* column a table1 & ea* pair ( lett]s a pair ( p>tn]s4 ,a ,b -- ,i ,l ,e ,j -- ,g ,k ,f ,h -- ,c ,d ,a ,c -- ,j ,b ,f ,k -- ,h ,l ,g ,i -- ,d ,e ,a ,d -- ,k ,c ,g ,l -- ,i ,b ,h ,j -- ,e ,f ,a ,e -- ,l ,d ,h ,b -- ,j ,c ,i ,k -- ,f ,g ,a ,f -- ,b ,e ,i ,c -- ,k ,d ,j ,l -- ,g ,h ,a ,g -- ,c ,f ,j ,d -- ,l ,e ,k ,b -- ,h ,i ,a ,h -- ,d ,g ,k ,e -- ,b ,f ,l ,c -- ,i ,j ,a ,i -- ,e ,h ,l ,f -- ,c ,g ,b ,d -- ,j ,k ,a ,j -- ,f ,i ,b ,g -- ,d ,h ,c ,e -- ,k ,l ,a ,k -- ,g ,j ,c ,h -- ,e ,i ,d ,f -- ,l ,b ,a ,l -- ,h ,k ,d ,i -- ,f ,j ,e ,g -- ,b ,c ,x w be se5 t ! lett]s ,b1 ,c1 ,d 444,l desc5d cyclically4 ,! solution giv5 above is absolutely p]fect 9 all respects4 ,x w be f\nd t e play] has e o!r play] once z 8 p>tn] & twice z 8 oppon5t4 #bff4--;,a ,,t5nis ,,t\rnam5t4 ,call ! m5 ,a1 ,b1 ,d1 ,e1 & _! wives a1 b1 d1 e4 ,!n !y may play z foll[s )\t any p]son ev] play+ twice ) or aga9/ any o!r p]son3-- ,fir/ ,c\rt4 ,second ,c\rt4 #a/ ,day ,a d aga9/ ,b e ,d a aga9/ ,e b #bnd ,day ,a e ,d b ,e a ,b d #crd ,day ,a b ,e d ,b a ,d e ,x w be se5 t no man ev] plays ) or aga9/ 8 own wife--an ideal arrangem5t4 ,if ! read] wants a h>d puzzle1 let him try to arrange eight m>ri$ c\ples 79 f\r c\rts on sev5 days7 "u exactly simil> conditions4 ,x c be done1 b ,i leave ! read] 9 ?is case ! pleasure ( seek+ ! answ] & ! g5]al solution4 #bfg4--,,! ,,wrong ,,hats4 ,! numb] ( diff]5t ways 9 :i* eight p]sons1 ) eight hats1 c ea* take ! wrong hat1 is #ad1hcc4 ,"h are ! successive solutions = any numb] ( p]sons f "o to eight3-- #a #j #b #a #c #b #d #i #e #dd #f #bfe #g #a1hed #h #ad1hcc ,to get ^! numb]s1 multiply successively by #b1 #c1 #d1 #e1 etc4 ,:5 ! multipli] is ev51 add #a2 :5 odd1 d$uct #a4 ,?us1 #c #a - #a #b2 #d #b #a #i2 #e #i - #a #dd2 & s on4 ,or y c multiply ! sum ( ! numb] ( ways = n - #a & n - #b p]sons by n - #a1 & s get ! solution = n p]sons4 ,?us1 #d7b #i7 #dd2 #e7i #dd7 #bfe2 & s on4 #bfh4--,,! ,,peal ,,( ,,bells4 ,! bells %\ld be rung z foll[s3-- #a #b #c #d #b #a #d #c #b #d #a #c #d #b #c #a #d #c #b #a #c #d #a #b #c #a #d #b #a #c #b #d #c #a #b #d #a #c #d #b #a #d #c #b #d #a #b #c #d #b #a #c #b #d #c #a #b #c #d #a #c #b #a #d #b #c #a #d #c #b #d #a #c #d #b #a #d #c #a #b #d #a #c #b #a #d #b #c #a #b #d #c #b #a #c #d ,i h con/ruct$ peals = five & six bells respectively1 & a solution is possible = any numb] ( bells "u ! conditions previ\sly /at$4 #bfi4--,,?ree ,,m5 ,,9 ,a ,,boat4 ,if !re 7 no conditions :atev]1 except t ! m5 7 all to g out toge!r1 9 ?rees1 !y c\ld r[ 9 an imm5se numb] ( diff]5t ways4 ,if ! read] wi%es to "k h[ _m1 ! numb] is #deeg4 ,& ) ! condition t no two may ev] be toge!r m ?an once1 !re are no few] ?an #ae1efg1eeb1jjj diff]5t solutions--t is1 diff]5t ways ( arrang+ ! m54 ,) "o solution be=e him1 ! read] w realize :y ?is mu/ be1 = al?\gh1 z an example1 ,a mu/ g out once ) ,b & once ) ,c1 x does n necess>ily foll[ t he mu/ g out ) ,c on ! same occasion t he goes ) ,b4 ,he might take any o!r lett] ) him on t occasion1 ?\gh ! fact ( 8 tak+ o!r ?an ,b w\ld h its effect on ! arrangem5t ( ! o!r triplets4 ,( c\rse only a c]ta9 numb] ( all ^! arrangem5ts are available :5 we h t o!r condition ( us+ ! smalle/ possible numb] ( boats4 ,z a matt] ( fact we ne$ employ only t5 diff]5t boats4 ,"h is "o ! arrangem5ts3-- #a #b #c #d #e #a/ ,day 7,,abc7 7,,dbf7 7,,ghi7 7,,jkl7 7,,mno7 #h #f #g #i #aj #bnd ,day 7,,adg7 7,,bkn7 7,,col7 7,,jei7 7,,mhf7 #c #e #d #a #b #crd ,day 7,,ajm7 7,,beh7 7,,cfi7 7,,dko7 7,,gnl7 #g #f #h #i #a #d? ,day 7,,aek7 7,,cgm7 7,,boi7 7,,dhl7 7,,jnf7 #d #e #c #aj #b #e? ,day 7,,ahn7 7,,cdj7 7,,bfl7 7,,geo7 7,,mki7 #f #g #h #aj #a #f? ,day 7,,afo7 7,,bgj7 7,,ckh7 7,,dni7 7,,mel7 #e #d #c #i #b #g? ,day 7,,ail7 7,,bdm7 7,,c57 7,,gkf7 7,,jho7 ,x w be f\nd t no two m5 ev] g out twice toge!r1 & t no man ev] goes out twice 9 ! same boat4 ,?is is an ext5sion ( ! well-kn[n problem ( ! ,fifte5 ,s*oolgirls1 by ,kirkman4 ,! orig9al conditions 7 simply t fifte5 girls walk$ out on sev5 days 9 triplets )\t any girl ev] walk+ twice 9 a triplet ) ano!r girl4 ,attempts at a g5]al solution ( ?is puzzle _h ex]cis$ ! +5uity ( ma!maticians s9ce #ahej1 :5 ! "q 0 fir/ prop\nd$1 until rec5tly4 ,9 #aijh & ! two foll[+ ye>s ,i indicat$ 7see ,educational ,times ,repr9ts1 ,vols4 ,,xiv41 ,,xv41 & ,,xvii47 t all our tr\ble _h aris5 f a failure to discov] t #ae is a special case 7too small to ent] into ! g5]al law = all high] numb]s ( girls ( ! =m #fn#c71 & %[$ :at t g5]al law is & h[ ! gr\ps %\ld be pos$ = any numb] ( girls4 ,i gave actual arrangem5ts = numb]s t _h previ\sly baffl$ all attempts to manipulate1 & ! problem may n[ be consid]$ g5]ally solv$4 ,read]s w f9d an excell5t full acc\nt ( ! puzzle 9 ,w4,w4 ,r\se ,ball's ,ma!matical ,recreations1 #e? edition4 #bgj4--,,! ,,glass ,,balls4 ,!re are1 9 all1 sixte5 balls to be brok51 or sixte5 places 9 ! ord] ( break+4 ,call ! f\r /r+s ,a1 ,b1 ,c1 & ,d--ord] is "h ( no importance4 ,! break+ ( ! balls on ,a may occupy any #d out ( ^! #af places--t is1 ! comb9ations ( #af ?+s1 tak5 #d toge!r1 w be #ac #ad #ae #af ----------------- #a1hbj #a #b #c #d ways = ,a4 ,9 e "o ( ^! cases ,b may occupy any #d out ( ! rema9+ #ab places1 mak+ #i #aj #aa #ab ----------------- #die #a #b #c #d ways4 ,?us #a1hbj #die #ijj1ijj diff]5t plac+s are op5 to ,a & ,b4 ,b = e "o ( ^! cases ,c may occupy #e #f #g #h ------------- #gj #a #b #c #d diff]5t places2 s t #ijj1ijj #gj #fc1jfc1jjj diff]5t plac+s are op5 to ,a1 ,b1 & ,c4 ,9 e "o ( ^! cases1 ,d has no *oice b to take ! f\r places t rema94 ,!re=e ! correct answ] is t ! balls may be brok5 9 #fc1jfc1jjj diff]5t ways "u ! conditions4 ,read]s %\ld comp>e ?is problem ) ,no4 #cde1 ,! ,two ,pawns1 :i* !y w !n "k h[ to solve = cases :]e !re are ?ree1 f\r1 or m pawns on ! bo>d4 #bga4--;,,fifte5 ,,lett] ,,puzzle4 ,! foll[+ w be f\nd to comply ) ! conditions ( gr\p+3-- ,,ale ,,met ,,mop ,,blm ,,bag ,,cap ,,y ,,clt ,,ire ,,oil ,,lug ,,lnr ,,nay ,,bit ,,bun ,,bpr ,,aim ,,bey ,,rum ,,gmy ,,o> ,,g9 ,,ply ,,cgr ,,peg ,,icy ,,try ,,cmn ,,cue ,,cob ,,tau ,,pnt ,,"o ,,got ,,piu ,! fifte5 lett]s us$ are ,a1 ,e1 ,i1 ,o1 ,u1 ,y1 & ,b1 ,c1 ,g1 ,l1 ,m1 ,n1 ,p1 ,r1 ,t4 ,! numb] ( words is #bg1 & ^! are all %[n 9 ! fir/ ?ree columns4 ,! la/ ^w1 ,,piu1 is a musical t]m 9 common use2 b al?\gh x has crept into "s ( our diction>ies1 x is ,italian1 mean+ a little2 slightly4 ,! rema9+ tw5ty-six are good words4 ,( c\rse a ,,tau-cross is a ,t-%ap$ cross1 also call$ ! cross ( ,/4 ,an?ony1 & borne on a badge 9 ! ,bi%op's ,palace at ,exet]4 ,x is also a "n = ! toad-fi%4 ,we ?us h tw5ty-six good words & "o d\btful1 obta9$ "u ! requir$ conditions1 & ,i d n ?9k x w be easy to improve on ?is answ]4 ,( c\rse we are n b\nd by diction>ies b by common usage4 ,if we w5t by ! diction>y only 9 a case ( ?is k9d1 we %\ld f9d ourselves involv$ 9 prefixes1 contractions1 & su* absurdities z ,i4,o4,u41 :i* ,nuttall actually gives z a ^w4 #bgb4--,,! ,,n9e ,,s*oolboys4 ,! boys c walk out z foll[s3-- #a/ ,day4 #bnd ,day4 #crd ,day4 ,a ,b ,c ,b ,f ,h ,f ,a ,g ,d ,e ,f ,e ,i ,a ,i ,d ,b ,g ,h ,i ,c ,g ,d ,h ,c ,e #d? ,day4 #e? ,day4 #f? ,day4 ,a ,d ,h ,g ,b ,i ,d ,c ,a ,b ,e ,g ,c ,f ,d ,e ,h ,b ,f ,i ,c ,h ,a ,e ,i ,g ,f ,e boy w !n h walk$ by ! side ( e o!r boy once & once only4 ,deal+ ) ! problem g5]ally1 #abn#i boys may walk out 9 triplets "u ! conditions on #in#f days1 :]e n may be n\ght or any integ]4 ,e possible pair w occur once4 ,call ! numb] ( boys m4 ,!n e boy w pair m-#a times1 ( :i* 7m-#a7_/d times he w be 9 ! middle ( a triplet & 7m-#a7_/b times on ! outside4 ,?us1 if we ref] to ! solution above1 we f9d t e boy is 9 ! middle twice 7mak+ #d pairs7 & f\r times on ! outside 7mak+ ! rema9+ #d pairs ( 8 #h74 ,! read] may n[ l to try 8 h& at solv+ ! two next cases ( #ba boys on #ae days1 & #cc boys on #bd days4 ,x is1 p]haps1 int]e/+ to note t a s*ool ( #dhi boys c\ld ?us walk out daily 9 "o leap ye>1 b x w\ld take #gca girls 7ref]r$ to 9 ! solution to ,no4 #bfi7 to p]=m _! p>ticul> feat by a daily walk 9 a ye> ( #cfe days4 #bgc4--,,! ,,r\nd ,,table4 ,! hi/ory ( ?is problem w be f\nd 9 ,! ,cant]bury ,puzzles 7,no4 #ij74 ,s9ce ! publication ( t book 9 #aijg1 s f> z ,i "k1 nobody has succe$$ 9 solv+ ! case = t unlucky numb] ( p]sons1 #ac1 seat$ at a table on #ff occasions4 ,a solution is possible = any numb] ( p]sons1 & ,i h record$ s*$ules = e numb] up to #be p]sons inclusive & = #cc4 ,b z ,i "k a good _m ma!maticians are /ill consid]+ ! case ( #ac1 ,i w n at ?is /age rob !m ( ! pleasure ( solv+ x by %[+ ! answ]4 ,b ,i w n[ display ! solutions = all ! cases up to #ab p]sons inclusive4 ,"s ( ^! solutions are n[ publi%$ = ! fir/ "t1 & !y may af=d useful clues to inve/igators4 ,! solution = ! case ( #c p]sons seat$ on #a occasion ne$s no rem>k4 ,a solution = ! case ( #d p]sons on #c occasions is z foll[s3-- #a #b #c #d #a #c #d #b #a #d #b #c ,ea* l9e repres5ts ! ord] = a sitt+1 & ! p]son repres5t$ by ! la/ numb] 9 a l9e mu/1 ( c\rse1 be reg>d$ z sitt+ next to ! fir/ p]son 9 ! same l9e1 :5 plac$ at ! r\nd table4 ,! case ( #e p]sons on #f occasions may be solv$ z foll[s3-- #a #b #c #d #e #a #b #d #e #c #a #b #e #c #d --------- #a #c #b #e #d #a #d #b #c #e #a #e #b #d #c ,! case = #f p]sons on #aj occasions is solv$ ?us3-- #a #b #c #f #d #e #a #c #d #b #e #f #a #d #e #c #f #b #a #e #f #d #b #c #a #f #b #e #c #d ----------- #a #b #d #e #f #c #a #c #e #f #b #d #a #d #f #b #c #e #a #e #b #c #d #f #a #f #c #d #e #b ,x w n[ no long] be necess>y to give ! solutions 9 full1 = reasons t ,i w expla94 ,x w be se5 9 ! examples above t ! #a 7&1 9 ! case ( #e p]sons1 also ! #b7 is repeat$ d[n ! column4 ,su* a numb] ,i call a repeat]4 ,! o!r numb]s desc5d 9 cyclical ord]4 ,?us1 = #f p]sons we get ! cycle1 #b1 #c1 #d1 #e1 #f1 #b1 & s on1 9 e column4 ,s x is only necess>y to give ! two l9es #a #b #c #f #d #e & #a #b #d #e #f #c1 & d5ote ! cycle & repeat]s1 to enable any "o to write out ! full solution /raight away4 ,! read] may wond] :y ,i d n />t ! la/ solution ) ! numb]s 9 _! natural ord]1 #a #b #c #d #e #f4 ,if ,i did s ! numb]s 9 ! desc5d+ cycle w\ld n be 9 _! natural ord]1 & x is m conv5i5t to h a regul> cycle ?an to consid] ! ord] 9 ! fir/ l9e4 ,! difficult case ( #g p]sons on #ae occasions is solv$ z foll[s1 & 0 giv5 by me 9 ,! ,cant]bury ,puzzles3-- #a #b #c #d #e #g #f #a #f #b #g #e #c #d #a #c #e #b #f #g #d #a #e #g #d #c #f #b #a #e #b #g #c #d #f ,9 ?is case ! #a is a repeat]1 & !re are two sep>ate cycles1 #b1 #c1 #d1 #b1 & #e1 #f1 #g1 #e4 ,we ?us get five gr\ps ( ?ree l9es ea*1 = a f\r? l9e 9 any gr\p w m]ely repeat ! fir/ l9e4 ,a solution = #h p]sons on #ba occasions is z foll[s3-- #a #h #f #c #d #e #b #g #a #h #d #e #g #b #c #f #a #h #b #g #c #f #d #e ,! #a is "h a repeat]1 & ! cycle #b1 #c1 #d1 #e1 #f1 #g1 #h4 ,e "o ( ! #c gr\ps w give #g l9es4 ,"h is my solution = #i p]sons on #bh occasions3-- #b #a #i #g #d #e #f #c #h #b #i #e #a #f #h #c #d #g #b #i #c #a #h #d #g #e #f #b #i #a #e #f #d #g #h #c ,!re are "h two repeat]s1 #a & #b1 & ! cycle is #c1 #d1 #e1 #f1 #g1 #h1 #i4 ,we ?us get #d gr\ps ( #g l9es ea*4 ,! case ( #aj p]sons on #cf occasions is solv$ z foll[s3-- #a #aj #h #c #f #e #d #g #b #i #a #aj #f #e #b #i #g #d #c #h #a #aj #b #i #c #h #f #e #g #d #a #aj #g #d #h #c #b #i #e #f ,! repeat] is #a1 & ! cycle1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #aj4 ,we "h h #d gr\ps ( #i l9es ea*4 ,my solution = #aa p]sons on #de occasions is z foll[s3-- #b #aa #i #d #g #f #e #a #h #c #aj #b #a #aa #g #f #c #aj #h #e #d #i #b #aa #aj #c #i #d #h #e #a #g #f #b #aa #e #h #a #c #aj #f #g #i #d #b #aa #a #aj #c #d #i #f #g #e #h ,!re are two repeat]s1 #a & #b1 & ! cycle is1 #c1 #d1 #e1444 #aa4 ,we ?us get #e gr\ps ( #i l9es ea*4 ,! case ( #ab p]sons on #ee occasions is solv$ ?us3-- #a #b #c #ab #d #aa #e #aj #f #i #g #h #a #b #d #aa #f #i #h #g #aj #e #ab #c #a #b #e #aj #h #g #aa #d #c #ab #f #i #a #b #f #i #aj #e #c #ab #g #h #aa #d #a #b #g #h #ab #c #f #i #aa #d #e #aj ,"h #a is a repeat]1 & ! cycle is #b1 #c1 #d1 #e1444 #ab4 ,we ?us get #e gr\ps ( #aa l9es ea*4 #bgd4--,,! ,,m\se-,,trap ,,puzzle4 ,if we int]*ange c>ds #f & #ac & beg9 our c\nt at #ad1 we may take up all ! tw5ty-"o c>ds--t is1 make tw5ty-"o cat*es--9 ! foll[+ ord]3 #f1 #h1 #ac1 #b1 #aj1 #a1 #aa1 #d1 #ad1 #c1 #e1 #g1 #ba1 #ab1 #ae1 #bj1 #i1 #af1 #ah1 #ag1 #ai4 ,we may also ex*ange #aj & #ad & />t at #af1 or ex*ange #f & #h & />t at #ai4 #bge4--,,! ,,sixte5 ,,%eep4 ,! six diagrams on next page %[ solutions = ! cases :]e we replace #b1 #c1 #d1 #e1 #f1 & #g hurdles4 ,! d>k l9es indicate ! hurdles t h be5 replac$4 ,!re are1 ( c\rse1 o!r ways ( mak+ ! removals4 #bgf4--,,! ,,eight ,,villas4 ,!re are sev]al ways ( solv+ ! puzzle1 b !re is v little diff]5ce betwe5 !m4 ,! solv] %\ld1 h[ev]1 fir/ ( all be> 9 m9d t 9 mak+ 8 calculations he ne$ only consid] ! f\r villas t /& at ! corn]s1 because ! int]m$iate villas c nev] v>y :5 ! corn]s are kn[n4 ,"o way is to place ! numb]s n\ght to #i "o at a "t 9 ! top left-h& corn]1 & !n consid] ea* case 9 turn4 ,n[1 if we place #i 9 ! corn] z %[n 9 ! ,diagram ,a1 two ( ! corn]s _c be occupi$1 :ile ! corn] t is diagonally opposite may be fill$ by #j1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 or #i p]sons4 ,we ?us see t !re are #aj ,illu/ration3 ------ ------ ------ ,o ,,oho ,o ,,oho ,o ,o ,o ,,oho ,o ,h ,o ,,oho ,o ,,oho ,o ,o ,o ,,ohoho - -- ------ --- ,o,o ,o,o ,o,o ,o ,o ,o ,o ,o,o --- -- - ,o ,o ,o ,o ,o ,o ,,oho ,o ,o,o ,o ------- ------- ------- #b #c #d ------ ------ ------- ,o ,o ,,oho ,,oho ,o ,o ,o ,o ,o ,o ,o ,,oho ,o ,,ohoho ,o ,,ohoho ,o -- - ,o,o ,o,o ,o,o ,o,o ,o,,oho ,o ,o ,o ,o ,o ,,oho ,o ,o ,o ,o,o ,o ------- ------ ------ #e #f #g ,,! ,,sixte5 ,,%eep solutions ) a #i 9 ! corn]4 ,if1 h[ev]1 we sub/itute #h1 ! two corn]s 9 ! same r[ & column may conta9 #j1 #j1 or #a1 #a1 or #j1 #a1 or #a1 #j4 ,9 ! case ( ,b1 t5 diff]5t selections may be made = ! f\r? corn]2 b 9 ea* ( ! cases ,c1 ,d1 & ,e1 only n9e selections are possible1 because we _c use ! #i4 ,!re=e ) #h 9 ! top left-h& corn] !re are #aj 7#c #i7 #cg diff]5t solutions4 ,if we !n try #g 9 ! corn]1 ! result w be #aj #bg #dj1 or #gg solutions4 ,) #f we get #aj #bg #dj #di #abf2 ) #e1 #aj #bg #dj #di #ed #ahj2 ) #d1 ! same z ) #e1 #ee #bce 2 ) #c1 ! same z ) #d1 #eb #bhg2 ) #b1 ! same z ) #c1 #de #ccb2 ) #a1 ! same z ) #b1 #cd #cff1 & ) n\ght 9 ! top left-h& corn] ! numb] ( solutions w be f\nd to be #aj #bg #dj #di #ed #ee #eb #de #cd #ai #che4 ,z !re is no o!r numb] to be plac$ 9 ! top left-h& corn]1 we h n[ only to add ^! totals toge!r ?us1 #aj #cg #gg #abf #ahj #bce #bhg #ccb #cff #che #b1jce4 ,we !re=e f9d t ! total numb] ( ways 9 :i* t5ants may occupy "s or all ( ! eight villas s t !re %all be always n9e p]sons liv+ along ea* side ( ! squ>e is #b1jce4 ,( c\rse1 ?is me?od mu/ obvi\sly cov] all ! rev]sals & reflections1 s9ce ea* corn] 9 turn is occupi$ by e numb] 9 all possible comb9ations ) ! o!r two corn]s t are 9 l9e ) x4 ,illu/ration3 ,a ,b ,c ,d ,e --- --- --- --- --- #i #j #h #j #h #a #h #j #h #a --- --- --- --- --- 99 99 99 99 99 --- --- --- --- --- #j #j #a #a #j --- --- --- --- --- ,"h is a g5]al =mula = solv+ ! puzzle3 7n #cn #b77n #cn #c7_/f4 ,:atev] may be ! /ipulat$ numb] ( resid5ts along ea* ( ! sides 7:i* numb] is repres5t$ by n71 ! total numb] ( diff]5t arrangem5ts may be ?us asc]ta9$4 ,9 our p>ticul> case ! numb] ( resid5ts 0 n9e4 ,!re=e 7#ha #bg #b7 7#ha #bg #c7 & ! product1 divid$ by #f1 gives #b1jce4 ,if ! numb] ( resid5ts _h be5 #j1 #a1 #b1 #c1 #d1 #e1 #f1 #g1 or #h1 ! total arrangem5ts w\ld be #a1 #g1 #bf1 #gj1 #aee1 #cja1 #ecb1 #hgf1 or #a1cfe respectively4 #bgg4--;,,c\nt] ,,crosses4 ,let u fir/ deal ) ! ,greek ,cross4 ,!re are j eighte5 =ms 9 :i* ! numb]s may be pair$ = ! two arms4 ,"h !y are3-- #abigh #acifh #adieh #cdief #bdieg #bcifg #bcieh #acgfi #adgei #adifg #bdgeh #bcgfh #abehi #bcgei #acegi #cdefg #adgfh #bdefh #adefi #bcefi #adcgi #bcegh #adegh #becfh #aecfi #bdcfi #bcahi #bdcgh #aecgh #deafg #bdagi #beafi #cdafi #ceafh #cdagh #beagh ,( c\rse1 ! numb] 9 ! middle is common to bo? arms4 ,! fir/ pair is ! "o ,i gave z an example4 ,i w suppose t we h writt5 out all ^! crosses1 always plac+ ! fir/ r[ ( a pair 9 ! upright & ! second r[ 9 ! horizontal arm4 ,n[1 if we leave ! c5tral figure fix$1 !re are #bd ways 9 :i* ! numb]s 9 ! upright may be v>i$1 = ! f\r c\nt]s may be *ang$ 9 #a #b #c #d #bd ways4 ,& z ! f\r 9 ! horizontal may also be *ang$ 9 #bd ways = e arrangem5t on ! o!r arm1 we f9d t !re are #bd #bd #egf v>iations = e =m2 !re=e1 z !re are #ah =ms1 we get #ah #egf #aj1cfh ways4 ,b ?is w include half ! f\r rev]sals & half ! f\r reflections t we b>r$1 s we mu/ divide ?is by #d to obta9 ! correct answ] to ! ,greek ,cross1 :i* is ?us #b1eib diff]5t ways4 ,! division is by #d & n by #h1 because we provid$ aga9/ half ! rev]sals & reflections by always res]v+ "o numb] = ! upright & ! o!r = ! horizontal4 ,9 ! case ( ! ,lat9 ,cross1 x is obvi\s t we h to deal ) ! same #ah =ms ( pair+4 ,! total numb] ( diff]5t ways 9 ?is case is ! full numb]1 #ah #egf4 ,ow+ to ! fact t ! upp] & l[] arms are unequal 9 l5g?1 p]mutations w repeat by reflection1 b n by rev]sal1 = we _c rev]se4 ,!re=e ?is fact only entails division by #b4 ,b 9 e pair we may ex*ange ! figures 9 ! upright ) ^? 9 ! horizontal 7:i* we c\ld n d 9 ! case ( ! ,greek ,cross1 z ! arms are !re all alike72 consequ5tly we mu/ multiply by #b4 ,?is multiplication by #b & division by #b cancel "o ano!r4 ,h5ce #aj1cfh is "h ! correct answ]4 #bgh4--;,a ,,dormitory ,,puzzle4 ,illu/ration3 ,,mon4 ,,tues4 ,,w$4 --------- --------- --------- #a #b #a #a #c #a #a #d #a --------- --------- --------- #b #b #a #a #a #a --------- --------- --------- #a #bb #a #c #ai #c #d #af #d --------- --------- --------- ,,?urs4 ,,fri4 ,,sat4 --------- --------- --------- #a #e #a #b #f #b #d #d #d --------- --------- --------- #b #b #a #a #d #d --------- --------- --------- #d #ac #d #g #f #g #d #d #d --------- --------- --------- ,arrange ! nuns f day to day z %[n 9 ! six diagrams4 ,! smalle/ possible numb] ( nuns w\ld be ?irty-two1 & ! arrangem5ts on ! la/ ?ree days admit ( v>iation4 #bgi4--,,! ,,b>rels ,,( ,,balsam4 ,?is is q easy to solve = any numb] ( b>rels--if y "k h[4 ,?is is ! way to d x4 ,!re are five b>rels 9 ea* r[ ,multiply ! numb]s #a1 #b1 #c1 #d1 #e toge!r2 & also multiply #f1 #g1 #h1 #i1 #aj toge!r4 ,divide "o result by ! o!r1 & we get ! numb] ( diff]5t comb9ations or selections ( t5 ?+s tak5 five at a "t4 ,?is is "h #beb4 ,n[1 if we divide ?is by #f 7#a m ?an ! numb] 9 ! r[7 we get #db1 :i* is ! correct answ] to ! puzzle1 = !re are #db diff]5t ways ( arrang+ ! b>rels4 ,try ?is me?od ( solution 9 ! case ( six b>rels1 ?ree 9 ea* r[1 & y w f9d ! answ] is #e ways4 ,if y *eck ?is by trial1 y w discov] ! five arrangem5ts ) #abc1 #abd1 #abe1 #acd1 #ace respectively 9 ! top r[1 & y w f9d no o!rs4 ,! g5]al solution to ! problem is1 9 fact1 ?is3 n ,c #bn ----- n #a :]e #bn equals ! numb] ( b>rels4 ,! symbol ,c1 ( c\rse1 implies t we h to f9d h[ _m comb9ations1 or selections1 we c make ( #bn ?+s1 tak5 n at a "t4 #bhj4--;,,build+ ,,! ,,tetrah$ron4 ,take y\r con/ruct$ pyramid & hold x s t "o /ick only lies on ! table4 ,n[1 f\r /icks mu/ bran* (f f x 9 diff]5t directions--two at ea* end4 ,any "o ( ! five /icks may be left out ( ?is connection2 !re=e ! f\r may be select$ 9 #e diff]5t ways4 ,b ^! f\r mat*es may be plac$ 9 #bd diff]5t ord]s4 ,& z any mat* may be jo9$ at ei!r ( its ends1 !y may fur!r be v>i$ 7aft] _! situations are settl$ = any p>ticul> arrangem5t7 9 #af diff]5t ways4 ,9 e arrangem5t ! six? /ick may be add$ 9 #b diff]5t ways4 ,n[ multiply ^! results toge!r1 & we get #e #bd #af #b #c1hdj z ! exact numb] ( ways 9 :i* ! pyramid may be con/ruct$4 ,?is me?od excludes all possibility ( error4 ,a common cause ( error is ?is4 ,if y calculate y\r comb9ations by work+ upw>ds f a basic triangle ly+ on ! table1 y w get half ! correct numb] ( ways1 because y ov]look ! fact t an equal numb] ( pyramids may be built on t triangle d[nw>ds1 s to speak1 ?r\gh ! table4 ,!y are1 9 fact1 reflections ( ! o!rs1 & examples f ! two sets ( pyramids _c be set up to resemble "o ano!r--except "u f\r? dim5sional conditions6 #bha4--,,pa9t+ ,a ,,pyramid4 ,x w be conv5i5t to imag9e t we are pa9t+ our pyramids on ! flat c>dbo>d1 z 9 ! diagrams1 be=e fold+ up4 ,n[1 if we take any f\r col\rs 7say r$1 blue1 gre51 & yell[71 !y may be appli$ 9 only #b di/9ctive ways1 z %[n 9 ,figs1 #a & #b4 ,any o!r way w only result 9 "o ( ^! :5 ! pyramids are fold$ up4 ,if we take any ?ree col\rs1 !y may be appli$ 9 ! #c ways %[n 9 ,figs4 #c1 #d1 & #e4 ,if we take any two col\rs1 !y may be appli$ 9 ! #c ways %[n 9 ,figs4 #f1 #g1 & #h4 ,if we take any s+le col\r1 x may obvi\sly be appli$ 9 only #a way4 ,b f\r col\rs may be select$ 9 #ce ways out ( sev52 ?ree 9 #ce ways2 two 9 #ba ways2 & "o col\r 9 #g ways4 ,!re=e #ce appli$ 9 #b ways #gj2 #ce 9 #c ways #aje2 #ba 9 #c ways #fc2 & #g 9 #a way #g4 ,consequ5tly ! pyramid may be pa9t$ 9 #bde diff]5t ways 7#gj #aje #fc #g71 us+ ! sev5 col\rs ( ! sol> spectrum 9 accordance ) ! conditions ( ! puzzle4 ,illu/ration3 #a #b --------------- --------------- ,r _/ ,b _/ ,b _/ ,r _/ _/ _/ _/ _/ _/ ,g _/ _/ ,g _/ -------_/ -------_/ _/ _/ ,y _/ ,y _/ _/ _/ ' ' #c #d #e --------------- --------------- --------------- ,r _/ ,r _/ ,r _/ ,g _/ ,y _/ ,r _/ _/ _/ _/ _/ _/ _/ _/ ,g _/ _/ ,g _/ _/ ,g _/ -------_/ -------_/ -------_/ _/ _/ _/ ,y _/ ,y _/ ,y _/ _/ _/ _/ ' ' ' #f #g #h --------------- --------------- --------------- ,g _/ ,y _/ ,y _/ ,y _/ ,g _/ ,g _/ _/ _/ _/ _/ _/ _/ _/ ,g _/ _/ ,g _/ _/ ,g _/ -------_/ -------_/ -------_/ _/ _/ _/ ,y _/ ,y _/ ,y _/ _/ _/ _/ ' ' ' #bhb4--,,! ,,antiqu>y',s ,,*a94 ,illu/ration ,,! numb] ( ways 9 :i* n9e ?+s may be arrang$ 9 a r[ )\t any re/rictions is #a #b #c #d #e #f #g #h #i #cfb1hhj4 ,b we are told t ! two circul> r+s mu/ nev] be toge!r2 !re=e we mu/ d$uct ! numb] ( times t ?is w\ld occur4 ,! numb] is #a #b #c #d #e #f #g #h #dj1cbj #b #hj1fdj1 because if we consid] ! two circul> l9ks to be insep>ably jo9$ toge!r !y become z "o l9k1 & eight l9ks are capable ( #dj1cbj arrangem5ts2 b z ^! two l9ks may always be put on 9 ! ord]s ,,ab or ,,ba1 we h to d\ble ?is numb]1 x be+ a "q ( arrangem5t & n ( design4 ,! d$uction requir$ r$uces our total to #bhb1bdj4 ,!n "o ( our l9ks is ( a peculi> =m1 l an #h4 ,we h !re=e ! option ( jo9+ on ei!r "o end or ! o!r on e occasion1 s we mu/ d\ble ! la/ result4 ,?is br+s up our total to #efd1dhj4 ,we n[ come to ! po9t to :i* ,i direct$ ! read]'s att5tion--t e l9k may be put on 9 "o ( two ways4 ,if we jo9 ! fir/ f+] & ?umb ( our left h& horizontally1 & !n l9k ! fir/ f+] & ?umb ( ! "r h&1 we see t ! "r ?umb may be ei!r above or bel[4 ,b 9 ! case ( our *a9 we mu/ rememb] t al?\gh t #h-%ap$ l9k has two indep5d5t ends x is l e o!r l9k 9 hav+ only two sides--t is1 y _c turn ov] "o end )\t turn+ ! o!r at ! same "t4 ,we w1 = conv5i5ce1 assume t ea* l9k has a black side & a side pa9t$ :ite4 ,n[1 if x 7 /ipulat$ t 7) ! *a9 ly+ on ! table1 & e successive l9k fall+ ov] its pr$ecessor 9 ! same way1 z 9 ! diagram7 only ! :ite sides %\ld be upp]mo/ z 9 ,a1 !n ! answ] w\ld be #efd1dhj1 z above--ignor+ = ! pres5t all rev]sals ( ! complet$ *a94 ,if1 h[ev]1 ! fir/ l9k 7 all[$ to be plac$ ei!r side up1 !n we c\ld h ei!r ,a or ,b1 & ! answ] w\ld be #b #efd1dhj #a1abh1ifj2 if two l9ks might be plac$ ei!r way up1 ! answ] w\ld be #d #efd1dhj2 if ?ree l9ks1 !n #h #efd1dhj1 & s on4 ,s9ce1 !re=e1 e l9k may be plac$ ei!r side up1 ! numb] w be #efd1dhj multipli$ by #bi1 or by #eab4 ,?is raises our total to #bhi1jac1gfj4 ,b !re is /ill "o m po9t to be consid]$4 ,we h n yet all[$ = ! fact t ) any giv5 arrangem5t ?ree ( ! o!r arrangem5ts may be obta9$ by simply turn+ ! *a9 ov] ?r\gh its entire l5g? & by rev]s+ ! ends4 ,?us ,c is really ! same z ,a1 & if we turn ?is page upside d[n1 !n ,a & ,c give two o!r arrangem5ts t are /ill really id5tical4 ,?us to get ! correct answ] to ! puzzle we mu/ divide our la/ total by #d1 :5 we f9d t !re are j #gb1bec1ddj diff]5t ways 9 :i* ! smi? might h put ^? l9ks toge!r4 ,9 o!r words1 if ! n9e l9ks _h orig9ally =m$ a piece ( *a91 & x 0 kn[n t ! two circul> l9ks 7 sep>at$1 !n x w\ld be #gb1bec1dci *ances to #a t ! smi? w\ld n h put ! l9ks toge!r aga9 precisely z !y 7 arrang$ be=e6 #bhc4--,,! ,,fifte5 ,,dom9oes4 ,! read] may h notic$ t at ea* end ( ! l9e ,i give is a f\r1 s t1 if we l1 we c =m a r+ in/ead ( a l9e4 ,x c easily be prov$ t ?is mu/ always be s4 ,e l9e arrangem5t w make a circul> arrangem5t if we l to jo9 ! ends4 ,n[1 curi\s z x may at fir/ appe>1 ! foll[+ diagram exactly repres5ts ! conditions :5 we leave ! d\bles out ( ! "q & devote our att5tion to =m+ circul> arrangem5ts4 ,ea* numb]1 or half dom9o1 is 9 l9e ) e o!r numb]1 s t if we />t at any "o ( ! five numb]s & g ov] all ! l9es ( ! p5tagon once & once only we %all come back to ! />t+ place1 & ! ord] ( our r\te w give u "o ( ! circul> arrangem5ts = ! t5 dom9oes4 ,take y\r p5cil & foll[ out ! foll[+ r\te1 />t+ at ! #d3 #dacjdbajbcd4 ,y h be5 ov] all ! l9es once only1 & by repeat+ all ^! figures 9 ?is way1 #da--ac--cj--jd--db--ba--aj--jb--bc--cd1 y get an arrangem5t ( ! dom9oes 7)\t ! d\bles7 :i* w be p]fectly cle>4 ,take o!r r\tes & y w get o!r arrangem5ts4 ,if1 !re=e1 we c asc]ta9 j h[ _m ( ^! circul> r\tes are obta9able f ! p5tagon1 !n ! re/ is v easy4 ,well1 ! numb] ( diff]5t circul> r\tes ov] ! p5tagon is #bfd4 ,h[ ,i arrive at ^! figures ,i w n at pres5t expla91 because x w\ld take a lot ( space4 ,! dom9oes may1 !re=e1 be arrang$ 9 a circle 9 j #bfd diff]5t ways1 leav+ out ! d\bles4 ,n[1 9 any "o ( ^! circles ! five d\bles may be ins]t$ 9 #be #cb diff]5t ways4 ,!re=e :5 we include ! d\bles !re are #bfd #cb #h1ddh diff]5t circul> arrangem5ts4 ,b ea* ( ^? circles may be brok5 7s z to =m our /raight l9e7 9 any "o ( #ae diff]5t places4 ,consequ5tly1 #h1ddh #ae gives #abf1gbj diff]5t ways z ! correct answ] to ! puzzle4 ,illu/ration3 ----- _/ _/ ----- _/ 4 4 ----- 4 4 ----- 4 4 o o o -4--------4--- 4 4 4 o o ----- 4 4 44 ----- 4 4 4 4 _/ ----- 44 ----- o 4 4 o --------- o o 4 4 o ----- ----- ,i purposely refra9$ f ask+ ! read] to discov] 9 j h[ _m diff]5t ways ! full set ( tw5ty-eight dom9oes may be arrang$ 9 a /raight l9e 9 accordance ) ! ord9>y rules ( ! game1 left to "r & "r to left ( any arrangem5t c\nt+ z diff]5t ways4 ,x is an exce$+ly difficult problem1 b ! correct answ] is #g1iei1bbi1ica1ebj ways4 ,! me?od ( solv+ is v complex4 #bhd4--,,! ,,cross ,,t>get4 ,illu/ration3 -- -- 7,,cd77 7 -- -- 7,,ae77,a 7 -- -- -- -- -- -- 7,,ce77,e 77,a 77,,ab77,c 77,d 7 -- -- -- -- -- -- 7,d 77 77,b 77,e 77,,eb77 7 -- -- -- -- -- -- 7,c 77,b 7 -- -- 7 77,,ed7 -- -- ,tw5ty-"o diff]5t squ>es may be select$4 ,( ^! n9e w be ( ! size %[n by ! f\r ,a's 9 ! diagram1 f\r ( ! size %[n by ! ,b's1 f\r ( ! size %[n by ! ,c's1 two ( ! size %[n by ! ,d's1 & two ( ! size indicat$ by ! upp] s+le ,a1 ! upp] s+le ,e1 ! l[] s+le ,c1 & ! ,,eb4 ,x is an int]e/+ fact t y _c =m any "o ( ^! tw5ty-"o squ>es )\t us+ at lea/ "o ( ! six circles m>k$ ,e4 #bhe4--,,! ,,f\r ,,po/age ,,/amps4 ,ref]r+ to ! orig9al diagram1 ! f\r /amps may be giv5 9 ! %ape #a1 #b1 #c1 #d1 9 ?ree ways2 9 ! %ape #a1 #b1 #e1 #f1 9 six ways2 9 ! %ape #a1 #b1 #c1 #e1 or #a1 #b1 #c1 #g1 or #a1 #e1 #f1 #g1 or #c1 #e1 #f1 #g1 9 tw5ty-eight ways2 9 %ape #a1 #b1 #c1 #f1 or #b1 #e1 #f1 #g1 9 f\rte5 ways2 9 %ape #a1 #b1 #f1 #g1 or #b1 #c1 #e1 #f1 or #a1 #e1 #f1 #aj1 or #b1 #e1 #f1 #i1 9 f\rte5 ways4 ,?us !re are sixty-five ways 9 all4 #bhf4--,,pa9t+ ,,! ,,die4 ,! #a c be m>k$ on any "o ( six diff]5t sides4 ,= e side occupi$ by #a we h a selection ( f\r sides = ! #b4 ,= e situation ( ! #b we h two places = ! #c4 7,! #f1 #e1 & #d ne$ n be consid]$1 z _! positions are det]m9$ by ! #a1 #b1 & #c47 ,!re=e #f1 #d1 & #b multipli$ toge!r make #dh diff]5t ways--! correct answ]4 #bhg4--;,,an ,,acro/ic ,,puzzle4 ,!re are tw5ty-six lett]s 9 ! alphabet1 giv+ #cbe diff]5t pairs4 ,e "o ( ^! pairs may be rev]s$1 mak+ #fej ways4 ,b e initial lett] may be repeat$ z ! f9al1 produc+ #bf o!r ways4 ,! total is !re=e #fgf diff]5t pairs4 ,9 o!r words1 ! answ] is ! squ>e ( ! numb] ( lett]s 9 ! alphabet4 #bhh4--;,,*equ]$ ,,bo>d ,,divisions4 ,!re are #bee diff]5t ways ( cutt+ ! bo>d into two pieces ( exactly ! same size & %ape4 ,e way mu/ involve "o ( ! five cuts %[n 9 ,diagrams ,a1 ,b1 ,c1 ,d1 & ,e4 ,to avoid repetitions by rev]sal & reflection1 we ne$ only consid] cuts t ent] at ! po9ts a1 b1 & c4 ,b ! exit mu/ always be at a po9t 9 a /raight l9e f ! entry ?r\gh ! c5tre4 ,?is is ! mo/ important condition to rememb]4 ,9 case ,b y _c ent] at a1 or y w get ! cut provid$ = 9 ,e4 ,simil>ly 9 ,c or ,d1 y mu/ n ent] ! key-l9e 9 ! same direction z itself1 or y w get ,a or ,b4 ,if y are work+ on ,a or ,c & ent]+ at a1 y mu/ consid] jo9s at "o end only ( ! key-l9e1 or y w get repetitions4 ,9 o!r cases y mu/ consid] jo9s at bo? ends ( ! key2 b aft] leav+ a 9 case ,d1 turn always ei!r to "r or left--use "o direction only4 ,figs4 #a & #b are examples "u ,a2 #c & #d are examples "u ,b2 #e & #f come "u ,c2 ,illu/ration & #g is a pretty example ( ,d4 ,( c\rse1 ,e is a peculi> type1 & obvi\sly admits ( only "o way ( cutt+1 = y cle>ly _c ent] at b or c4 ,"h is a table ( ! results3-- a b c ,ways4 ,a #h #ag #ba #df ,b #j #ag #ba #ch ,c #ae #ca #ci #he ,d #ag #bi #ci #he ,e #a #j #j #a -- -- -- --- #da #id #abj #bee ,i h n attempt$ ! task ( enum]at+ ! ways ( divid+ a bo>d #h #h--t is1 an ord9>y *essbo>d4 ,:atev] ! me?od adopt$1 ! solution w\ld entail consid]able lab\r4 #bhi4--,,lions ,,& ,,cr[ns4 ,illu/ration ,"h is ! solution4 ,x w be se5 t ea* ( ! f\r pieces 7aft] mak+ ! cuts along ! ?ick l9es7 is ( exactly ! same size & %ape1 & t ea* piece conta9s a lion & a cr[n4 ,two ( ! pieces are %ad$ s z to make ! solution q cle> to ! eye4 #bij4--;,,bo>ds ,,) ,,an ,,odd ,,numb] ,,( ,,squ>es4 ,!re are fifte5 diff]5t ways ( cutt+ ! #e #e bo>d 7) ! c5tral squ>e remov$7 into two pieces ( ! same size & %ape4 ,limitations ( space w n all[ me to give diagrams ( all ^!1 b ,i w enable ! read] to draw !m all out = himself )\t ! slighte/ difficulty4 ,at :atev] po9t on ! edge y\r cut ent]s1 x mu/ always end at a po9t on ! edge1 exactly opposite 9 a l9e ?r\gh ! c5tre ( ! squ>e4 ,?us1 if y ent] at po9t #a 7see ,fig4 #a7 at ! top1 y mu/ leave at po9t #a at ! bottom4 ,n[1 #a & #b are ! only two really diff]5t po9ts ( entry2 if we use any o!rs !y w simply produce simil> solutions4 ,! directions ( ! cuts 9 ! foll[+ fifte5 ,illu/ration3 ,fig4 #a4 ,fig4 #b4 solutions are indicat$ by ! numb]s on ! diagram4 ,! duplication ( ! numb]s c lead to no confusion1 s9ce e successive numb] is contigu\s to ! previ\s "o4 ,b :i*ev] direction y take f ! top d[nw>ds y mu/ repeat f ! bottom upw>ds1 "o direction be+ an exact reflection ( ! o!r4 #a1 #d1 #h4 #a1 #d1 #c1 #g1 #h4 #a1 #d1 #c1 #g1 #aj1 #i4 #a1 #d1 #c1 #g1 #aj1 #f1 #e1 #i4 #a1 #d1 #e1 #i4 #a1 #d1 #e1 #f1 #aj1 #i4 #a1 #d1 #e1 #f1 #aj1 #g1 #h4 #b1 #c1 #d1 #h4 #b1 #c1 #d1 #e1 #i4 #b1 #c1 #d1 #e1 #f1 #aj1 #i4 #b1 #c1 #d1 #e1 #f1 #aj1 #g1 #h4 #b1 #c1 #g1 #h4 #b1 #c1 #g1 #aj1 #i4 #b1 #c1 #g1 #aj1 #f1 #e1 #i4 #b1 #c1 #g1 #aj1 #f1 #e1 #d1 #h4 ,x w be se5 t ! f\r? direction 7#a1 #d1 #c1 #g1 #aj1 #f1 #e1 #i7 produces ! solution %[n 9 ,fig4 #b4 ,! ?irte5? produces ! solution giv5 9 prop\nd+ ! puzzle1 :]e ! cut ent]$ at ! side in/ead ( at ! top4 ,! pieces1 h[ev]1 w be ( ! same %ape if turn$ ov]1 :i*1 z x 0 /at$ 9 ! conditions1 w\ld n con/itute a diff]5t solution4 #bia4--,,! ,,gr& ,,lama',s ,,problem4 ,! me?od ( divid+ ! *essbo>d s t ea* ( ! f\r p>ts %all be ( exactly ! same size & %ape1 & conta9 "o ( ! gems1 is %[n 9 ! diagram4 ,! me?od ( %ad+ ! squ>es is adopt$ to make ! %ape ( ! pieces cle> to ! eye4 ,two ( ! pieces are %ad$ & two left :ite4 ,! read] may f9d x int]e/+ to comp>e ?is puzzle ) t ( ! ,weav] 7,no4 #ad1 ,cant]bury ,puzzles74 ,illu/ration3 ,,! ,,gr& ,,lama',s ,,problem4 3o3 3 3 3 3 3 3 3 ,i444,i444 333 o 33333333333333333333333 ,i444,i444,i444444,i 333 3o3 3 3 3 333 ,i444,i444,i444,i444,i444,i444,i 333 333 o 3333333 333 ,i444,i444,i444,i444,i444,i444,i 333 3333333 333 333 ,i444,i444444444,i444,i444,i 333 3 3 3 333 333 ,i444,i444,i444,i444,i 33333333333333333333333 333 444,i444,i 3 3 3 3 3 3 333 #bib4--,,! ,,abbot',s ,,w9d[4 ,,! man :o 0 le>n$ 9 /range my/]ies po9t$ out to ,fa!r ,john t ! ord]s ( ! ,"l ,abbot ( ,/4 ,edmondsbury might be easily c>ri$ out by block+ up twelve ( ! lights 9 ! w9d[ z %[n by ! d>k squ>es 9 ! foll[+ sket*3-- ,illu/ration3 3 3 3 3 3 3 3 ,i444444444444444444,i ,,iiiii 3 3 3 ,,iiiii ,i444444444444,i 3 ,,iiiii 3 ,,iiiii 3 ,i444444444444,i 3 3 ,,iiiiiiiii 3 3 ,i444444444444444444,i 3 3 ,,iiiiiiiii 3 3 ,i444444444444,i 3 ,,iiiii 3 ,,iiiii 3 ,i444444444444,i ,,iiiii 3 3 3 ,,iiiii ,i444444444444444444,i 3 3 3 3 3 3 3 ,fa!r ,john held t ! f\r corn]s %\ld also be d>k5$1 b ! sage expla9$ t x 0 desir$ to ob/ruct no m light ?an 0 absolutely necess>y1 & he said1 anticipat+ ,"l ,dundre>y1 ,a s+le pane c no m be 9 a l9e ) itself ?an "o bird c g into a corn] & flock 9 solitude4 ,! ,abbot's condition 0 t no diagonal l9es %\ld conta9 an odd numb] ( lights4 ,n[1 :5 ! holy man saw :at _h be5 done he 0 well pleas$1 & said1 ,truly1 ,fa!r ,john1 ?\ art a man ( deep wisdom1 9 t ?\ ha/ done t :i* seem$ impossible1 & yet )al adorn$ our w9d[ ) a device ( ! cross ( ,/4 ,&rew1 ^: "n ,i receiv$ f my godfa!rs & godmo!rs4 ,!reaft] he slept well & arose refre%$4 ,! w9d[ might be se5 intact to-day 9 ! mona/]y ( ,/4 ,edmondsbury1 if x exi/$1 :i*1 alas6 ! w9d[ does n4 #bic4--,,! ,,*9ese ,,*essbo>d4 ,i,i 3333 #b 3333 #c 333 #e 3#f3 ,i444444444,i444,i444,i 3333 #a 333 3333 #d 333 #g ,i444444,i,i444,i,i 3333 333 3333 #i 333 ,i,i444,i,i444,i444,i 3333 #aa3333 33333 #aj333 #h ,i,i,i,i444,i444,i 33333 #ab3333 #ac3333 333 ,i444,i444,i 3333 #ad333 333 #af3333 #ag ,i444444,i,i444444444,i 3333 33333 #ae333 3333 ,i,i 3333 33333 #ah33333 33333 ,i,i,i 3333 3333 333 3333 ,i444,i444,i,i444,i444,i 333 3333 3333 333 ,i444,i,i,i,i444,i 3333 3333 33333 333 ,i,i,i,i,i,i 333 3333 3333 3333 ,i444,i444,i444444,i444444,i 3333 3333 333 3333 ,i444,i444,i444,i444,i 333 3333 3333 333 ,i,i444,i444,i,i 3333 3333 33333 333 ,i444444,i444444,i 3333 3333 333 33333 ,eighte5 is ! maximum numb] ( pieces4 ,i give two solutions4 ,! numb]$ diagram is s cut t ! eighte5? piece has ! l>ge/ area--eight squ>es--t is possible "u ! conditions4 ,! second diagram 0 prep>$ "u ! add$ condition t no piece %\ld conta9 m ?an five squ>es4 ,no4 #gd 9 ,! ,cant]bury ,puzzles %[s h[ to cut ! bo>d into twelve pieces1 all diff]5t1 ea* conta9+ five squ>es1 ) "o squ>e piece ( f\r squ>es4 #bid4--,,! ,,*essbo>d ,,s5t5ce4 ,i,i,i,i,i 333 333 3333 3333 ,i,i444,i,i444,i444,i444,i 333 33333 333 33333 444444,i444,i444444,i 333 333 3333 3333 444444,i,i,i,i 3333 33333 333 33333 ,i,i,i444,i,i444 33333 333 333 333 444444444444444,i444 333 333 333 3333 444444444444,i444444 333 333 33333 333 ,i444,i444,i444 3333 3333 33333 3333 ,i ,! pieces may be fitt$ toge!r1 z %[n 9 ! illu/ration1 to =m a p]fect *essbo>d4 #bie4--,,! ,,eight ,,rooks4 ,obvi\sly !re mu/ be a rook 9 e r[ & e column4 ,/>t+ ) ! top r[1 x is cle> t we may put our fir/ rook on any "o ( eight diff]5t squ>es4 ,:]ev] x is plac$1 we h ! option ( sev5 squ>es = ! second rook 9 ! second r[4 ,!n we h six squ>es f :i* to select ! ?ird r[1 five 9 ! f\r?1 & s on4 ,!re=e ! numb] ( our diff]5t ways mu/ be #h #g #f #e #d #c #b #a #dj1cbj 7t is #h671 :i* is ! correct answ]4 ,h[ _m ways !re are if m]e rev]sals & reflections are n c\nt$ z diff]5t has n yet be5 det]m9$2 x is a difficult problem4 ,b ?is po9t1 on a small] squ>e1 is consid]$ 9 ! next puzzle4 #bif4--,,! ,,f\r ,,lions4 ,!re are only sev5 diff]5t ways "u ! conditions4 ,!y are z foll[s3 #a #b #c #d1 #a #b #d #c1 #a #c #b #d1 #a #c #d #b1 #a #d #c #b1 #b #a #d #c1 #b #d #a #c4 ,tak+ ! la/ example1 ?is notation means t we place a lion 9 ! second squ>e ( fir/ r[1 f\r? squ>e ( second r[1 fir/ squ>e ( ?ird r[1 & ?ird squ>e ( f\r? r[4 ,! fir/ example is1 ( c\rse1 ! "o we gave :5 sett+ ! puzzle4 #big4--;,,bi%ops--,,ungu>d$4 444444444444444444444444 3 33333 33333 33333 33333 444444444444444444444444 33333 33333 33333 33333 3 444444444444444444444444 3 33333 33333 33333 33333 444444444444444444444444 33,b33 ,b 33,b33 ,b 33,b33 ,b 33,b33 ,b 3 444444444444444444444444 3 33333 33333 33333 33333 444444444444444444444444 33333 33333 33333 33333 3 444444444444444444444444 33333 33333 33333 33333 3 444444444444444444444444 3 33333 33333 33333 33333 444444444444444444444444 ,?is _c be done ) few] bi%ops ?an eight1 & ! simple/ solution is to place ! bi%ops 9 l9e along ! f\r? or fif? r[ ( ! bo>d 7see diagram74 ,b x w be notic$ t no bi%op is "h gu>d$ by ano!r1 s we consid] t po9t 9 ! next puzzle4 #bih4--;,,bi%ops--,,gu>d$4 44444444444444444444444444 3 33333 33333 33333 33333 444444444444444444444444 33333 33333 33333 33333 3 444444444444444444444444 3 33333 33333 33333 33333 4444444444444444444444444 33333 ,b 33,b33 ,b 33333 ,b 33,b33 3 44444444444444444444444444 3 33,b33 ,b 33,b33 33,b33 ,b 33333 444444444444444444444444 33333 33333 33333 33333 3 444444444444444444444444 3 33333 33333 33333 33333 4444444444444444444444444 33333 33333 33333 33333 3 4444444444444444444444444 ,?is puzzle is q easy if y fir/ ( all give x a little ?\ght4 ,y ne$ only consid] squ>es ( "o col\r1 = :atev] c be done 9 ! case ( ! :ite squ>es c always be repeat$ on ! black1 & !y are "h q indep5d5t ( "o ano!r4 ,?is equality1 ( c\rse1 is 9 consequ5ce ( ! fact t ! numb] ( squ>es on an ord9>y *essbo>d1 sixty-f\r1 is an ev5 numb]4 ,if a squ>e *equ]$ bo>d has an odd numb] ( squ>es1 !n !re w always be "o m squ>e ( "o col\r ?an ( ! o!r4 ,t5 bi%ops are necess>y 9 ord] t e squ>e %all be attack$ & e bi%op gu>d$ by ano!r bi%op4 ,i give "o way ( arrang+ !m 9 ! diagram4 ,x w be notic$ t ! two c5tral bi%ops 9 ! gr\p ( six on ! left-h& side ( ! bo>d s]ve no purpose1 except to protect ^? bi%ops t are on adjo9+ squ>es4 ,ano!r solution w\ld !re=e be obta9$ by simply rais+ ! upp] "o ( ^! "o squ>e & plac+ ! o!r a squ>e l[] d[n4 #bii4--;,,bi%ops ,,9 ,,convocation4 ,! f\rte5 bi%ops may be plac$ 9 #bef diff]5t ways4 ,b e bi%op mu/ always be plac$ on "o ( ! sides ( ! bo>d--t is1 some:]e on a r[ or file on ! extreme edge4 ,! puzzle1 !re=e1 consi/s 9 c\nt+ ! numb] ( diff]5t ways t we c arrange ! f\rte5 r\nd ! edge ( ! bo>d )\t attack4 ,?is is n a difficult matt]4 ,on a *essbo>d ( n squ>es #bn - #b bi%ops 7! maximum numb]7 may always be plac$ 9 #bn ways )\t attack+4 ,on an ord9>y *essbo>d n w\ld be #h2 !re=e #ad bi%ops may be plac$ 9 #bef diff]5t ways4 ,x is r curi\s t ! g5]al result %\ld come out 9 s simple a =m4 ,illu/ration #cjj4--,,! ,,eight ,,que5s4 ,illu/ration ,! solution to ?is puzzle is %[n 9 ! diagram4 ,x w be f\nd t no que5 attacks ano!r1 & also t no ?ree que5s are 9 a /raight l9e 9 any oblique direction4 ,?is is ! only arrangem5t out ( ! twelve fundam5tally diff]5t ways ( plac+ eight que5s )\t attack t fulfils ! la/ condition4 #cja4--,,! ,,eight ,,/>s4 ,! solution ( ?is puzzle is %[n 9 ! fir/ diagram4 ,x is ! only possible solution )9 ! conditions /at$4 ,b if "o ( ! eight />s _h n already be5 plac$ z %[n1 !re w\ld !n h be5 eight ways ( arrang+ ! />s accord+ to ?is s*eme1 if we c\nt rev]sals & reflections z diff]5t4 ,if y turn ?is page r\nd s t ea* side is 9 turn at ! bottom1 y w get ! f\r rev]sals2 & if y reflect ea* ( ^! 9 a mirror1 y w get ! f\r reflections4 ,^! are1 !re=e1 m]ely eight aspects ( "o fundam5tal solution4 ,b )\t t fir/ /> be+ s plac$1 !re is ano!r fundam5tal solution1 z %[n 9 ! second diagram4 ,b ?is arrangem5t be+ 9 a way symmetrical1 only produces f\r diff]5t aspects by rev]sal & reflection4 ,illu/ration #cjb4--;,a ,,problem ,,9 ,,mosaics4 ,illu/ration ,! diagram %[s h[ ! tiles may be re>rang$4 ,z be=e1 "o yell[ & "o purple tile are disp5s$ )4 ,i w "h po9t out t 9 ! previ\s arrangem5t ! yell[ & purple tiles 9 ! sev5? r[ might h *ang$ places1 b no o!r arrangem5t 0 possible4 #cjc4--,,"u ,,! ,,veil4 ,"s s*emes give m diagonal read+s ( f\r lett]s ?an o!rs1 & we are at fir/ tempt$ to fav\r ^!2 b ?is is a false sc5t1 because :at y appe> to ga9 9 ?is direction y lose 9 o!rs4 ,( c\rse x imm$iately occurs to ! solv] t e ,,live or ,,evil is wor? twice z mu* z any o!r ^w1 s9ce x reads bo? ways & always c\nts z #b4 ,?is is an important consid]ation1 ?\gh sometimes ^? arrangem5ts t conta9 mo/ read+s ( ^! two words are fruitless 9 o!r words1 & we lose 9 ! g5]al c\nt4 ,illu/ration3 ,i ,v ,e ,l ,e ,v ,l ,i ,l ,i ,v ,e ,i ,v ,e ,l ,e ,l ,v ,i ,l ,i ,i ,e ,v _/,v ,e ,l ,i ,i ,v ,e ,l ,! above diagram is 9 accordance ) ! conditions requir+ no lett] to be 9 l9e ) ano!r simil> lett]1 & x gives tw5ty read+s ( ! five words--six horizontally1 six v]tically1 f\r 9 ! diagonals indicat$ by ! arr[s on ! left1 & f\r 9 ! diagonals indicat$ by ! arr[s on ! "r4 ,?is is ! maximum4 ,f\r sets ( eight lett]s may be plac$ on ! bo>d ( sixty-f\r squ>es 9 z _m z #fjd diff]5t ways1 )\t any lett] ev] be+ 9 l9e ) a simil> "o4 ,?is does n c\nt rev]sals & reflections z diff]5t1 & x does n take into consid]ation ! actual p]mutations ( ! lett]s among !mselves2 t is1 = example1 mak+ ! ,l's *ange places ) ! ,e's4 ,n[ x is a s+ul> fact t n only d ! tw5ty ^w-read+s t ,i h giv5 prove to be ! real maximum1 b !re is actually only t "o arrangem5t f :i* ?is maximum may be obta9$4 ,b if y make ! ,v's *ange places ) ! ,i's1 & ! ,l's ) ! ,e's1 9 ! solution giv51 y /ill get tw5ty read+s--! same numb] z be=e 9 e direction4 ,!re=e !re are two ways ( gett+ ! maximum f ! same arrangem5t4 ,! m9imum numb] ( read+s is z]o--t is1 ! lett]s c be s arrang$ t no ^w c be read 9 any ( ! directions4 #cjd4--;,,ba*et',s ,,squ>e4 ,illu/ration3 #a ,illu/ration3 #b ,illu/ration3 #c ,illu/ration3 #d ,let u use ! lett]s ,a1 ,k1 ,q1 ,j1 to d5ote ace1 k+1 que51 jack2 & ,d1 ,s1 ,h1 ,c1 to d5ote diamonds1 spades1 he>ts1 clubs4 ,9 ,diagrams #a & #b we h ! two available ways ( arrang+ ei!r gr\p ( lett]s s t no two simil> lett]s %all be 9 l9e--?\gh a qu>t]-turn ( #a w give u ! arrangem5t 9 #b4 ,if we sup]impose or comb9e ^! two squ>es1 we get ! arrangem5t ( ,diagram #c1 :i* is "o solution4 ,b 9 ea* squ>e we may put ! lett]s 9 ! top l9e 9 tw5ty-f\r diff]5t ways )\t alt]+ ! s*eme ( arrangem5t4 ,?us1 9 ,diagram #d ! ,s's are simil>ly plac$ to ! ,d's 9 #b1 ! ,h's to ! ,s's1 ! ,c's to ! ,h's1 & ! ,d's to ! ,c's4 ,x cle>ly foll[s t !re mu/ be #bdbd #egf ways ( comb9+ ! two primitive arrangem5ts4 ,b ! error t ,labosne fell into 0 t ( assum+ t ! ,a1 ,k1 ,q1 ,j mu/ be arrang$ 9 ! =m #a1 & ! ,d1 ,s1 ,h1 ,c 9 ! =m #b4 ,he ?us includ$ reflections & half-turns1 b n qu>t]-turns4 ,!y may obvi\sly be int]*ang$4 ,s t ! correct answ] is #b #egf #a1aeb1 c\nt+ reflections & rev]sals z diff]5t4 ,put 9 ano!r mann]1 ! pairs 9 ! top r[ may be writt5 9 #af #i #d #a #egf diff]5t ways1 & ! squ>e !n complet$ 9 #b ways1 mak+ #a1aeb ways 9 all4 #cje4--,,! ,,?irty-,,six ,,lett] ,,blocks4 ,i po9t$ out t x 0 impossible to get all ! lett]s into ! box "u ! conditions1 b ! puzzle 0 to place z _m z possible4 ,?is requires a little judgm5t & c>eful inve/igation1 or we are liable to jump to ! ha/y conclusion t ! prop] way to solve ! puzzle mu/ be fir/ to place all six ( "o lett]1 !n all six ( ano!r lett]1 & s on4 ,z !re is only "o s*eme 7) its rev]sals7 = plac+ six simil> lett]s s t no two %all be 9 a l9e 9 any direction1 ! read] w f9d t aft] he has plac$ f\r diff]5t k9ds ( lett]s1 six times ea*1 e place is occupi$ except ^? twelve t =m ! two long diagonals4 ,he is1 !re=e1 unable to place m ?an two ea* ( 8 la/ two lett]s1 & !re are eight blanks left4 ,i give su* an arrangem5t 9 ,diagram #a4 ,illu/ration3 #a ,illu/ration3 #b ,! secret1 h[ev]1 consi/s 9 n try+ ?us to place all six ( ea* lett]4 ,x w be f\nd t if we cont5t ourselves ) plac+ only five ( ea* lett]1 ?is numb] 7?irty 9 all7 may be got into ! box1 & !re w be only six blanks4 ,b ! correct solution is to place six ( ea* ( two lett]s & five ( ea* ( ! rema9+ f\r4 ,an exam9ation ( ,diagram #b w %[ t !re are six ea* ( ,c & ,d1 & five ea* ( ,a1 ,b1 ,e1 & ,f4 ,!re are1 !re=e1 only f\r blanks left1 & no lett] is 9 l9e ) a simil> lett] 9 any direction4 #cjf4--,,! ,,cr[d$ ,,*essbo>d4 ,illu/ration ,"h is ! solution4 ,only #h que5s or #h rooks c be plac$ on ! bo>d )\t attack1 :ile ! greate/ numb] ( bi%ops is #ad1 & ( knights #cb4 ,b z all ^! knights mu/ be plac$ on squ>es ( ! same col\r1 :ile ! que5s occupy f\r ( ea* col\r & ! bi%ops #g ( ea* col\r1 x foll[s t only #ba knights c be plac$ on ! same col\r 9 ?is puzzle4 ,m ?an #ba knights c be plac$ alone on ! bo>d if we use bo? col\rs1 b ,i h n succe$$ 9 plac+ m ?an #ba on ! cr[d$ *essbo>d4 ,i believe ! above solution conta9s ! maximum numb] ( pieces1 b possibly "s +5i\s read] may succe$ 9 gett+ 9 ano!r knight4 #cjg4--,,! ,,col\r$ ,,c\nt]s4 ,! c\nt]s may be arrang$ 9 ?is ord]3-- ,r#a1 ,b#b1 ,y#c1 ,o#d1 ,,gs4 ,y#d1 ,o#e1 ,g#a1 ,r#b1 ,b#c4 ,g#b1 ,r#c1 ,b#d1 ,y#e1 ,o#a4 ,b#e1 ,y#a1 ,o#b1 ,g#c1 ,r#d4 ,o#c1 ,g#d1 ,r#e1 ,b#a1 ,y#b4 #cjh4--,,! ,,g5tle ,,art ,,( ,,/amp-,,lick+4 ,! foll[+ arrangem5t %[s h[ sixte5 /amps may be /uck on ! c>d1 "u ! conditions1 ( a total value ( fifty p5ce1 or #ds4 #b;d43-- ,illu/ration ,if1 aft] plac+ ! f\r #e;d4 /amps1 ! read] is tempt$ to place f\r #d;d4 /amps also1 he c aft]w>ds only place two ( ea* ( ! ?ree o!r d5om9ations1 ?us los+ two spaces & c\nt+ no m ?an =ty-eight p5ce1 or #ds4 ,?is is ! pitfall t 0 h9t$ at4 7,comp>e ) ,no4 #dc1 ,cant]bury ,puzzles47 #cji4--,,! ,,=ty-,,n9e ,,c\nt]s4 ,! c\nt]s may be arrang$ 9 ?is ord]3-- ,a#a1 ,b#b1 ,c#c1 ,d#d1 ,e#e1 ,f#f1 ,g#g4 ,f#d1 ,g#e1 ,a#f1 ,b#g1 ,c#a1 ,d#b1 ,e#c4 ,d#g1 ,e#a1 ,f#b1 ,g#c1 ,a#d1 ,b#e1 ,c#f4 ,b#c1 ,c#d1 ,d#e1 ,e#f1 ,f#g1 ,g#a1 ,a#b4 ,g#f1 ,a#g1 ,b#a1 ,c#b1 ,d#c1 ,e#d1 ,f#e4 ,e#b1 ,f#c1 ,g#d1 ,a#e1 ,b#f1 ,c#g1 ,d#a4 ,c#e1 ,d#f1 ,e#g1 ,f#a1 ,g#b1 ,a#c1 ,b#d4 #caj4--,,! ,,?ree ,,%eep4 ,! numb] ( diff]5t ways 9 :i* ! ?ree %eep may be plac$ s t e p5 %all always be ei!r occupi$ or 9 l9e ) at lea/ "o %eep is =ty-sev54 ,! foll[+ table1 if us$ ) ! key 9 ,diagram #a1 w enable ! read] to place !m 9 all ^! ways3-- ---------------------------------------- --------- ,no4 ( ,two ,%eep4 ,?ird ,%eep4 ,ways4 ---------------------------------------- --------- ,a & ,b ,c1 ,e1 ,g1 ,k1 ,l1 ,n1 or ,p #g ,a & ,c ,i1 ,j1 ,k1 or ,o #d ,a & ,d ,m1 ,n1 or ,j #c ,a & ,f ,j1 ,k1 ,l1 or ,p #d ,a & ,g ,h1 ,j1 ,k1 ,n1 ,o1 or ,p #f ,a & ,h ,k1 ,l1 ,n1 or ,o #d ,a & ,o ,k or ,l #b ,b & ,c ,n #a ,b & ,e ,f1 ,h1 ,k1 or ,l #d ,b & ,f ,g1 ,j1 ,n1 or ,o #d ,b & ,g ,k1 ,l1 or ,n #c ,b & ,h ,j or ,n #b ,b & ,j ,k or ,l #b ,f & ,g ,j #a ---- #dg ---------------------------------------- --------- ,?is1 ( c\rse1 means t if y place %eep 9 ! p5s m>k$ ,a & ,b1 !n !re are sev5 diff]5t p5s 9 :i* y may place ! ?ird %eep1 giv+ sev5 diff]5t solutions4 ,x 0 und]/ood t rev]sals & reflections d n c\nt z diff]5t4 ,if "o p5 at lea/ is to be n 9 l9e ) a %eep1 !re w\ld be ?irty solutions to t problem4 ,if we c\nt$ all ! rev]sals & reflections ( ^! #dg & #cj cases respectively z diff]5t1 _! total w\ld be #efj1 :i* is ! numb] ( diff]5t ways 9 :i* ! %eep may be plac$ 9 ?ree p5s )\t any conditions4 ,i w rem>k t !re are ?ree ways 9 :i* two %eep may be plac$ s t e p5 is occupi$ or 9 l9e1 z 9 ,diagrams #b1 #c1 & #d1 b 9 e case ea* %eep is 9 l9e ) its companion4 ,!re are only two ways 9 :i* ?ree %eep may be s plac$ t e p5 %all be occupi$ or 9 l9e1 b no %eep 9 l9e ) ano!r4 ,^! ,i %[ 9 ,diagrams #e & #f4 ,f9ally1 !re is only "o way 9 :i* ?ree %eep may be plac$ s t at lea/ "o p5 %all n be 9 l9e ) a %eep & yet no %eep 9 l9e ) ano!r4 ,place ! %eep 9 ,c1 ,e1 ,l4 ,?is is practically all !re is to be said on ?is pleasant pa/oral subject4 ,illu/ration #caa4--,,! ,,five ,,dogs ,,puzzle4 ,! diagrams %[ f\r fundam5tally diff]5t solutions4 ,9 ! case ( ,a we c rev]se ! ord]1 s t ! s+le dog is 9 ! bottom r[ & ! o!r f\r %ift$ up two squ>es4 ,also we may use ! next column to ! "r & bo? ( ! two c5tral horizontal r[s4 ,?us ,a gives #h solutions4 ,!n ,b may be rev]s$ & plac$ 9 ei!r diagonal1 giv+ #d solutions4 ,simil>ly ,c w give #d solutions4 ,! l9e 9 ,d be+ symmetrical1 its rev]sal w n be diff]5t1 b x may be dispos$ 9 #d diff]5t directions4 ,we ?us h 9 all #bj diff]5t solutions4 ,illu/ration #cab4--,,! ,,five ,,cresc5ts ,,( ,,byzantium4 ,illu/ration ,if t anci5t ar*itect _h arrang$ 8 five cresc5t tiles 9 ! mann] %[n 9 ! foll[+ diagram1 e tile w\ld h be5 wat*$ ov] by1 or 9 a l9e )1 at lea/ "o cresc5t1 & space w\ld h be5 res]v$ = a p]fectly squ>e c>pet equal 9 area to exactly half ( ! pavem5t4 ,x is a v curi\s fact t1 al?\gh !re are two or ?ree solutions all[+ a c>pet to be laid d[n )9 ! conditions s z to cov] an area ( ne>ly tw5ty-n9e ( ! tiles1 ?is is ! only possible solution giv+ exactly half ! area ( ! pavem5t1 :i* is ! l>ge/ space obta9able4 #cac4--,,que5s ,,& ,,bi%op ,,puzzle4 ,illu/ration3 ,,fig4 #a4 ,illu/ration3 ,,fig4 #b4 ,! bi%op is on ! squ>e orig9ally occupi$ by ! rook1 & ! f\r que5s are s plac$ t e squ>e is ei!r occupi$ or attack$ by a piece4 7,fig4 #a47 ,i po9t$ out 9 #ahii t if f\r que5s are plac$ z %[n 9 ! diagram 7,fig4 #b71 !n ! fif? que5 may be plac$ on any "o ( ! twelve squ>es m>k$ a1 b1 c1 d1 & e2 or a rook on ! two squ>es1 c2 or a bi%op on ! eight squ>es1 a1 b1 & e2 or a pawn on ! squ>e b2 or a k+ on ! f\r squ>es1 b1 c1 & e4 ,! only kn[n arrangem5t = f\r que5s & a knight is t giv5 by ,mr4 ,j4 ,wallis 9 ,! ,/r& ,magaz9e = ,augu/ #aijh1 "h reproduc$4 7,fig4 #c47 ,illu/ration3 ,,fig4 #c4 ,i h record$ a l>ge numb] ( solutions ) f\r que5s & a rook1 or bi%op1 b ! only arrangem5t1 ,i believe1 ) ?ree que5s & two rooks 9 :i* all ! pieces are gu>d$ is t ( :i* ,i give an illu/ration 7,fig4 #d71 fir/ publi%$ by ,dr4 ,c4 ,planck4 ,b ,i h s9ce f\nd ! accompany+ solution ) ?ree que5s1 a rook1 & a bi%op1 ?\gh ! pieces d n protect "o ano!r4 7,fig4 #e47 ,illu/ration3 ,,fig4 #d4 ,illu/ration3 ,,fig4 #e4 #cad4--,,! ,,s\!rn ,,cross4 ,my read]s h be5 s famili>iz$ ) ! fact t x requires at lea/ five planets to attack e "o ( a squ>e arrangem5t ( sixty-f\r />s t _m ( !m h1 p]haps1 got to believe t a l>g] squ>e arrangem5t ( />s mu/ ne$ an increase ( planets4 ,x 0 to correct ?is possible error ( reason+1 & s w>n read]s aga9/ ano!r ( ^? num]\s little pitfalls 9 ! _w ( puzzl$om1 t ,i devis$ ?is new /ell> problem4 ,let me !n /ate at once t1 9 ! case ( a squ>e arrangem5t ( eighty "o />s1 !re are sev]al ways ( plac+ five planets s t e /> %all be 9 l9e ) at lea/ "o planet v]tically1 horizontally1 or diagonally4 ,"h is ! solution to ! ,s\!rn ,cross3 -- ,x w be rememb]$ t ,i said t ! five planets 9 _! new positions w1 ( c\rse1 obscure five o!r />s 9 place ( ^? at pres5t cov]$4 ,?is 0 to exclude an easi] solution 9 :i* only f\r planets ne$ be mov$4 #cae4--,,! ,,hat-,,peg ,,puzzle4 ,! moves w be made q cle> by a ref]5ce to ! diagrams1 :i* %[ ! position on ! bo>d aft] ea* ( ! f\r moves4 ,! d>ts indicate ! successive removals t h be5 made4 ,x w be se5 t at e /age all ! squ>es are ei!r attack$ or occupi$1 & t aft] ! f\r? move no que5 attacks any o!r4 ,9 ! case ( ! la/ move ! que5 9 ! top r[ might also h be5 mov$ "o squ>e f>!r to ! left4 ,?is is1 ,i believe1 ! only solution to ! puzzle4 ,illu/ration3 #a ,illu/ration3 #b ,illu/ration3 #c ,illu/ration3 #d #caf4--,,! ,,amazons4 ,x w be se5 t only ?ree que5s h be5 remov$ f _! positions on ! edge ( ! bo>d1 & t1 z a consequ5ce1 elev5 squ>es 7indicat$ by ! black dots7 are left unattack$ by any que54 ,i w haz>d ! /atem5t t eight que5s _c be plac$ on ! *essbo>d s z to leave m ?an elev5 squ>es unattack$4 ,x is true t we h no rigid pro( ( ?is yet1 b ,i h entirely conv9c$ myself ( ! tru? ( ! /atem5t4 ,!re are at lea/ five diff]5t ways ( arrang+ ! que5s s z to leave elev5 squ>es unattack$4 ,illu/ration #cag4--;,a ,,puzzle ,,) ,,pawns4 ,sixte5 pawns may be plac$ s t no ?ree %all be 9 a /raight l9e 9 any possible direction1 z 9 ! diagram4 ,we reg>d1 z ! conditions requir$1 ! pawns z m]e po9ts on a plane4 ,illu/ration #cah4--,,lion-,,hunt+4 ,!re are #f1dhj ways ( plac+ ! man & ! lion1 if !re are no re/rictions :atev] except t !y mu/ be on diff]5t spots4 ,?is is obvi\s1 because ! man may be plac$ on any "o ( ! #ha spots1 & 9 e case !re are #hj spots rema9+ = ! lion2 !re=e #ha #hj #f1dhj4 ,n[1 if we d$uct ! numb] ( ways 9 :i* ! lion & ! man may be plac$ on ! same pa?1 ! result mu/ be ! numb] ( ways 9 :i* !y w n be on ! same pa?4 ,! numb] ( ways 9 :i* !y may be 9 l9e is f\nd )\t mu* difficulty to be #haf4 ,consequ5tly1 #f1dhj - #haf #e1ffd1 ! requir$ answ]4 ,! g5]al solution is ?is3 #a_/cn7n - #a77cn - n #b74 ,?is is1 ( c\rse1 equival5t to say+ t if we call ! numb] ( squ>es on ! side ( a *essbo>d n1 !n ! =mula %[s ! numb] ( ways 9 :i* two bi%ops may be plac$ )\t attack+ "o ano!r4 ,only 9 ?is case we mu/ divide by two1 because ! two bi%ops h no di/9ct individuality1 & _c produce a diff]5t solution by m]e ex*ange ( places4 #cai4--,,! ,,knight-,,gu>ds4 ,illu/ration3 ,,diagram #a4 ,illu/ration3 ,,diagram #b4 ,! smalle/ possible numb] ( knights ) :i* ?is puzzle c be solv$ is f\rte54 ,x has sometimes be5 assum$ t !re are a great _m diff]5t solutions4 ,z a matt] ( fact1 !re are only ?ree arrangem5ts--n c\nt+ m]e rev]sals & reflections z diff]5t4 ,curi\sly en\gh1 nobody seems ev] to h hit on ! foll[+ simple pro(1 or to h ?\ght ( deal+ ) ! black & ! :ite squ>es sep>ately4 ,illu/ration3 ,,diagram #c4 ,illu/ration3 ,,diagram #d4 ,illu/ration3 ,,diagram #e4 ,sev5 knights c be plac$ on ! bo>d on :ite squ>es s z to attack e black squ>e 9 two ways only4 ,^! are %[n 9 ,diagrams #a & #b4 ,note t ?ree knights occupy ! same position 9 bo? arrangem5ts4 ,x is !re=e cle> t if we turn ! bo>d s t a black squ>e %all be 9 ! top left-h& corn] in/ead ( a :ite1 & place ! knights 9 exactly ! same positions1 we %all h two simil> ways ( attack+ all ! :ite squ>es4 ,i w assume ! read] has made ! two la/ describ$ diagrams on transp>5t pap]1 & m>k$ !m #a;a & #b;a4 ,n[1 by plac+ ! transp>5t ,diagram #a;a ov] #a y w be able to obta9 ! solution 9 ,diagram #c1 by plac+ #b;a ov] #b y w get ,diagram #d1 & by plac+ #b;a ov] #a y w get ,diagram #e4 ,y may n[ try all possible comb9ations ( ^? two pairs ( diagrams1 b y w only get ! ?ree arrangem5ts ,i h giv51 or _! rev]sals & reflections4 ,!re=e ^! ?ree solutions are all t exi/4 #cbj4--,,! ,,rook',s ,,t\r4 ,illu/ration ,! only possible m9imum solutions are %[n 9 ! two diagrams1 :]e x w be se5 t only sixte5 moves are requir$ to p]=m ! feat4 ,mo/ p f9d x difficult to r$uce ! numb] ( moves bel[ sev5te5994 ,illu/ration3 ,,! ,,rook',s ,,t\r4 #cba4--,,! ,,rook',s ,,j\rney4 ,illu/ration ,i %[ ! r\te 9 ! diagram4 ,x w be se5 t ! t5? move l&s u at ! squ>e m>k$ #aj1 & t ! la/ move1 ! tw5ty-fir/1 br+s u to a halt on squ>e #ba4 #cbb4--,,! ,,langui%+ ,,maid54 ,! dott$ l9e %[s ! r\te 9 tw5ty-two /raight pa?s by :i* ! knight may rescue ! maid54 ,x is necess>y1 aft] ent]+ ! fir/ cell1 imm$iately to return be=e ent]+ ano!r4 ,o!rwise a solution w\ld n be possible4 7,see ,! ,gr& ,t\r1 p4 #bjj47 #cbc4--;,a ,,dungeon ,,puzzle4 ,if ! prison] takes ! r\te %[n 9 ! diagram--:]e = cle>ness ! doorways are omitt$--he w succe$ 9 visit+ e cell once1 & only once1 9 z _m z fifty-sev5 /raight l9es4 ,no rook's pa? ov] ! *essbo>d c exce$ ?is numb] ( moves4 ,illu/ration3 ,,! ,,langui%+ ,,maid5 ,illu/ration3 ,a ,,dungeon ,,puzzle4 #cbd4--,,! ,,lion ,,& ,,! ,,man4 ,fir/ ( all1 ! fewe/ possible /raight l9es 9 ea* case are tw5ty-two1 & 9 ord] t no cell may be visit$ twice x is absolutely necess>y t ea* %\ld pass into "o cell & !n imm$iately visit ! "o f :i* he />t$1 aft]w>ds proce$+ by way ( ! second available cell4 ,9 ! foll[+ diagram ! man's r\te is indicat$ by ! unbrok5 l9es1 & ! lion's by ! dott$ l9es4 ,x w be f\nd1 if ! two r\tes are foll[$ cell by cell ) two p5cil po9ts1 t ! lion & ! man nev] meet4 ,b !re 0 "o little po9t t ought n to be ov]look$--!y occasionally got glimpses ( "o ano!r4 ,n[1 if we take "o r\te = ! man & m]ely rev]se x = ! lion1 we inv>iably f9d t1 go+ at ! same spe$1 !y nev] get a glimpse ( "o ano!r4 ,b 9 our diagram x w be f\nd t ! man & ! lion are 9 ! cells m>k$ ,a at ! same mom5t1 & may see "o ano!r ?r\gh ! op5 doorways2 :ile ! same happ5s :5 !y are 9 ! two cells m>k$ ,b1 ! upp] lett]s indicat+ ! man & ! l[] ! lion4 ,9 ! fir/ case ! lion goes /raight = ! man1 :ile ! man appe>s to attempt to get 9 ! re> ( ! lion2 9 ! second case x looks suspici\sly l runn+ away f "o ano!r6 ,illu/ration #cbe4--;,,an ,,episcopal ,,visitation4 ,illu/ration ,9 ! diagram ,i %[ h[ ! bi%op may be made to visit e "o ( 8 :ite p>i%es 9 sev5te5 moves4 ,x is obvi\s t we mu/ />t f "o corn] squ>e & end at ! "o t is diagonally opposite to x4 ,! puzzle _c be solv$ 9 few] ?an sev5te5 moves4 #cbf4--;,a ,,new ,,c\nt] ,,puzzle4 ,play z foll[s3 #b--c1 #i--d1 #aj--g1 #c--h1 #d--b1 #g--e1 #h--f1 #e--aj1 #f--i1 #b--e1 #a--f1 #f--d1 #e--c1 #aj--h1 #d--g1 #c--b1 #h--a1 #g--aj4 ,! :ite c\nt]s h n[ *ang$ places ) ! r$ ones1 9 eighte5 moves1 )\t break+ ! conditions4 #cbg4--;,a ,,new ,,bi%op',s ,,puzzle4 ,illu/ration3 ,a ,illu/ration3 ,b ,play z foll[s1 us+ ! notation indicat$ by ! numb]$ squ>es 9 ,diagram ,a3-- ,:ite4 ,black4 ,:ite4 ,black4 #a4 #ah--ae #a4 #c--f #aj4 #bj--aj #aj4 #a--aa #b4 #ag--h #b4 #d--ac #aa4 #c--i #aa4 #ah--ab #c4 #ai--ad #c4 #b--g #ab4 #aj--ac #ab4 #aa--h #d4 #ae--e #d4 #f--af #ac4 #ai--af #ac4 #b--e #e4 #h--c #e4 #ac-ah #ad4 #af--a #ad4 #e--bj #f4 #ad--i #f4 #g--ab #ae4 #i--f #ae4 #ab--ae #g4 #e--aj #g4 #af-aa #af4 #ac-g #af4 #h--ad #h4 #i--ai #h4 #ab--b #ag4 #f--c #ag4 #ae-ah #i4 #aj--d #i4 #aa-ag #ah4 #g--b #ah4 #ad--ai ,diagram ,b %[s ! position aft] ! n9? move4 ,bi%ops at #a & #bj h n yet mov$1 b #b & #ai h salli$ =? & return$4 ,9 ! end1 #a & #ai1 #b & #bj1 #c & #ag1 & #d & #ah w h ex*ang$ places4 ,note ! position aft] ! ?irte5? move4 #cbh4--,,! ,,que5',s ,,t\r4 ,illu/ration ,! annex$ diagram %[s a second way ( p]=m+ ! ,que5's ,t\r4 ,if y break ! l9e at ! po9t ,j & erase ! %ort] portion ( t l9e1 y w h ! requir$ pa? solution = any ,j squ>e4 ,if y break ! l9e at ,i1 y w h a non-re-entrant solution />t+ f any ,i squ>e4 ,& if y break ! l9e at ,g1 y w h a solution = any ,g squ>e4 ,! ,que5's ,t\r previ\sly giv5 may be simil>ly brok5 at ?ree diff]5t places1 b ,i seiz$ ! opportunity ( exhibit+ a second t\r4 #cbi4--,,! ,,/> ,,puzzle4 ,! illu/ration expla9s itself4 ,! />s are all /ruck out 9 f\rte5 /raight /rokes1 />t+ & end+ at a :ite />4 ,illu/ration #ccj4--,,! ,,ya*t ,,race4 ,! diagram expla9s itself4 ,! numb]s w %[ ! direction ( ! l9es 9 _! prop] ord]1 & x w be se5 t ! sev5? c\rse ends at ! flag-buoy1 z /ipulat$4 ,illu/ration #cca4--,,! ,,sci5tific ,,skat]4 ,9 ?is case we g beyond ! b\nd>y ( ! squ>e4 ,ap>t f t1 ! moves are all que5 moves4 ,!re are ?ree or f\r ways 9 :i* x c be done4 ,"h is "o way ( p]=m+ ! feat3-- ,illu/ration ,x w be se5 t ! skat] /rikes out all ! />s 9 "o cont9u\s j\rney ( f\rte5 /raight l9es1 return+ to ! po9t f :i* he />t$4 ,to foll[ ! skat]'s c\rse 9 ! diagram x is necess>y always to g z f> z we c 9 a /raight l9e be=e turn+4 #ccb4--,,! ,,=ty-,,n9e ,,/>s4 ,! illu/ration %[s h[ all ! />s may be /ruck out 9 twelve /raight /rokes1 beg9n+ & end+ at a black />4 ,illu/ration #ccc4--,,! ,,que5',s ,,j\rney4 ,! correct solution to ?is puzzle is %[n 9 ! diagram by ! d>k l9e4 ,! five moves indicat$ w take ! que5 ! greate/ di/ance t x is possible = h] to g 9 five moves1 )9 ! conditions4 ,! dott$ l9e %[s ! r\te t mo/ p sugge/1 b x is n q s long z ! o!r4 ,let u assume t ! di/ance f ! c5tre ( any squ>e to ! c5tre ( ! next 9 ! same horizontal or v]tical l9e is #b in*es1 & t ! que5 travels f ! c5tre ( h] orig9al squ>e to ! c5tre ( ! "o at :i* %e re/s4 ,!n ! fir/ r\te w be f\nd to exce$ #fg4i in*es1 :ile ! dott$ r\te is less ?an #fg4h in*es4 ,! diff]5ce is small1 b x is suffici5t to settle ! po9t z to ! long] r\te4 ,all o!r r\tes are %ort] /ill ?an ^! two4 ,illu/ration #ccd4--,,/4 ,,george ,,& ,,! ,,dragon4 ,we select = ! solution ( ?is puzzle "o ( ! prettie/ designs t c be =m$ by repres5t+ ! moves ( ! knight by l9es f squ>e to squ>e4 ,! *equ]+ ( ! squ>es is omitt$ to give great] cle>ness4 ,/4 ,george ?us slays ! ,dragon 9 /rict accordance ) ! conditions & 9 ! elegant mann] we %\ld expect ( him4 ,illu/ration3 ,/4 ,george & ! ,dragon4 #cce4--;,,f>m] ,,lawr5ce',s ,,cornfields4 ,!re are num]\s solutions to ?is little agricultural problem4 ,! v]sion ,i give 9 ! next column is r curi\s on acc\nt ( ! long p>allel /raight l9es =m$ by "s ( ! moves4 ,illu/ration3 ,f>m] ,lawr5ce's ,cornfields4 #ccf4--,,! ,,greyh\nd ,,puzzle4 ,!re are sev]al int]e/+ po9ts involv$ 9 ?is "q4 ,9 ! fir/ place1 if we _h made no /ipulation z to ! positions ( ! two ends ( ! /r+1 x is q impossible to =m any su* /r+ unless we beg9 & end 9 ! top & bottom r[ ( k5nels4 ,we may beg9 9 ! top r[ & end 9 ! bottom 7or1 ( c\rse1 ! rev]se71 or we may beg9 9 "o ( ^! r[s & end 9 ! same4 ,b we c nev] beg9 or end 9 "o ( ! two c5tral r[s4 ,our places ( />t+ & end+1 h[ev]1 7 fix$ = u4 ,yet ! fir/ half ( our r\te mu/ be conf9$ entirely to ^? squ>es t are di/+ui%$ 9 ! foll[+ diagram by circles1 & ! second half w !re=e be conf9$ to ! squ>es t are n circl$4 ,! squ>es res]v$ = ! two half-/r+s w be se5 to be symmetrical & simil>4 ,! next po9t is t ! fir/ half-/r+ mu/ end 9 "o ( ! c5tral r[s1 & ! second half-/r+ mu/ beg9 9 "o ( ^! r[s4 ,?is is n[ obvi\s1 because !y h to l9k toge!r to =m ! complete /r+1 & e squ>e on an outside r[ is connect$ by a knight's move ) simil> squ>es only--t is1 circl$ or non-circl$ z ! case may be4 ,! half-/r+s c1 !re=e1 only be l9k$ 9 ! two c5tral r[s4 ,illu/ration ,n[1 !re are j eight diff]5t fir/ half-/r+s1 & consequ5tly also eight second half-/r+s4 ,we %all see t ^! comb9e to =m twelve complete /r+s1 :i* is ! total numb] t exi/ & ! correct solution ( our puzzle4 ,i d n propose to give all ! r\tes at l5g?1 b ,i w s f> indicate !m t if ! read] has dropp$ any he w be able to discov] :i* !y are & "w !m out = himself )\t any difficulty4 ,! foll[+ numb]s apply to ^? 9 ! above diagram4 ,! eight fir/ half-/r+s are3 #a to #f 7#b r\tes72 #a to #h 7#a r\te72 #a to #aj 7#c r\tes72 #a to #ab 7#a r\te72 & #a to #ad 7#a r\te74 ,! eight second half-/r+s are3 #g to #bj 7#a r\te72 #i to #bj 7#a r\te72 #aa to #bj 7#c r\tes72 #ac to #bj 7#a r\te72 & #ae to #bj 7#b r\tes74 ,e diff]5t way 9 :i* y c l9k "o half-/r+ to ano!r gives a diff]5t solution4 ,^! l9k+s w be f\nd to be z foll[s3 #f to #ac 7#b cases72 #aj to #ac 7#c cases72 #h to #aa 7#c cases72 #h to #ae 7#b cases72 #ab to #i 7#a case72 & #ad to #g 7#a case74 ,!re are1 !re=e1 twelve diff]5t l9k+s & twelve diff]5t answ]s to ! puzzle4 ,! r\te giv5 9 ! illu/ration ) ! greyh\nd w be f\nd to consi/ ( "o ( ! ?ree half-/r+s #a to #aj1 l9k$ to ! half-/r+ #ac to #bj4 ,x %\ld be not$ t t5 ( ! solutions are produc$ by five di/9ctive r\tes & _! rev]sals--t is1 if y indicate ^! five r\tes by l9es & !n turn ! diagrams upside d[n y w get ! five o!r r\tes4 ,! rema9+ two solutions are symmetrical 7^! are ! cases :]e #ab to #i & #ad to #g are ! l9ks71 & consequ5tly !y d n produce new solutions by rev]sal4 #ccg4--,,! ,,f\r ,,kang>oos4 ,illu/ration ,a pretty symmetrical solution to ?is puzzle is %[n 9 ! diagram4 ,ea* ( ! f\r kang>oos makes 8 little excursion & returns to 8 corn]1 )\t ev] ent]+ a squ>e t has be5 visit$ by ano!r kang>oo & )\t cross+ ! c5tral l9e4 ,x w at once occur to ! read]1 z a possible improvem5t ( ! puzzle1 to divide ! bo>d by a c5tral v]tical l9e & make ! condition t ?is also %all n be cross$4 ,?is w\ld mean t ea* kang>oo _h to conf9e himself to a squ>e #d by #d1 b x w\ld be q impossible1 z ,i %all expla9 9 ! next two puzzles4 #cch4--,,! ,,bo>d ,,9 ,,comp>tm5ts4 ,illu/ration ,9 attempt+ to solve ?is problem x is fir/ necess>y to take ! two di/9ctive comp>tm5ts ( tw5ty & twelve squ>es respectively & analyse !m ) a view to det]m9+ :]e ! necess>y po9ts ( entry & exit lie4 ,9 ! case ( ! l>g] comp>tm5t x w be f\nd t to complete a t\r ( x we mu/ beg9 & end on two ( ! outside squ>es on ! long sides4 ,b ?\gh y may />t at any "o ( ^! t5 squ>es1 y are re/rict$ z to ^? at :i* y c end1 or 7:i* is ! same ?+7 y may end at :i*ev] ( ^! y l1 provid$ y beg9 y\r t\r at c]ta9 p>ticul> squ>es4 ,9 ! case ( ! small] comp>tm5t y are compell$ to beg9 & end at "o ( ! six squ>es ly+ at ! two n>r[ ends ( ! comp>tm5ts1 b simil> re/rictions apply z 9 ! o!r in/ance4 ,a v little ?\ght w %[ t 9 ! case ( ! two small comp>tm5ts y mu/ beg9 & f9i% at ! ends t lie toge!r1 & x !n foll[s t ! t\rs 9 ! l>g] comp>tm5ts mu/ also />t & end on ! contigu\s sides4 ,9 ! diagram giv5 ( "o ( ! possible solutions x w be se5 t !re are eight places at :i* we may />t ?is p>ticul> t\r2 b !re is only "o r\te 9 ea* case1 because we mu/ complete ! comp>tm5t 9 :i* we f9d ourself be=e pass+ into ano!r4 ,9 any solution we %all f9d t ! squ>es di/+ui%$ by />s mu/ be ent]+ or exit po9ts1 b ! law ( rev]sals leaves u ! option ( mak+ ! o!r connections ei!r at ! diamonds or at ! circles4 ,9 ! solution work$ out ! diamonds are us$1 b o!r v>iations occur 9 :i* ! circle squ>es are employ$ in/ead4 ,i ?9k ^! rem>ks expla9 all ! ess5tial po9ts 9 ! puzzle1 :i* is di/9ctly in/ructive & int]e/+4 #cci4--,,! ,,f\r ,,knights' ,,t\rs4 ,illu/ration ,x w be se5 9 ! illu/ration h[ a *essbo>d may be divid$ into f\r p>ts1 ea* ( ! same size & %ape1 s t a complete re-entrant knight's t\r may be made on ea* portion4 ,!re is only "o possible r\te = ea* knight & its rev]sal4 #cdj4--,,! ,,cubic ,,knight',s ,,t\r4 ,illu/ration ,if ! read] %\ld cut out ! above diagram1 fold x 9 ! =m ( a cube1 & /ick x toge!r by ! /rips left = t purpose at ! edges1 he w\ld h an int]e/+ little curiosity4 ,or he c make "o on a l>g] scale = himself4 ,x w be f\nd t if we imag9e ! cube to h a complete *essbo>d on ea* ( its sides1 we may />t ) ! knight on any "o ( ! #chd squ>es1 & make a complete t\r ( ! cube1 always return+ to ! />t+-po9t4 ,! me?od ( pass+ f "o side ( ! cube to ano!r is easily und]/ood1 b1 ( c\rse1 ! difficulty consi/$ 9 f9d+ ! prop] po9ts ( entry & exit on ea* bo>d1 ! ord] 9 :i* ! diff]5t bo>ds %\ld be tak51 & 9 gett+ arrangem5ts t w\ld comply ) ! requir$ conditions4 #cda4--,,! ,,f\r ,,frogs4 ,! fewe/ possible moves1 c\nt+ e move sep>ately1 are sixte54 ,b ! puzzle may be solv$ 9 sev5 plays1 z foll[s1 if any numb] ( successive moves by "o frog c\nt z a s+le play4 ,all ! moves conta9$ )9 a bracket are a s+le play2 ! numb]s ref] to ! toad/ools3 7#a--e71 7#c--g1 #g--a71 7#h--d1 #d--c1 #c--g71 7#f--b1 #b--h1 #h--d1 #d--c71 7#e--f1 #f--b1 #b--h71 7#a--e1 #e--f71 7#g--a74 ,?is is ! famili> old puzzle by ,gu>9i1 prop\nd$ 9 #aeab1 & ,i give x "h 9 ord] to expla9 my buttons & /r+ me?od ( solv+ ?is class ( mov+-c\nt] problem4 ,diagram ,a %[s ! old way ( pres5t+ ,gu>9i's puzzle1 ! po9t be+ to make ! :ite knights *ange places ) ! black ones4 ,9 ,! ,f\r ,frogs pres5tation ( ! idea ! possible directions ( ! moves are indicat$ by l9es1 to obviate ! necessity ( ! read]'s und]/&+ ! nature ( ! knight's move 9 *ess4 ,b x w at once be se5 t ! two problems are id5tical4 ,! c5tral squ>e c1 ( c\rse1 be ignor$1 s9ce no knight c ev] ent] x4 ,n[1 reg>d ! toad/ools z buttons & ! connect+ l9es z /r+s1 z 9 ,diagram ,b4 ,!n by dis5tangl+ ^! /r+s we c cle>ly pres5t ! diagram 9 ! =m %[n 9 ,diagram ,c1 :]e ! relation%ip betwe5 ! buttons is precisely ! same z 9 ,b4 ,any solution on ,c w be applicable to ,b1 & to ,a4 ,place y\r :ite knights on #a & #c & y\r black knights on #f & #h 9 ! ,c diagram1 & ! simplicity ( ! solution w be v evid5t4 ,y h simply to move ! knights r\nd ! circle 9 "o direction or ! o!r4 ,play ov] ! moves giv5 above1 & y w f9d t e little difficulty has disappe>$4 ,illu/rations3 ,a ,b ,c ,d ,e ,9 ,diagram ,d ,i give ano!r famili> puzzle t fir/ appe>$ 9 a book publi%$ 9 ,brussels 9 #aghi1 ,les ,petites ,av5tures de ,j]ome ,%>p4 ,place sev5 c\nt]s on sev5 ( ! eight po9ts 9 ! foll[+ mann]4 ,y mu/ always t\* a po9t t is vacant ) a c\nt]1 & !n move x along a /raight l9e lead+ f t po9t to ! next vacant po9t 79 ei!r direction71 :]e y deposit ! c\nt]4 ,y proce$ 9 ! same way until all ! c\nt]s are plac$4 ,rememb] y always t\* a vacant place & slide ! c\nt] f x to ! next place1 :i* mu/ be also vacant4 ,n[1 by ! buttons & /r+ me?od ( simplification we c trans=m ! diagram into ,e4 ,!n ! solution becomes obvi\s4 ,always move to ! po9t t y la/ mov$ f4 ,?is is n1 ( c\rse1 ! only way ( plac+ ! c\nt]s1 b x is ! simple/ solution to c>ry 9 ! m9d4 ,!re are sev]al puzzles 9 ?is book t ! read] w f9d l5d !mselves readily to ?is me?od4 #cdb4--,,! ,,m&>9',s ,,puzzle4 ,! r p]plex+ po9t t ! solv] has to decide = himself 9 attack+ ?is puzzle is :e!r ! %ad$ numb]s 7^? t are %[n 9 _! "r places7 are m]e dummies or n4 ,n9ety-n9e p]sons out ( a hundr$ might =m ! op9ion t !re c be no advantage 9 mov+ any ( !m1 b if s !y w\ld be wrong4 ,! %orte/ solution )\t mov+ any %ad$ numb] is 9 ?irty-two moves4 ,b ! puzzle c be solv$ 9 ?irty moves4 ,! trick lies 9 mov+ ! #f1 or ! #ae1 on ! second move & replac+ x on ! n9ete5? move4 ,"h is ! solution3 #b1 #f1 #ac1 #d1 #a1 #ba1 #d1 #a1 #aj1 #b1 #ba1 #aj1 #b1 #e1 #bb1 #af1 #a1 #ac1 #f1 #ai1 #aa1 #b1 #e1 #bb1 #af1 #e1 #ac1 #d1 #aj1 #ba4 ,?irty moves4 #cdc4--;,,ex]cise ,,= ,,prison]s4 ,!re are eighty diff]5t arrangem5ts ( ! numb]s 9 ! =m ( a p]fect knight's pa?1 b only =ty ( ^! c be rea*$ )\t two m5 ev] be+ 9 a cell at ! same "t4 ,two is ! greate/ numb] ( m5 t c be giv5 a complete re/1 & ?\gh ! knight's pa? c be arrang$ s z to leave ei!r #g & #ac1 #h & #ac1 #e & #g1 or #e & #ac 9 _! orig9al positions1 ! foll[+ f\r arrangem5ts1 9 :i* #g & #ac are unmov$1 are ! only ones t c be rea*$ "u ! mov+ conditions4 ,x !re=e resolves itself into f9d+ ! fewe/ possible moves t w lead up to "o ( ^! positions4 ,?is is c]ta9ly no easy matt]1 & no rigid rules c be laid d[n = arriv+ at ! correct answ]4 ,x is l>gely a matt] = individual judgm5t1 pati5t exp]im5t1 & a %>p eye = revolutions & position4 ,a -------- #f #aajae -------- #iab #g #d -------- #b #eadaa -------- #ac #h #c9999 -------- ,b -------- #f #aajae -------- #aaad #g #d -------- #b #eab #i -------- #ac #h #c9999 -------- ,c -------- #f #i #dae -------- #aab #gaj -------- #h #ead #c -------- #ac #baa9999 -------- ,d -------- #faa #dae -------- #aad #gaj -------- #h #eab #c -------- #ac #b #i9999 -------- ,illu/ration3 ,a1 ,b1 ,c1 ,d ,z a matt] ( fact1 ! position ,c c be rea*$ 9 z few z sixty-six moves 9 ! foll[+ mann]3 #ab1 #aa1 #ae1 #ab1 #aa1 #h1 #d1 #c1 #b1 #f1 #e1 #a1 #f1 #e1 #aj1 #ae1 #h1 #d1 #c1 #b1 #e1 #aj1 #ae1 #h1 #d1 #c1 #b1 #e1 #aj1 #ae1 #h1 #d1 #ab1 #aa1 #c1 #b1 #e1 #aj1 #ae1 #f1 #a1 #h1 #d1 #i1 #h1 #a1 #f1 #d1 #i1 #ab1 #b1 #e1 #aj1 #ae1 #d1 #i1 #ab1 #b1 #e1 #c1 #aa1 #ad1 #b1 #e1 #ad1 #aa #ff moves4 ,?\gh ?is is ! %orte/ t ,i "k (1 & ,i d n ?9k x c be beat51 ,i _c /ate positively t !re is n a %ort] way yet to be discov]$4 ,! mo/ tempt+ arrangem5t is c]ta9ly ,a2 b ?+s are n :at !y seem1 & ,c is really ! easie/ to rea*4 ,if ! bottom left-h& corn] cell might be left vacant1 ! foll[+ is a solution 9 =ty-five moves by ,mr4 ,r4 ,elrick3 #ae1 #aa1 #aj1 #i1 #ac1 #ad1 #aa1 #aj1 #g1 #h1 #d1 #c1 #h1 #f1 #i1 #g1 #ab1 #d1 #f1 #i1 #e1 #ac1 #g1 #e1 #ac1 #a1 #b1 #ac1 #e1 #g1 #a1 #b1 #ac1 #h1 #c1 #f1 #i1 #ab1 #g1 #aa1 #ad1 #a1 #aa1 #ad1 #a4 ,b e man has mov$4 #cdd4--,,! ,,k5nel ,,puzzle4 ,! fir/ po9t is to make a *oice ( ! mo/ promis+ knight's /r+ & !n consid] ! "q ( rea*+ ! arrangem5t 9 ! fewe/ moves4 ,i am /rongly ( op9ion t ! be/ /r+ is ! "o repres5t$ 9 ! foll[+ diagram1 9 :i* x w be se5 t ea* successive numb] is a knight's move f ! prec$+ "o1 & t five ( ! dogs 7#a1 #e1 #aj1 #ae1 & #bj7 nev] leave _! orig9al k5nels4 ----------------------------- #a #b #c #d #e #a #ah #i #ad #e ----------------------------- #f #g #h #i #aj #h #ac #d #ai #aj ----------------------------- #aa #ab #ac #ad #ae #ag #b #aa #f #ae ----------------------------- #af #ag #ah #ai #bj #ab #g #af #c #bj ----------------------------- #ba #bb #bc #bd #be ----------------------------- ,illu/ration ,?is position may be arriv$ at 9 z few z =ty-six moves1 z foll[s3 #af--ba1 #af--bb1 #af--bc1 #ag--af1 #ab--ag1 #ab--bb1 #ab--ba1g--ab1 #g--ag1 #g--bb1 #aa--ab1 #aa--ag1 #b--g1 #b--ab1 #f--aa1 #h--g1 #h--f1 #ac--h1 #ah--ac1 #aa--ah1 #b--ag1 #ah--ab1 #ah--g1 #ah--b1 #ac--g1 #c--h1 #c--ac1 #d--c1 #d--h1 #i--d1 #i--c1 #ad--i1 #ad--d1 #ai--ad1 #ai--i1 #c--ad1 #c--ai1 #f--ab1 #f--ac1 #f--ad1 #ag--aa1 #ab--af1 #b--ab1 #g--ag1 #aa--ac1 #af--ah #df moves4 ,i am1 ( c\rse1 n able to say positively t a solution _c be discov]$ 9 few] moves1 b ,i believe x w be f\nd a v h>d task to r$uce ! numb]4 #cde4--,,! ,,two ,,pawns4 ,call "o pawn ,a & ! o!r ,b4 ,n[1 ow+ to t optional fir/ move1 ei!r pawn may make ei!r #e or #f moves 9 rea*+ ! eigh? squ>e4 ,!re are1 !re=e1 f\r cases to be consid]$3 7#a7 ,a #f moves & ,b #f moves2 7#b7 ,a #f moves & ,b #e moves2 7#c7 ,a #e moves & ,b #f moves2 7#d7 ,a #e moves & ,b #e moves4 ,9 case 7#a7 !re are #ab moves1 & we may select any #f ( ^! = ,a4 ,!re=e #ghiajaaab divid$ by #abcdef gives u ! numb] ( v>iations = ?is case--t is1 #ibd4 ,simil>ly = case 7#b71 #f selections out ( #aa w be #dfb2 9 case 7#c71 #e selections out ( #aa w also be #dfb2 & 9 case 7#d71 #e selections out ( #aj w be #beb4 ,add ^! f\r numb]s toge!r & we get #b1ajj1 :i* is ! correct numb] ( diff]5t ways 9 :i* ! pawns may advance "u ! conditions4 7,see ,no4 #bgj1 on p4 #bjd47 #cdf4--,,sett+ ,,! ,,bo>d4 ,! ,:ite pawns may be arrang$ 9 #dj1cbj ways1 ! ,:ite rooks 9 #b ways1 ! bi%ops 9 #b ways1 & ! knights 9 #b ways4 ,multiply ^! numb]s toge!r1 & we f9d t ! ,:ite pieces may be plac$ 9 #cbb1efj diff]5t ways4 ,! ,black pieces may1 ( c\rse1 be plac$ 9 ! same numb] ( ways4 ,!re=e ! m5 may be set up 9 #cbb1efj #cbb1efj #ajd1jdd1iec1fjj ways4 ,b ! po9t t ne>ly ev]ybody ov]looks is t ! bo>d may be plac$ 9 two diff]5t ways = e arrangem5t4 ,!re=e ! answ] is d\bl$1 & is #bjh1jhi1ijg1bjj diff]5t ways4 #cdg4--;,,c\nt+ ,,! ,,rectangles4 ,!re are #a1bif diff]5t rectangles 9 all1 #bjd ( :i* are squ>es1 c\nt+ ! squ>e bo>d itself z "o1 & #a1jib rectangles t are n squ>es4 ,! g5]al =mula is t a bo>d ( n squ>es conta9s 77n n77_/#d rectangles1 ( :i* 7#bn #cn n7_/#f are squ>es & 7#cn#d #bn - #cn - #bn7_/#ab are rectangles t are n squ>es4 ,x is curi\s & int]e/+ t ! total numb] ( rectangles is always ! squ>e ( ! triangul> numb] ^: side is n4 #cdh4--,,! ,,rook]y4 ,! answ] involves ! little po9t t 9 ! f9al position ! numb]$ rooks mu/ be 9 num]ical ord] 9 ! direction contr>y to t 9 :i* !y appe> 9 ! orig9al diagram1 o!rwise x _c be solv$4 ,play ! rooks 9 ! foll[+ ord] ( _! numb]s4 ,z !re is nev] m ?an "o squ>e to :i* a rook c move 7except on ! f9al move71 ! notation is obvi\s--#e1 #f1 #g1 #e1 #f1 #d1 #c1 #f1 #d1 #g1 #e1 #d1 #g1 #c1 #f1 #g1 #c1 #e1 #d1 #c1 #a1 #h1 #c1 #d1 #e1 #f1 #g1 #a1 #h1 #b1 #a1 & rook takes bi%op1 *eckmate4 ,^! are ! fewe/ possible moves--?irty-two4 ,! ,black k+'s moves are all =c$1 & ne$ n be giv54 #cdi4--,,/alemate4 ,work+ indep5d5tly1 ! same position 0 arriv$ at by ,messrs4 ,s4 ,loyd1 ,e4,n4 ,frank5/e91 ,w4,h4 ,?ompson1 & myself4 ,s ! foll[+ may be accept$ z ! be/ solution possible to ?is curi\s problem 3-- ,:ite4 ,black4 #a4 ,p--,q#d #a4 ,p--,k#d #b4 ,q--,q#c #b4 ,q--,r#e #c4 ,q--,k,kt#c #c4 ,b--,kt#e * #d4 ,kt--,q#b #d4 ,p--,,qr#d #e4 ,p--,r#d #e4 ,p--,q#c #f4 ,p--,r#c #f4 ,b--,k#c #g4 ,r--,r#c #g4 ,p--,,kb#d #h4 ,q--,r#b #h4 ,p--,b#d #i4 ,r--,k,kt#c #i4 ,b--,kt#f #aj4 ,p--,,qb#d #aj4 ,p--,b#e #aa4 ,p--,b#c #aa4 ,p--,k#e #ab4 ,p--,q#e #ab4 ,p--,k#f ,& ,:ite is /alemat$4 ,we give a diagram ( ! curi\s position arriv$ at4 ,x w be se5 t n "o ( ,:ite's pieces may be mov$4 ,illu/ration -------- rn k nr -------- p pp -------- p -------- p p,p -------- ,pb,p p q -------- b p,p,r,p -------- ,p ,n,p ,p,q -------- ,b ,k,b,n,r -------- #cej4--,,! ,,=sak5 ,,k+4 ,play z foll[s3-- ,:ite4 ,black4 #a4 ,p to ,k #d? #a4 ,any move #b4 ,q to ,kt #d? #b4 ,any move except on ,,kb file 7a7 #c4 ,q to ,kt #g? #c4 ,k moves to royal r[ #d4 ,b to ,kt #e? #d4 ,any move #e4 ,mate 9 two moves ,if #c4 ,k o!r ?an to royal r[ #d4 ,p to ,q #d? #d4 ,any move #e4 ,mate 9 two moves 7a7 ,if #b4 ,any move on ,,kb file #c4 ,q to ,q #g? #c4 ,k moves to royal r[ #d4 ,p to ,q ,kt #crd #d4 ,any move #e4 ,mate 9 two moves ,if #c4 ,k o!r ?an to royal r[ #d4 ,p to ,q #d? #d4 ,any move #e4 ,mate 9 two moves ,( c\rse1 by royal r[ is meant ! r[ on :i* ! k+ orig9ally /&s at ! beg9n+ ( a game4 ,?\gh1 if ,black plays badly1 he may1 9 c]ta9 positions1 be mat$ 9 few] moves1 ! above provides = e v>iation he c possibly br+ ab\t4 #cea4--,,! ,,crusad]4 ,:ite4 ,black4 #a4 ,kt to ,,qb #crd #a4 ,p to ,q #d? #b4 ,kt takes ,,qp #b4 ,kt to ,,qb #crd #c4 ,kt takes ,,kp #c4 ,p to ,k,kt #d? #d4 ,kt takes ,b #d4 ,kt to ,,kb #crd #e4 ,kt takes ,p #e4 ,kt to ,k #e? #f4 ,kt takes ,kt #f4 ,kt to ,b #f? #g4 ,kt takes ,q #g4 ,r to ,k,kt sq #h4 ,kt takes ,,bp #h4 ,r to ,k,kt #crd #i4 ,kt takes ,p #i4 ,r to ,k #crd #aj4 ,kt takes ,p #aj4 ,kt to ,kt #h? #aa4 ,kt takes ,b #aa4 ,r to ,r #f? #ab4 ,kt takes ,r #ab4 ,p to ,kt #d? #ac4 ,kt takes ,p 7*7 #ac4 ,k to ,b #bnd #ad4 ,kt takes ,p #ad4 ,k to ,kt #crd #ae4 ,kt takes ,r #ae4 ,k to ,r #d? #af4 ,kt takes ,kt #af4 ,k to ,r #e? ,:ite n[ mates 9 ?ree moves4 #ag4 ,p to ,q #d? #ag4 ,k to ,r #d? #ah4 ,q to ,q #crd #ah4 ,k moves #ai4 ,q to ,,kr #crd 7mate7 ,if #ag4 ,k to ,kt #e? #ah4 ,p to ,k #d? 7dis4 *7 #ah4 ,k moves #ai4 ,p to ,k,kt #crd 7mate7 ,! position aft] ! sixte5? move1 ) ! mate 9 ?ree moves1 0 fir/ giv5 by ,s4 ,loyd 9 ,*ess ,nuts4 #ceb4--;,,immovable ,,pawns4 #a4 ,kt to ,,kb #c #b4 ,kt to ,,kr #d #c4 ,kt to ,kt #f #d4 ,kt takes ,r #e4 ,kt to ,kt #f #f4 ,kt takes ,b #g4 ,k takes ,kt #h4 ,kt to ,,qb #c #i4 ,kt to ,r #d #aj4 ,kt to ,kt #f #aa4 ,kt takes ,r #ab4 ,kt to ,kt #f #ac4 ,kt takes ,b #ad4 ,kt to ,q #f #ae4 ,q to ,k sq #af4 ,kt takes ,q #ag4 ,k takes ,kt1 & ! position is rea*$4 ,black plays precisely ! same moves z ,:ite1 & !re=e we give "o set ( moves only4 ,! above sev5te5 moves are ! fewe/ possible4 #cec4--,,?irty-,,six ,,mates4 ,place ! rema9+ eight ,:ite pieces ?us3 ,k at ,,kb #d?1 ,q at ,q,kt #f?1 ,r at ,q #f?1 ,r at ,k,kt #g?1 ,b at ,q #e?1 ,b at ,,kr #h?1 ,kt at ,,qr #e?1 & ,kt at ,,qb #e?4 ,! foll[+ mates c !n be giv53-- ,by discov]y f ,q #h ,by discov]y f ,r at ,q #f? #ac ,by discov]y f ,b at ,r #h? #aa ,giv5 by ,kt at ,r #e? #b ,giv5 by pawns #b -- ,total #cf ,is x possible to con/ruct a position 9 :i* m ?an ?irty-six diff]5t mates on ! move c be giv58 ,s f> z ,i "k1 nobody has yet beat5 my arrangem5t4 #ced4--;,,an ,,amaz+ ,,dilemma4 ,mr ,black left 8 k+ on 8 que5's knight's #g?1 & no matt] :at piece ,:ite *ooses = 8 pawn1 ,black _c be *eckmat$4 ,z we said1 ! ,black k+ takes no notice ( *ecks & nev] moves4 ,:ite may que5 8 pawn1 capture ! ,black rook1 & br+ 8 ?ree pieces up to ! attack1 b mate is q impossible4 ,! ,black k+ _c be left on any o!r squ>e )\t a *eckmate be+ possible4 ,! late ,sam ,loyd fir/ po9t$ out ! peculi>ity on :i* ?is puzzle is bas$4 #cee4--;,,*eckmate6 ,remove ! ,:ite pawn f ,b #f? to ,k #d? & place a ,black pawn on ,black's ,,kb #bnd4 ,n[1 ,:ite plays ,p to ,k #e?1 *eck1 & ,black mu/ play ,p to ,b #d?4 ,!n ,:ite plays ,p takes ,p en passant1 *eckmate4 ,?is 0 !re=e ,:ite's la/ move1 & leaves ! position giv54 ,x is ! only possible solution4 #cef4--,,que] ,,*ess4 -------- -------- ,rk,r,n -------- -------- ,if y place ! pieces z foll[s 7:]e only a portion ( ! bo>d is giv51 to save space71 ! ,black k+ is 9 *eck1 ) no possible move op5 to him4 ,! read] w n[ see :y ,i avoid$ ! t]m *eckmate1 ap>t f ! fact t !re is no ,:ite k+4 ,! position is impossible 9 ! game ( *ess1 because ,black c\ld n be giv5 *eck by bo? rooks at ! same "t1 nor c\ld he h mov$ into *eck on 8 la/ move4 ,i believe ! position 0 fir/ publi%$ by ! late ,s4 ,loyd4 #ceg4--;,,anci5t ,,*9ese ,,puzzle4 ,play z foll[s3-- #a4 ,r--,q #f #b4 ,k--,r #g #c4 ,r 7,r #f7--;,b #f 7mate74 ,black's moves are =c$1 s ne$ n be giv54 #ceh4--,,! ,,six ,,pawns4 ,! g5]al =mula = six pawns on all squ>es great] ?an #b is ?is3 ,six times ! squ>e ( ! numb] ( comb9ations ( n ?+s tak5 ?ree at a "t1 :]e n repres5ts ! numb] ( squ>es on ! side ( ! bo>d4 ,( c\rse1 :]e n is ev5 ! unoccupi$ squ>es 9 ! r[s & columns w be ev51 & :]e n is odd ! numb] ( squ>es w be odd4 ,"h n is #h1 s ! answ] is #ah1haf diff]5t ways4 ,?is is ,! ,dy]'s ,puzzle 7,cant]bury ,puzzles1 ,no4 #bg7 9 ano!r =m4 ,i repeat x "h 9 ord] to expla9 a me?od ( solv+ t w be readily grasp$ by ! novice4 ,fir/ ( all1 x is evid5t t if we put a pawn on any l9e1 we mu/ put a second "o 9 t l9e 9 ord] t ! rema9d] may be ev5 9 numb]4 ,we _c put f\r or six 9 any r[ )\t mak+ x impossible to get an ev5 numb] 9 all ! columns int]f]$ )4 ,we h1 !re=e1 to put two pawns 9 ea* ( ?ree r[s & 9 ea* ( ?ree columns4 ,n[1 !re are j six s*emes or arrangem5ts t fulfil ^! conditions1 & ^! are %[n 9 ,diagrams ,a to ,f1 inclusive1 on next page4 ,illu/ration ,i w j rem>k 9 pass+ t ,a & ,b are ! only di/9ctive arrangem5ts1 because1 if y give ,a a qu>t]-turn1 y get ,f2 & if y give ,b ?ree qu>t]-turns 9 ! direction t a clock h& moves1 y w get successively ,c1 ,d1 & ,e4 ,no matt] h[ y may place y\r six pawns1 if y h compli$ ) ! conditions ( ! puzzle !y w fall "u "o ( ^! arrangem5ts4 ,( c\rse x w be und]/ood t m]e expansions d n de/roy ! ess5tial "* ( ! arrangem5ts4 ,?us ,g is only an expansion ( =m ,a4 ,! solution !re=e consi/s 9 f9d+ ! numb] ( ^! expansions4 ,suppos+ we conf9e our op]ations to ! fir/ ?ree r[s1 z 9 ,g1 !n ) ! pairs a & b plac$ 9 ! fir/ & second columns ! pair c may be dispos$ 9 any "o ( ! rema9+ six columns1 & s give six solutions4 ,n[ slide pair b into ! ?ird column1 & !re are five possible positions = c4 ,slide b into ! f\r? column1 & c may produce f\r new solutions4 ,& s on1 until 7/ill leav+ a 9 ! fir/ column7 y h b 9 ! sev5? column1 & !re is only "o place = c--9 ! eigh? column4 ,!n y may put a 9 ! second column1 b 9 ! ?ird1 & c 9 ! f\r?1 & />t slid+ c & b z be=e = ano!r s]ies ( solutions4 ,we f9d ?us t1 by us+ =m ,a alone & conf9+ our op]ations to ! ?ree top r[s1 we get z _m answ]s z !re are comb9ations ( #h ?+s tak5 #c at a "t4 ,?is is 7#h #g #f7_/7a #b #c7 #ef4 ,& x w at once /rike ! read] t if !re are #ef diff]5t ways ( elect+ ! columns1 !re mu/ be = ea* ( ^! ways j #ef ways ( select+ ! r[s1 = we may simultane\sly "w t slid+ process d[nw>ds to ! v bottom 9 exactly ! same way z we h work$ f left to "r4 ,!re=e ! total numb] ( ways 9 :i* =m ,a may be appli$ is #ef #f #c1acf4 ,b !re are1 z we h se51 six arrangem5ts1 & we h only dealt ) "o ( ^!1 ,a4 ,we mu/1 !re=e1 multiply ?is result by #f1 :i* gives u #c1acf #f #ah1haf1 :i* is ! total numb] ( ways1 z we h already /at$4 #cei4--;,,c\nt] ,,solitaire4 ,play z foll[s3 #c--aa1 #i--aj1 #a--b1 #g--ae1 #h--af1 #h--g1 #e--ac1 #a--d1 #h--e1 #f--ad1 #c--h1 #f--c1 #f--ab1 #a--f1 #a--i1 & all ! c\nt]s w h be5 remov$1 ) ! exception ( ,no4 #a1 z requir$ by ! conditions4 #cfj4--;,,*essbo>d ,,solitaire4 ,play z foll[s3 #g--ae1 #h--af1 #h--g1 #b--aj1 #a--i1 #a--b1 #e--ac1 #c--d1 #f--c1 #aa--a1 #ad--h1 #f--ab1 #e--f1 #e--aa1 #ca--bc1 #cb--bd1 #cb--ca1 #bf--ah1 #be--ag1 #be--bf1 #bb--cb1 #ad--bb1 #bi--ba1 #ad--bi1 #bg--bh1 #cj--bg1 #be--ad1 #cj--bj1 #be--cj1 #be--e4 ,! two c\nt]s left on ! bo>d are #be & #ai--;bo? belong+ to ! same gr\p1 z /ipulat$--& #ai has nev] be5 mov$ f its orig9al place4 ,i d n ?9k any solution is possible 9 :i* only "o c\nt] is left on ! bo>d4 #cfa4--,,! ,,mon/rosity4 ,:ite ,black1 #a4 ,p to ,,kb #d ,p to ,,qb #c #b4 ,k to ,b #b ,q to ,r #d #c4 ,k to ,k #c ,k to ,q sq #d4 ,p to ,b #e ,k to ,b #b #e4 ,q to ,k sq ,k to ,kt #c #f4 ,q to ,kt #c ,kt to ,,qr #c #g4 ,q to ,kt #h ,p to ,,kr #d #h4 ,kt to ,,kb #c ,r to ,r #c #i4 ,kt to ,k #e ,r to ,kt #c #aj4 ,q takes ,b ,r to ,kt #f1 * #aa4 ,p takes ,r ,k to ,kt #d #ab4 ,r to ,r #d ,p to ,b #c #ac4 ,r to ,q #d ,p takes ,kt #ad4 ,p to ,q,kt #d ,p takes ,r1 * #ae4 ,k to ,b #d ,p to ,r #e #af4 ,q to ,k #h ,p to ,r #f #ag4 ,kt to ,b #c1 * ,p takes ,kt #ah4 ,b to ,r #c ,p to ,r #g #ai4 ,r to ,kt sq ,p to ,r #h 7,q7 #bj4 ,r to ,kt #b ,p takes ,r #ba4 ,k to ,kt #e ,q to ,k,kt #h #bb4 ,q to ,r #e ,k to ,r #e #bc4 ,p to ,kt #e ,r to ,b sq #bd4 ,p to ,kt #f ,r to ,b #b #be4 ,p takes ,r ,p to ,kt #h 7,b7 #bf4 ,p to ,b #h 7,r7 ,q to ,b #b #bg4 ,b to ,q #f ,kt to ,kt #e #bh4 ,k to ,kt #f ,k to ,r #f #bi4 ,r to ,r #h ,k to ,kt #g #cj4 ,p to ,r #d ,q 7,kt #h7 to ,kt #c #ca4 ,p to ,r #e ,k to ,b #h #cb4 ,p takes ,q ,k to ,q #h #cc4 ,p takes ,q ,k to ,k #h #cd4 ,k to ,b #g ,kt to ,,kr #c1 * #ce4 ,k to ,k #h ,b to ,r #g #cf4 ,p to ,b #f ,b to ,kt sq #cg4 ,p to ,b #g ,k takes ,b #ch4 ,p to ,b #h 7,b7 ,kt to ,q #d #ci4 ,b to ,kt #h ,kt to ,b #c1 * #dj4 ,k to ,q #h ,kt to ,k sq #da4 ,p takes ,kt 7,r7 ,kt to ,b #b1 * #db4 ,k to ,b #g ,kt to ,q sq #dc4 ,q to ,b #g1 * ,k to ,kt #h ,& ! position is rea*$4 ,! ord] ( ! moves is immat]ial1 & ?is ord] may be greatly v>i$4 ,b1 al?\gh _m attempts h be5 made1 nobody has succe$$ 9 r$uc+ ! numb] ( my moves4 #cfb4--,,! ,,wassail ,,b[l4 ,! division ( ! twelve p9ts ( ale c be made 9 elev5 manipulations1 z bel[4 ,! six columns %[ at a glance ! quantity ( ale 9 ! b>rel1 ! five-p9t jug1 ! ?ree-p9t jug1 & ! tramps ,x1 ,y1 & ,z respectively aft] ea* manipulation4 ,b>rel4 #e-p9t4 #c-p9t4 ,x4 ,y4 ,z4 #g 44 #e 44 #j 44 #j 44 #j 44 #j #g 44 #b 44 #c 44 #j 44 #j 44 #j #g 44 #j 44 #c 44 #b 44 #j 44 #j #g 44 #c 44 #j 44 #b 44 #j 44 #j #d 44 #c 44 #c 44 #b 44 #j 44 #j #j 44 #c 44 #c 44 #b 44 #d 44 #j #j 44 #e 44 #a 44 #b 44 #d 44 #j #j 44 #e 44 #j 44 #b 44 #d 44 #a #j 44 #b 44 #c 44 #b 44 #d 44 #a #j 44 #j 44 #c 44 #d 44 #d 44 #a #j 44 #j 44 #j 44 #d 44 #d 44 #d ,& ea* man has receiv$ 8 f\r p9ts ( ale4 #cfc4--,,! ,,doctor',s ,,qu]y4 ,! mixture ( spirits ( w9e & wat] is 9 ! proportion ( #dj to #a1 j z 9 ! o!r bottle x 0 9 ! proportion ( #a to #dj4 #cfd4--,,! ,,b>rel ,,puzzle4 ,illu/ration3 ,figs4 #a1 #b1 & #c ,all t is necess>y is to tilt ! b>rel z 9 ,fig4 #a1 & if ! edge ( ! surface ( ! wat] exactly t\*es ! lip a at ! same "t t x t\*es ! edge ( ! bottom b1 x w be j half full4 ,to be m exact1 if ! bottom is an in* or s f ! gr\nd1 !n we c all[ = t1 & ! ?ickness ( ! bottom1 at ! top4 ,if :5 ! surface ( ! wat] rea*$ ! lip a x _h ris5 to ! po9t c 9 ,fig4 #b1 !n x w\ld be m ?an half full4 ,if1 z 9 ,fig4 #c1 "s portion ( ! bottom 7 visible & ! level ( ! wat] fell to ! po9t d1 !n x w\ld be less ?an half full4 ,?is me?od applies to all symmetrically con/ruct$ vessels4 #cfe4--,,new ,,measur+ ,,puzzle4 ,! foll[+ solution 9 elev5 manipulations %[s ! cont5ts ( e vessel at ! />t & aft] e manipulation3-- #aj-qu>t4 #aj-qu>t4 #e-qu>t4 #d-qu>t4 #aj 44 #aj 44 #j 44 #j #e 44 #aj 44 #e 44 #j #e 44 #aj 44 #a 44 #d #i 44 #aj 44 #a 44 #j #i 44 #f 44 #a 44 #d #i 44 #g 44 #j 44 #d #i 44 #g 44 #d 44 #j #i 44 #c 44 #d 44 #d #i 44 #c 44 #e 44 #c #i 44 #h 44 #j 44 #c #d 44 #h 44 #e 44 #c #d 44 #aj 44 #c 44 #c #cff4--,,! ,,hone/ ,,dairyman4 ,:atev] ! respective quantities ( milk & wat]1 ! relative proportion s5t to ,london w\ld always be ?ree p>ts ( wat] to "o ( milk4 ,b !re are "o or two po9ts to be obs]v$4 ,!re mu/ orig9ally be m wat] ?an milk1 or !re w be no wat] 9 ,a to d\ble 9 ! second transaction4 ,& ! wat] mu/ n be m ?an ?ree times ! quantity ( milk1 or !re w n be en\gh liquid 9 ,b to effect ! second transaction4 ,! ?ird transaction has no effect on ,a1 z ! relative proportions 9 x mu/ be ! same z aft] ! second transaction4 ,x 0 introduc$ to prev5t a quibble if ! quantity ( milk & wat] 7 orig9ally ! same2 = ?\gh d\ble no?+ w\ld be no?+1 yet ! ?ird transaction 9 su* a case c\ld n take place4 #cfg4--,,w9e ,,& ,,wat]4 ,! w9e 9 small glass 0 "o-six? ( ! total liquid1 & ! w9e 9 l>ge glass two-n9?s ( total4 ,add ^! toge!r1 & we f9d t ! w9e 0 sev5-eighte5?s ( total fluid1 & !re=e ! wat] elev5-eighte5?s4 #cfh4--,,! ,,keg ,,( ,,w9e4 ,! capacity ( ! jug mu/ h be5 a little less ?an ?ree gallons4 ,to be m exact1 x 0 #b4ic gallons4 #cfi4--,,mix+ ,,! ,,tea4 ,!re are ?ree ways ( mix+ ! teas4 ,tak+ !m 9 ! ord] ( quality1 #bs4 #f;d41 #bs4 #c;d41 #as4 #ip41 mix #af lbs41 #a lb41 #c lbs42 or #ad lbs41 #d lbs41 #b lbs42 or #ab lbs41 #g lbs41 #a lb4 ,9 e case ! tw5ty p\nds mixture %\ld be wor? #bs4 #d;d4 p] p\nd2 b ! la/ case requires ! smalle/ quantity ( ! be/ tea1 !re=e x is ! correct answ]4 #cgj4--;,a ,,pack+ ,,puzzle4 ,on ! side ( ! box1 #ad by #bbd_/e1 we c arrange #ac r[s conta9+ alt]nately #g & #f balls1 or #he 9 all4 ,above ?is we c place ano!r lay] consi/+ ( #ab r[s ( #g & #f alt]nately1 or a total ( #gh4 ,9 ! l5g? ( #bdi_/aj in*es #ae su* lay]s may be pack$1 ! alt]nate lay]s conta9+ #he & #gh balls4 ,?us #h times #he add$ to #g times #gh gives u #a1bbf = ! full cont5ts ( ! box4 #cga4--;,,gold ,,pack+ ,,9 ,,russia4 ,! box %\ld be #ajj in*es by #ajj in*es by #aa in*es deep1 int]nal dim5sions4 ,we c lay flat at ! bottom a r[ ( eight slabs1 l5g?ways1 end to end1 :i* w j fill "o side1 & n9e ( ^! r[s w dispose ( sev5ty-two slabs 7all on ! bottom71 ) a space left ov] on ! bottom measur+ #ajj in*es by #a in* by #a in*4 ,n[ make elev5 dep?s ( su* sev5ty-two slabs1 & we h pack$ #gib1 & h a space #ajj in*es by #a in* by #aa in*es deep4 ,9 ?is we may exactly pack ! rema9+ eight slabs on edge1 end to end4 #cgb4--,,! ,,b>rels ,,( ,,honey4 ,! only way 9 :i* ! b>rels c\ld be equally divid$ among ! ?ree bro!rs1 s t ea* %\ld receive 8 #c b>rels ( honey & 8 #g b>rels1 is z foll[s3-- ,full4 ,half-full4 ,empty4 ,a #c #a #c ,b #b #c #b ,c #b #c #b ,!re is "o o!r way 9 :i* ! division c\ld be made1 7 x n = ! objection t all ! bro!rs made to tak+ m ?an f\r b>rels ( ! same description4 ,except = ?is difficulty1 !y might h giv5 ,b 8 quantity 9 exactly ! same way z ,a above1 & !n h left ,c "o full b>rel1 five half-full b>rels1 & "o empty b>rel4 ,x w ?us be se5 t 9 any case two bro!rs w\ld h to receive _! all[ance 9 ! same way4 #cgc4--;,,cross+ ,,! ,,/ream4 ,fir/1 ! two sons cross1 & "o returns ,!n ! man crosses & ! o!r son returns4 ,!n bo? sons cross & "o returns4 ,!n ! lady crosses & ! o!r son returns ,!n ! two sons cross & "o ( !m returns = ! dog4 ,elev5 cross+s 9 all4 ,x w\ld appe> t no g5]al rule c be giv5 = solv+ ^! riv]-cross+ puzzles4 ,a =mula c be f\nd = a p>ticul> case 7say on ,no4 #cge or #cgf7 t w\ld apply to any numb] ( individuals "u ! re/rict$ conditions2 b x is n ( mu* use1 = "s little add$ /ipulation w entirely upset x4 ,z 9 ! case ( ! measur+ puzzles1 we g5]ally h to rely on individual +5uity4 #cgd4--;,,cross+ ,,! ,,riv] ,,axe4 ,"h is ! solution3-- ,j #e7 ,g ,t#h #c #e 7 ,j ,g ,t#h #c #e ,g #c7 ,,jt#h #ec 7 ,g ,,jt#h #ec ,j ,t7 ,g #h ,j #e 7,t #c ,g #h ,j #e ,g #h7 ,t #c ,g #h 7,j #e ,t ,g #h ,j ,t7 #ec ,,jt#h 7 ,g #ec ,,jt#h ,g #c7 #e ,g ,t#h #c 7 ,j #e ,g ,t#h #c ,j #e7 ,g1 ,j1 & ,t /& = ,giles1 ,jasp]1 & ,timo?y2 & #h1 #e1 #c1 = #hjj1 #ejj1 & #cjj respectively4 ,! two side columns repres5t ! left bank & ! "r bank1 & ! middle column ! riv]4 ,?irte5 cross+s are necess>y1 & ea* l9e %[s ! position :5 ! boat is 9 mid-/ream dur+ a cross+1 ! po9t ( ! bracket indicat+ ! direction4 ,x w be f\nd t n only is no p]son left alone on ! l& or 9 ! boat ) m ?an 8 %>e ( ! spoil1 b t also no two p]sons are left ) m ?an _! jo9t %>es1 ?\gh ?is la/ po9t 0 n insi/$ ^u 9 ! conditions4 #cge4--;,,five ,,jeal\s ,,husb&s4 ,x is obvi\s t !re mu/ be an odd numb] ( cross+s1 & t if ! five husb&s _h n be5 jeal\s ( "o ano!r ! p>ty might h all got ov] 9 n9e cross+s4 ,b no wife 0 to be 9 ! company ( a man or m5 unless h] husb& 0 pres5t4 ,?is entails two m cross+s1 elev5 9 all4 ,! foll[+ %[s h[ x might h be5 done4 ,! capital lett]s /& = ! husb&s1 & ! small lett]s = _! respective wives4 ,! position ( affairs is %[n at ! />t1 & aft] ea* cross+ betwe5 ! left bank & ! "r1 & ! boat is repres5t$ by ! a/]isk4 ,s y c see at a glance t a1 b1 & c w5t ov] at ! fir/ cross+1 t b & c return$ at ! second cross+1 & s on4 ,,abcde abcde 9944 #a4 ,,abcde de 4499 abc #b4 ,,abcde bcde 9944 a #c4 ,,abcde e 4499 abcd #d4 ,,abcde de 9944 abc #e4 ,,de de 1199 ,,abc abc #f4 ,,cde cde 9944 ,,ab ab #g4 cde 4499 ,,abcde ab #h4 bcde 9944 ,,abcde a #i4 e 4499 ,,abcde abcd #aj4 bc e 9944 ,,abcde a d #aa4 4499 ,,abcde abcde ,!re is a little subtlety conceal$ 9 ! words %[ ! quicke/ way4 ,ev]ybody correctly assumes t1 z we are told no?+ ( ! r[+ capabilities ( ! p>ty1 we mu/ take x t !y all r[ equally well4 ,b x is obvi\s t two su* p]sons %\ld r[ m quickly ?an "o4 ,!re=e 9 ! second & ?ird cross+s two ( ! ladies %\ld take back ! boat to fet* d1 n "o ( !m only4 ,?is does n affect ! numb] ( l&+s1 s no "t is lo/ on t acc\nt4 ,a simil> opportunity occurs 9 cross+s #aj & #aa1 :]e ! p>ty aga9 _h ! option ( s5d+ ov] two ladies or "o only4 ,to ^? :o ?9k !y h solv$ ! puzzle 9 n9e cross+s ,i w\ld say t 9 e case !y w f9d t !y are wrong4 ,no su* jeal\s husb& w\ld1 9 ! circum/ances1 s5d 8 wife ov] to ! o!r bank to a man or m51 ev5 if %e assur$ him t %e 0 com+ back next "t 9 ! boat4 ,if read]s w h ?is fact 9 m9d1 !y w at once discov] _! errors4 #cgf4--,,! ,,f\r ,,elopem5ts4 ,if !re _h be5 only ?ree c\ples1 ! isl& might h be5 disp5s$ )1 b ) f\r or m c\ples x is absolutely necess>y 9 ord] to cross "u ! conditions laid d[n4 ,x c be done 9 sev5te5 passages f l& to l& 7?\gh ,fr5* ma!maticians h decl>$ 9 _! books t 9 su* circum/ances tw5ty-f\r are ne$$71 & x _c be done 9 few]4 ,i w give "o way4 ,a1 ,b1 ,c1 & ,d are ! "y m51 & a1 b1 c1 & d are ! girls to :om !y are respectively engag$4 ,! ?ree columns %[ ! positions ( ! diff]5t individuals on ! lawn1 ! isl&1 & ! opposite %ore be=e />t+ & aft] ea* passage1 :ile ! a/]isk indicates ! position ( ! boat on e occasion4 ,lawn4 ,isl&4 ,%ore4 ,a,b,c,dabcd 99 ,,abcd cd ab 99 ,,abcd bcd 99 a ,,abcd d bc 99 a ,,abcd cd 99 b a ,,cd cd b ,,ab a 99 ,,bcd cd 99 b ,a a ,,bcd bcd 99 ,a a ,,bcd d 99 bc ,a a ,d d bc ,,abc a 99 ,d d abc 99 ,,abc ,d d b ,,abc a c 99 ,b ,d d 99 b ,a ,c a c d b ,,abcd a c 99 d bc 99 ,,abcd a d ,,abcd abc 99 cd 99 ,,abcd ab ,,abcd abcd 99 ,hav+ f\nd ! fewe/ possible passages1 we %\ld consid] two o!r po9ts 9 decid+ on ! quicke/ me?od3 ,:i* p]sons 7 ! mo/ exp]t 9 h&l+ ! o>s1 & :i* me?od entails ! fewe/ possible delays 9 gett+ 9 & out ( ! boat8 ,we h no data ^u :i* to decide ! fir/ po9t1 ?\gh x is probable t1 z ! boat belong$ to ! girls' h\sehold1 !y w\ld be capable o>swom54 ,! o!r po9t1 h[ev]1 is important1 & 9 ! solution ,i h giv5 7:]e ! girls d #h-ac?s ( ! r[+ & ,a & ,d ne$ n r[ at all7 !re are only sixte5 gett+s-9 & sixte5 gett+s-out4 ,a man & a girl are nev] 9 ! boat toge!r1 & no man ev] l&s on ! isl&4 ,!re are o!r me?ods t require sev]al m ex*anges ( places4 #cgg4--,,/eal+ ,,! ,,ca/le ,,treasure4 ,"h is ! be/ answ]1 9 elev5 manipulations3-- ,treasure d[n4 ,boy d[n--treasure up4 ,y\? d[n--boy up4 ,treasure d[n4 ,man d[n--y\? & treasure up4 ,treasure d[n4 ,boy d[n--treasure up4 ,treasure d[n4 ,y\? d[n--boy up4 ,boy d[n--treasure up4 ,treasure d[n4 #cgh4--;,,dom9oes ,,9 ,,progression4 ,!re are tw5ty-?ree diff]5t ways4 ,y may />t ) any dom9o1 except ! #d--d & ^? t be> a #e or #f1 ?\gh only c]ta9 initial dom9oes may be play$ ei!r way r\nd4 ,if y are giv5 ! common diff]5ce & ! fir/ dom9o is play$1 y h no option z to ! o!r dom9oes4 ,!re=e all ,i ne$ d is to give ! initial dom9o = all ! tw5ty-?ree ways1 & /ate ! common diff]5ce4 ,?is ,i w d z foll[s3-- ,) a common diff]5ce ( #a1 ! fir/ dom9o may be ei!r ( ^!3 #j--j1 #j--a1 #a--j1 #j--b1 #a--a1 #b--j1 #j--c1 #a--b1 #b--a1 #c--j1 #j--d1 #a--c1 #b--b1 #c--a1 #a--d1 #b--c1 #c--b1 #b--d1 #c--c1 #c--d4 ,) a diff]5ce ( #b1 ! fir/ dom9o may be #j--j1 #j--b1 or #j--a4 ,take ! la/ case ( all z an example4 ,hav+ play$ ! #j--a1 & ! diff]5ce be+ #b1 we are compell$ to cont9ue ) #a--b1 #b--c1 #c--d4 #d--e1 #e--f4 ,!re are ?ree dom9oes t c nev] be us$ at all4 ,^! are #j--e1 #j--f1 & #a--f4 ,if we us$ a box ( dom9oes ext5d+ to #i--i1 !re w\ld be =ty diff]5t ways4 #cgi4--,,! ,,five ,,dom9oes4 ,!re are j t5 diff]5t ways ( arrang+ ! dom9oes4 ,"h is "o ( !m3-- 7#b--j7 7#j--j7 7#j--a7 7#a--d7 7#d--j74 ,i w leave my read]s to f9d ! rema9+ n9e = !mselves4 #chj4--,,! ,,dom9o ,,frame ,,puzzle4 ,illu/ration3 ---------------------------------------- ------------ #b #b #e #e #f #f #f #f #a #a #d - ---------------------------------------- -------- #b #d --- - #b #c - --- #f #c --- ,t ,h ,e - #f #c - --- #c #c --- - #c #a - ,d ,o ,m ,i ,n ,o ,f ,r ,a ,m ,e --- #a --- - #a - --- #e #a --- -,s-,o-,l-,u-,t-,i-,o-,n- - #e #d - --- #c #d --- - #c #f - --- #b #f ---------------------------------------- -------- - #b #a #a #e #e #e #e #d #d #d #d #b #b ---------------------------------------- ------------ ,! illu/ration is a solution4 ,x w be f\nd t all f\r sides ( ! frame add up #dd4 ,! sum ( ! pips on all ! dom9oes is #afh1 & if we wi% to make ! sides sum to #dd1 we mu/ take c>e t ! f\r corn]s sum to #h1 because ^! corn]s are c\nt$ twice1 & #afh add$ to #h w equal #d times #dd1 :i* is necess>y4 ,!re are _m diff]5t solutions4 ,ev5 9 ! example giv5 c]ta9 int]*anges are possible to produce diff]5t arrangem5ts4 ,= example1 on ! left-h& side ! /r+ ( dom9oes f #b--b d[n to #c--b may be rev]s$1 or f #b--f to #c--b1 or f #c--j to #e--c4 ,also1 on ! "r-h& side we may rev]se f #d--c to #a--d4 ,^! *anges w n affect ! correctness ( ! solution4 #cha4--,,! ,,c>d ,,frame ,,puzzle4 ,! sum ( all ! pips on ! t5 c>ds is #ee4 ,suppose we are try+ to get #ad pips on e side4 ,!n #d times #ad is #ef4 ,b ea* ( ! f\r corn] c>ds is add$ 9 twice1 s t #ee d$uct$ f #ef1 or #a1 mu/ repres5t ! sum ( ! f\r corn] c>ds4 ,?is is cle>ly impossible2 !re=e #ad is also impossible4 ,b suppose we came to try+ #ah4 ,!n #d times #ah is #gb1 & if we d$uct #ee we get #ag z ! sum ( ! corn]s4 ,we ne$ !n only try diff]5t arrangem5ts ) ! f\r corn]s always summ+ to #ag1 & we soon discov] ! foll[+ solution3-- ,illu/ration3 --------------------- #b #aj #f ------------ ------ #c #g --- --- #h #a ------------------- #e #i #d --------------------- ,! f9al trials are v limit$ 9 numb]1 & mu/ ) a little judgm5t ei!r br+ u to a correct solution or satisfy u t a solution is impossible "u ! conditions we are attempt+4 ,! two c5tre c>ds on ! upright sides c1 ( c\rse1 always be int]*ang$1 b ,i d n call ^! diff]5t solutions4 ,if y reflect 9 a mirror y get ano!r arrangem5t1 :i* also is n consid]$ diff]5t4 ,9 ! answ] giv51 h[ev]1 we may ex*ange ! #e ) ! #h & ! #d ) ! #a4 ,?is is a diff]5t solution4 ,!re are two solutions ) #ah1 f\r ) #ai1 two ) #bj1 & two ) #bb--t5 arrangem5ts 9 all4 ,read]s may l to f9d all ^! = !mselves4 #chb4--,,! ,,cross ,,( ,,c>ds4 ,!re are eighte5 fundam5tal arrangem5ts1 z foll[s1 :]e ,i only give ! numb]s 9 ! horizontal b>1 s9ce ! rema9d] mu/ naturally fall into _! places4 #e #f #a #g #d #b #d #e #f #h #c #e #a #f #h #c #d #e #f #g #c #d #a #g #h #a #d #g #f #h #b #e #a #g #h #b #c #g #f #h #b #e #c #f #h #b #d #g #e #h #a #e #c #g #h #c #d #i #e #f #b #d #c #g #h #b #d #i #e #g #a #d #e #g #h #a #d #i #f #g #b #c #e #g #h #b #c #i #f #g ,x w be notic$ t !re mu/ always be an odd numb] 9 ! c5tre1 t !re are f\r ways ea* ( add+ up #bc1 #be1 & #bg1 b only ?ree ways ea* ( summ+ to #bd & #bf4 #chc4--,,! ,t ,,c>d ,,puzzle4 ,if we remove ! ace1 ! rema9+ c>ds may he divid$ into two gr\ps 7ea* add+ up alike7 9 f\r ways2 if we remove #c1 !re are ?ree ways2 if #e1 !re are f\r ways2 if #g1 !re are ?ree ways2 & if we remove #i1 !re are f\r ways ( mak+ two equal gr\ps4 ,!re are ?us eighte5 diff]5t ways ( gr\p+1 & if we take any "o ( ^! & keep ! odd c>d 7t ,i h call$ remov$7 at ! head ( ! column1 !n "o set ( numb]s c be v>i$ 9 ord] 9 tw5ty-f\r ways 9 ! column & ! o!r f\r tw5ty-f\r ways 9 ! horizontal1 or toge!r !y may be v>i$ 9 #bd #bd #egf ways4 ,& z !re are eighte5 su* cases1 we multiply ?is numb] by #ah & get #aj1cfh1 ! correct numb] ( ways ( plac+ ! c>ds4 ,z ?is numb] includes ! reflections1 we mu/ divide by #b1 b we h also to rememb] t e horizontal r[ c *ange places ) a v]tical r[1 necessitat+ our multiply+ by #b2 s "o op]ation cancels ! o!r4 #chd4--;,,c>d ,,triangles4 ,! foll[+ arrangem5ts ( ! c>ds %[ 7#a7 ! smalle/ possible sum1 #ag2 & 7#b7 ! l>ge/ possible1 #bc4 #a #g #i #f #d #b #d #h #c #f #c #g #e #b #i #e #a #h ,x w be se5 t ! two c>ds 9 ! middle ( any side may always be int]*ang$ )\t affect+ ! conditions4 ,?us !re are eight ways ( pres5t+ e fundam5tal arrangem5t4 ,! numb] ( fundam5tals is eighte51 z foll[s3 two summ+ to #ag1 f\r summ+ to #ai1 six summ+ to #bj1 f\r summ+ to #ba1 & two summ+ to #bc4 ,^! eighte5 fundam5tals1 multipli$ by eight 7= ! reason /at$ above71 give #add z ! total numb] ( diff]5t ways ( plac+ ! c>ds4 #che4--,,/r& ,,pati5ce4 ,! read] may f9d a solution q easy 9 a little ov] #bjj moves1 b1 surpris+ z x may at fir/ appe>1 n m ?an #fb moves are requir$4 ,"h is ! play3 ,by #d ,c up ,i mean a transf] ( ! #d ( clubs ) all ! c>ds t re/ on x4 #a ,d on space1 #b ,s on space1 #c ,d on space1 #b ,s on #c ,d1 #a ,h on #b ,s1 #b ,c on space1 #a ,d on #b ,c1 #d ,s on space1 #c ,h on #d ,s 7#i moves s f>71 #b ,s up on #c ,h 7#c moves71 #e ,h & #e ,d ex*ang$1 & #d ,c on #e ,d 7#f moves71 #c ,d on #d ,c 7#a71 #f ,s 7) #e ,h7 on space 7#c71 #d ,c up on #e ,h 7#c71 #b ,c up on #c ,d 7#c71 #g ,d on space 7#a71 #f ,c up on #g ,d 7#c71 #h ,s on space 7#a71 #g ,h on #h ,s 7#a71 #h ,c on #i ,d 7#a71 #g ,h on #h ,c 7#a71 #h ,s on #i ,h 7#a71 #g ,h on #h ,s 7#a71 #g ,d up on #h ,c 7#e71 #d ,c up on #e ,d 7#i71 #f ,s up on #g ,h 7#c71 #d ,s up on #e ,h 7#g7 #fb moves 9 all4 ,?is is my record2 p]haps ! read] c beat x4 #chf4--;,a ,,trick ,,) ,,dice4 ,all y h to d is to d$uct #bej f ! result giv51 & ! ?ree figures 9 ! answ] w be ! ?ree po9ts ?r[n ) ! dice4 ,?us1 9 ! ?r[ we gave1 ! numb] giv5 w\ld be #chf2 & :5 we d$uct #bej we get #acf1 f :i* we "k t ! ?r[s 7 #a1 #c1 & #f4 ,! process m]ely consi/s 9 giv+ #ajj;a #aj;b c #bej1 :]e a1 b1 & c repres5t ! ?ree ?r[s4 ,! result is obvi\s4 #chg4--,,! ,,village ,,cricket ,,mat*4 ,illu/ration3 ,mr4 ,dumk9s -- ------------------------ ------------------- ------------------- #a ----------------------- ------------------------ ------------------- ------------------- ----------------------- -- ,mr4 ,podd] ,mr4 ,luffey -- ------------------------ ------------------- ----------------------- #b ----------------------- ------------------- ------------------------ -- ,mr4 ,/ruggles ,! diagram ,no4 #a w %[ t z nei!r ,mr4 ,podd] nor ,mr4 ,dumk9s c ev] h be5 )9 ! crease opposite to t f :i* he />t$1 ,mr4 ,dumk9s w\ld score no?+ by 8 p]=mance4 ,diagram ,no4 #b w1 h[ev]1 make x cle> t s9ce ,mr4 ,luffey & ,mr4 ,/ruggles h1 not)/&+ _! en]getic b c>eless movem5ts1 contriv$ to *ange places1 ! manoeuvre mu/ increase ,mr4 ,/ruggles's total by "o run4 #chh4--,,sl[ ,,cricket4 ,! capta9 mu/ h be5 n out & scor$ #ba4 ,?us3-- #b m5 7ea* lbw7 #ai #d m5 7ea* caught7 #ag #a man 7run out7 #j #c m5 7ea* b[l$7 #i #a man 7capta9--n out7 #ba -- -- #aa #ff ,! capta9 ?us scor$ exactly #ae m ?an ! av]age ( ! team4 ,! o!rs :o 7 b[l$ c\ld only ref] to ?ree m51 z ! elev5? man w\ld be n out4 ,! read] c discov] = himself :y ! capta9 mu/ h be5 t elev5? man4 ,x w\ld n necess>ily foll[ ) any figures4 #chi4--,,! ,,football ,,play]s4 ,! smalle/ possible numb] ( m5 is sev54 ,!y c\ld be acc\nt$ = 9 ?ree diff]5t ways3 #a4 ,two ) bo? arms s\nd1 "o ) brok5 "r arm1 & f\r ) bo? arms brok54 #b4 ,"o ) bo? arms s\nd1 "o ) brok5 left arm1 two ) brok5 "r arm1 & ?ree ) bo? arms brok54 #c4 ,two ) left arm brok51 ?ree ) "r arm brok51 & two ) bo? arms brok54 ,b if e man 0 injur$1 ! la/ case is ! only "o t w\ld apply4 #cij4--,,! ,,horse-,,race ,,puzzle4 ,! answ] is3 #ab on ,acorn1 #ae on ,bluebottle1 #bj on ,capsule4 #cia4--,,! ,,motor-,,c> ,,race4 ,! fir/ po9t is to appreciate ! fact t1 9 a race r\nd a circul> track1 !re are ! same numb] ( c>s beh9d "o z !re are be=e4 ,all ! o!rs are bo? beh9d & be=e4 ,!re 7 ?irte5 c>s 9 ! race1 includ+ ,gogglesmi?'s c>4 ,!n "o-?ird ( twelve add$ to ?ree-qu>t]s ( twelve w give u ?irte5--! correct answ]4 #cib4--,,! ,,pebble ,,game4 ,9 ! case ( fifte5 pebbles1 ! fir/ play] w9s if he fir/ takes two4 ,!n :5 he holds an odd numb] & leaves #a1 #h1 or #i he w9s1 & :5 he holds an ev5 numb] & leaves #d1 #e1 or #ab he also w9s4 ,he c always d "o or o!r ( ^! ?+s until ! end ( ! game1 & s defeat 8 oppon5t4 ,9 ! case ( ?irte5 pebbles ! fir/ play] mu/ lose if 8 oppon5t plays correctly4 ,9 fact1 ! only numb]s ) :i* ! fir/ play] ought to lose are #e & multiples ( #h add$ to #e1 su* z #ac1 #ba1 #bi1 etc4 #cic4--,,! ,,two ,,rooks4 ,! second play] c always w91 b to ensure 8 do+ s he mu/ always place 8 rook1 at ! />t & on e subsequ5t move1 on ! same diagonal z 8 oppon5t's rook4 ,he c !n =ce 8 oppon5t into a corn] & w94 ,suppos+ ! diagram to repres5t ! positions ( ! rooks at ! />t1 !n1 if ,black play$ fir/1 ,:ite might h plac$ 8 rook at ,a & won next move4 ,any squ>e on t diagonal f ,a to ,h w w91 b ! be/ play is always to re/rict ! moves ( ! oppos+ rook z mu* z possible4 ,if ,:ite play$ fir/1 !n ,black %\ld h plac$ 8 rook at ,b 7,f w\ld n be s good1 z x gives ,:ite m scope72 !n if ,:ite goes to ,c1 ,black moves to ,d2 ,:ite to ,e1 ,black to ,f2 ,:ite to ,g1 ,black to ,c2 ,:ite to ,h1 ,black to ,i2 & ,black mu/ w9 next move4 ,if at any "t ,black _h fail$ to move on to ! same diagonal z ,:ite1 !n ,:ite c\ld take ,black's diagonal & w94 r3 black rook ,r3 :ite rook -------- r -------- ,a -------- -------- -------- ,b,d,f -------- ,r,c,e -------- ,i,g -------- ,h -------- ,,! ,,two ,,rooks4 #cid4--,,puss ,,9 ,,! ,,corn]4 ,no matt] :e!r he plays fir/ or second1 ! play] ,a1 :o />ts ! game at #ee1 mu/ w94 ,assum+ t ,b adopts ! v be/ l9es ( play 9 ord] to prolong z mu* z possible 8 exi/5ce1 ,a1 if he has fir/ move1 c always on 8 #ab? move capture ,b2 & if he has ! second move1 ,a c always on 8 #ad? move make ! capture4 ,8 po9t is always to get diagonally 9 l9e ) 8 oppon5t1 & by go+ to #cc1 if he has fir/ move1 he prev5ts ,b gett+ diagonally 9 l9e ) himself4 ,"h are two good games4 ,! numb] 9 front ( ! hyph5 is always ,a's move2 t aft] ! hyph5 is ,b's3-- #cc-h1 #cb-ae1 #ca-bb1 #cj-ba1 #bi-ad1 #bb-g1 #ae-f1 #ad-b1 #g-c1 #f-d1 #aa-1 & ,a mu/ capture on 8 next 7#ab?7 move1 -#ac1 #ed-bj1 #ec-bg1 #eb-cd1 #ea-da1 #ej-cd1 #db-bg1 #ce-bj1 #bh-ac1 #ba-f1 #ad-b1 #g-c1 #f-d1 #aa-1 & ,a mu/ capture on 8 next 7#ad?7 move4 #cie4--;,a ,,w> ,,puzzle ,,game4 ,! ,briti%] c always cat* ! enemy1 no matt] h[ clev] & elusive t a/ute individual may be2 b curi\s ?\gh x may seem1 ! ,briti% g5]al c only d s aft] he has paid a some:at my/]i\s visit to ! p>ticul> t[n m>k$ #a 9 ! map1 go+ 9 by #c & leav+ by #b1 or ent]+ by #b & leav+ by #c4 ,! ?ree t[ns t are %ad$ & h no numb]s d n really come into ! "q1 z "s may suppose1 = ! simple reason t ! ,briti%] nev] ne$s to ent] any "o ( !m1 :ile ! enemy _c be =c$ to g into !m1 & w\ld be cle>ly ill-advis$ to d s volunt>ily4 ,we may !re=e leave ^! out ( consid]ation altoge!r4 ,no matt] :at ! enemy may d1 ! ,briti%] %\ld make ! foll[+ fir/ n9e moves3 ,he %\ld visit t[ns #bd1 #bj1 #ai1 #ae1 #aa1 #g1 #c1 #a1 #b4 ,if ! enemy takes x into 8 head also to g to t[n #a1 x w be f\nd t he w h to beat a precipitate retreat ! same way t he w5t 91 or ! ,briti%] w infallibly cat* him 9 t[ns #b or #c1 z ! case may be4 ,s ! enemy w be wise to avoid t nor?-we/ corn] ( ! map altoge!r4 ,illu/ration ,n[1 :5 ! ,briti% g5]al has made ! n9e moves t ,i h giv51 ! enemy w be1 aft] 8 own n9? move1 9 "o ( ! t[ns m>k$ #e1 #h1 #aa1 #ac1 #ad1 #af1 #ai1 #ba1 #bd1 or #bg4 ,( c\rse1 if he imprud5tly goes to #c or #f at ?is po9t he w be caught at once4 ,:]ev] he may happ5 to be1 ! ,briti%] goes = him1 & has no long] any difficulty 9 cat*+ him 9 eight m moves at mo/ 7sev5te5 9 all7 9 "o ( ! foll[+ ways4 ,! ,briti%] w get to #h :5 ! enemy is at #e1 & w9 next move2 or he w get to #ai :5 ! enemy is at #bb1 & w9 next move2 or he w get to #bd :5 ! enemy is at #bg1 & s w9 next move4 ,x w be f\nd t he c be =c$ into "o or o!r ( ^! fatal positions4 ,9 %ort1 ! /rategy really am\nts to ?is3 ! ,briti%] plays ! fir/ n9e moves t ,i h giv51 & al?\gh ! enemy does 8 v be/ to escape1 our g5]al goes aft] 8 antagoni/ & always driv+ him away f t nor?-we/ corn] ultimately closes 9 ) him1 & w9s4 ,z ,i h said1 ! ,briti%] nev] ne$ make m ?an sev5te5 moves 9 all1 & may w9 9 few] moves if ! enemy plays badly4 ,b aft] play+ ^? fir/ n9e moves x does n matt] ev5 if ! ,briti%] makes a few bad ones4 ,he may lose "t1 b _c lose 8 advantage s long z he n[ keeps ! enemy f t[n #a1 & mu/ ev5tually cat* him4 ,?is is a complete explanation ( ! puzzle4 ,x may seem a little complex 9 pr9t1 b 9 practice ! w9n+ play w n[ be q easy to ! read]4 ,make ^? n9e moves1 & !re ought to be no difficulty :atev] 9 f9d+ ! conclud+ l9e ( play4 ,inde$1 x might almo/ be said t !n x is difficult = ! ,briti% g5]al n to cat* ! enemy4 ,x is a "q ( :at 9 *ess we call ! opposition1 & ! visit by ! ,briti%] to t[n #a gives him ! jump on ! enemy1 z ! man 9 ! /reet w\ld say4 ,"h is an illu/rative example 9 :i* ! enemy avoids capture z long z x is possible = him to d s4 ,! ,briti%]'s moves are above ! l9e & ! enemy's bel[ x4 ,play !m alt]nately4 #bd #bj #ai #ae #aa #g #c #a #b #f #aj #ad #ah #ai #bj #bd ---------------------------------------- ------- #ac #i #ac #ag #ba #bj #bd #bc #ai #ae #ai #bc #bd #be #bg ,! enemy mu/ n[ g to #be or ,b1 9 ei!r ( :i* t[ns he is imm$iately captur$4 #cif4--;,a ,,mat* ,,my/]y4 ,if y =m ! ?ree heaps 7& are !re=e ! second to draw71 any "o ( ! foll[+ ?irte5 gr\p+s w give y a w9 if y play correctly3 #ae1 #ad1 #a2 #ae1 #ac1 #b2 #ae1 #ab1 #c2 #ae1 #aa1 #d2 #ae1 #aj1 #e2 #ae1 #i1 #f2 #ae1 #h1 #g2 #ad1 #ac1 #c2 #ad1 #aa1 #e2 #ad1 #i1 #g2 #ac1 #aa1 #f2 #ac1 #aj1 #g2 #ab1 #aa1 #g4 ,! beautiful g5]al solution ( ?is problem is z foll[s4 ,express ! numb] 9 e heap 9 p[]s ( #b1 avoid+ repetitions & rememb]+ t #bj #a4 ,!n if y s leave ! mat*es to y\r oppon5t t !re is an ev5 numb] ( e p[]1 y c w94 ,& if at ! />t y leave ! p[]s ev51 y c always cont9ue to d s ?r\gh\t ! game4 ,take1 z example1 ! la/ gr\p+ giv5 above--#ab1 #aa1 #g4 ,express$ 9 p[]s ( #b we h-- #ab #h #d - - #aa #h - #b #a #g - #d #b #a ------- #b #b #b #b ------- ,z !re are ?us two ( e p[]1 y mu/ w94 ,say y\r oppon5t takes #g f ! #ab heap4 ,he !n leaves-- #e - #d - #a #aa #h - #b #a #g - #d #b #a ------- #a #b #b #c ------- ,"h ! p[]s are n all ev5 9 numb]1 b by tak+ #i f ! #aa heap y imm$iately re/ore y\r w9n+ position1 ?us-- #e - #d - #a #b - - #b - #g - #d #b #a ------- - #b #b #b ------- ,& s on to ! end4 ,?is solution is q g5]al1 & applies to any numb] ( mat*es & any numb] ( heaps4 ,a correspond5t in=ms me t ?is puzzle game 0 fir/ prop\nd$ by ,mr4 ,w4,m4,f4 ,mellor1 b :5 or :]e x 0 publi%$ ,i h n be5 able to asc]ta94 #cig4--,,! ,,mont5egr9 ,,dice ,,game4 ,! play]s %\ld select ! pairs #e & #i1 & #ac & #ae1 if ! *ances ( w9n+ are to be q equal4 ,!re are #baf diff]5t ways 9 :i* ! ?ree dice may fall4 ,!y may add up #e 9 #f diff]5t ways & #i 9 #be diff]5t ways1 mak+ #ca *ances out ( #baf = ! play] :o selects ^! numb]s4 ,also ! dice may add up #ac 9 #ba diff]5t ways1 & #ae 9 #aj diff]5t ways1 ?us giv+ ! o!r play] also #ca *ances 9 #baf4 #cih4--,,! ,,cig> ,,puzzle4 ,n a s+le memb] ( ! club ma/]$ ?is puzzle1 & yet ,i %all %[ t x is s simple t ! m]e/ *ild c und]/& its solution--:5 x is po9t$ out to him6 ,! l>ge majority ( my fri5ds express$ _! entire bewild]m5t4 ,_m consid]$ t ! !oretical result1 9 any case1 is det]m9$ by ! relation%ip betwe5 ! table & ! cig>s2 o!rs1 reg>d+ x z a problem 9 ! !ory ( ,probabilities1 arriv$ at ! conclusion t ! *ances are slightly 9 fav\r ( ! fir/ or second play]1 z ! case may be4 ,"o man took a table & a cig> ( p>ticul> dim5sions1 divid$ ! table into equal sections1 & proce$$ to make ! two play]s fill up ^! sections s t ! second play] %\ld w94 ,b :y %\ld ! fir/ play] be s accommodat+8 ,at any /age he has only to ?r[ d[n a cig> obliquely across sev]al ( ^! sections entirely to upset ,mr4 #b's calculations6 ,we h to assume t ea* play] plays ! be/ possible2 n t "o accommodates ! o!r4 ,! !ories ( "s o!r fri5ds w\ld be q s\nd if ! %ape ( ! cig> 7 t ( a torp$o--p]fectly symmetrical & po9t$ at bo? ends4 ,i w %[ t ! fir/ play] %\ld infallibly w91 if he always plays 9 ! be/ possible mann]4 ,exam9e c>efully ! foll[+ diagram1 ,no4 #a1 & all w be cle>4 ,illu/ration3 #a ,illu/ration3 #b ,! fir/ play] mu/ place 8 fir/ cig> on end 9 ! exact c5tre ( ! table1 z indicat$ by ! little circle4 ,n[1 :atev] ! second play] may d ?r\gh\t1 ! fir/ play] mu/ always repeat x 9 an exactly diametrically opposite position4 ,?us1 if ! second play] places a cig> at ,a1 ,i put "o at ,,aa2 he places "o at ,b1 ,i put "o at ,,bb2 he places "o at ,c1 ,i put "o at ,,cc2 he places "o at ,d1 ,i put "o at ,,dd2 he places "o at ,e1 ,i put "o at ,,ee2 & s on until no m cig>s c be plac$ )\t t\*+4 ,z ! cig>s are suppos$ to be exactly alike 9 e respect1 x is p]fectly cle> t = e move t ! second play] may *oose to make1 x is possible exactly to repeat x on a l9e drawn ?r\gh ! c5tre ( ! table4 ,! second play] c always duplicate ! fir/ play]'s move1 no matt] :]e he may place a cig>1 or :e!r he places x on end or on its side4 ,z ! cig>s are all alike 9 e respect1 "o w obvi\sly balance ov] ! edge ( ! table at precisely ! same po9t z ano!r4 ,( c\rse1 z ea* play] is suppos$ to play 9 ! be/ possible mann]1 x becomes a matt] ( !ory4 ,x is no valid objection to say t 9 actual practice "o w\ld n be suffici5tly exact to be sure ( w9n+4 ,if z ! fir/ play] y did n w91 x w\ld be 9 consequ5ce ( y\r n hav+ play$ ! be/ possible4 ,! second diagram w s]ve to %[ :y ! fir/ cig> mu/ be plac$ on end4 7,& "h ,i w say t ! fir/ cig> t ,i select$ f a box ,i 0 able s to /& on end1 & ,i am all[$ to assume t all ! o!r cig>s w\ld d ! same47 ,if ! fir/ cig> 7 plac$ on its side1 z at ,f1 !n ! second play] c\ld place a cig> z at ,g--z ne> z possible1 b n actually t\*+ ,f4 ,n[1 9 ?is position y _c repeat 8 play on ! opposite side1 because ! two ends ( ! cig> are n alike4 ,x w be se5 t ,,gg1 :5 plac$ on ! opposite side 9 ! same relation to ! c5tre1 int]sects1 or lies on top (1 ,f1 :]eas ! cig>s are n all[$ to t\*4 ,y mu/ !re=e put ! cig> f>!r away f ! c5tre1 :i* w\ld result 9 y\r hav+ insuffici5t room betwe5 ! c5tre & ! bottom left-h& corn] to repeat ev]y?+ t ! o!r play] w\ld d betwe5 ,g & ! top "r-h& corn]4 ,!re=e ! result w\ld n be a c]ta9 w9 = ! fir/ play]4 #cii4--,,! ,,tr\blesome ,,eight4 ,illu/ration3 --------- #d #h #b --------- #c #e #g --------- #g #b #e --------- ,! conditions 7 to place a diff]5t numb] 9 ea* ( ! n9e cells s t ! ?ree r[s1 ?ree columns1 & two diagonals %\ld ea* add up #ae4 ,probably ! read] at fir/ set himself an impossible task ?r\gh read+ into ^! conditions some?+ :i* is n !re--a common error 9 puzzle-solv+4 ,if ,i _h said a diff]5t figure1 in/ead ( a diff]5t numb]1 x w\ld h be5 q impossible ) ! #h plac$ any:]e b 9 a corn]4 ,& x w\ld h be5 equally impossible if ,i _h said a diff]5t :ole numb]4 ,b a numb] may1 ( c\rse1 be fractional1 & !re9 lies ! secret ( ! puzzle4 ,! arrangem5t %[n 9 ! figure w be f\nd to comply exactly ) ! conditions3 all ! numb]s are diff]5t1 & ! squ>e adds up #ae 9 all ! requir$ eight ways4 #djj4--,,! ,,magic ,,/rips4 ,!re are ( c\rse six diff]5t places betwe5 ! sev5 figures 9 :i* a cut may be made1 & ! secret lies 9 keep+ "o /rip intact & cutt+ ea* ( ! o!r six 9 a diff]5t place4 ,aft] ! cuts h be5 made !re are a l>ge numb] ( ways 9 :i* ! ?irte5 pieces may be plac$ toge!r s z to =m a magic squ>e4 ,"h is "o ( !m3-- ,illu/ration3 ------------- #a #b #c #d #e #f #g ------------ #c #d #e #f #ga #b ----------- #e #f #ga #b #c #d ----------- #ga #b #c #d #e #f ----------- #b #c #d #e #f #ga ----------- #d #e #f #ga #b #c ----------- #f #ga #b #c #d #e ------------ ,! arrangem5t has "s r int]e/+ features4 ,x w be se5 t ! uncut /rip is at ! top1 b x w be f\nd t if ! bottom r[ ( figures be plac$ at ! top ! numb]s w /ill =m a magic squ>e1 & t e successive removal f ! bottom to ! top 7c>ry+ ! uncut /rip /age by /age to ! bottom7 w produce ! same result4 ,if we imag9e ! numb]s to be on sev5 complete p]p5dicul> /rips1 x w be f\nd t ^! columns c\ld also be mov$ 9 succession f left to "r or f "r to left1 ea* "t produc+ a magic squ>e4 #dja4--;,,eight ,,jolly ,,gaol-,,birds4 ,!re are eight ways ( =m+ ! magic squ>e--all m]ely diff]5t aspects ( "o fundam5tal arrangem5t4 ,?us1 if y give our fir/ squ>e a qu>t] turn y w get ! second squ>e2 & z ! f\r sides may be 9 turn br\ght to ! top1 !re are f\r aspects4 ,^! f\r 9 turn reflect$ 9 a mirror produce ! rema9+ f\r aspects4 ,n[1 ( ^! eight arrangem5ts only f\r c possibly be rea*$ "u ! conditions1 & only two ( ^! f\r c be rea*$ 9 ! fewe/ possible moves1 :i* is n9ete54 ,^! two arrangem5ts are %[n4 ,move ! m5 9 ! foll[+ ord]3 #e1 #c1 #b1 #e1 #g1 #f1 #d1 #a1 #e1 #g1 #f1 #d1 #a1 #f1 #d1 #h1 #c1 #b1 #g1 & y get ! fir/ squ>e4 ,move !m ?us3 #d1 #a1 #b1 #d1 #a1 #f1 #g1 #a1 #e1 #h1 #a1 #e1 #f1 #g1 #e1 #f1 #d1 #b1 #g1 & y h ! arrangem5t 9 ! second squ>e4 ,9 ! fir/ case e man has mov$1 b 9 ! second case ! man numb]$ #c has nev] left 8 cell4 ,!re=e ,no4 #c mu/ be ! ob/9ate prison]1 & ! second squ>e mu/ be ! requir$ arrangem5t4 ,illu/ration3 --------- --------- #e #g #g #d #c - -- -- - - -- -- - #f #d #b #d #h - -- -- - - -- -- - #a #h #c #e #f #a --------- --------- #djb4--,,n9e ,,jolly ,,gaol ,,birds4 ,!re is a pitfall set = ! unw>y 9 ?is little puzzle4 ,at ! />t "o man is all[$ to be plac$ on ! %\ld]s ( ano!r1 s z to give always "o empty cell to enable ! prison]s to move ab\t )\t any two ev] be+ 9 a cell toge!r4 ,! two unit$ prison]s are all[$ to add _! numb]s toge!r1 & are1 ( c\rse1 p]mitt$ to rema9 toge!r at ! completion ( ! magic squ>e4 ,b !y are obvi\sly n compell$ s to rema9 toge!r1 provid$ t "o ( ! pair on 8 f9al move does n break ! condition ( ent]+ a cell already occupi$4 ,aft] ! acute solv] has notic$ ?is po9t1 x is = him to det]m9e :i* me?od is ! bett] "o--= ! two to be toge!r at ! c\nt or to sep>ate4 ,z a matt] ( fact1 ! puzzle c be solv$ 9 sev5te5 moves if ! m5 are to rema9 toge!r2 b if !y sep>ate at ! end1 !y may actually save a move & p]=m ! feat 9 sixte56 ,! trick consi/s 9 plac+ ! man 9 ! c5tre on ! back ( "o ( ! corn] m51 & !n work+ ! pair into ! c5tre be=e _! f9al sep>ation4 ,illu/ration3 ,a ,b --------- --------- #b #i #d #f #g #b - -- -- - - -- -- - #g #e #c #a #e #i - -- -- - - -- -- - #f #a #h #h #c #d --------- --------- ,"h are ! moves = gett+ ! m5 into "o or o!r ( ! above two positions4 ,! numb]s are ^? ( ! m5 9 ! ord] 9 :i* !y move into ! cell t is = ! "t be+ vacant4 ,! pair is %[n 9 brackets3-- ,place #e on #a4 ,!n1 #f1 #i1 #h1 #f1 #d1 7#f71 #b1 #d1 #i1 #c1 #d1 #i1 7#f71 #g1 #f1 #a4 ,place #e on #i4 ,!n1 #d1 #a1 #b1 #d1 #f1 7#ad71 #h1 #f1 #a1 #g1 #f1 #a1 7#ad71 #c1 #d1 #i4 ,place #e on #c4 ,!n1 #f1 7#h71 #b1 #f1 #d1 #g1 #h1 #d1 #g1 #a1 #f1 #g1 7#h71 #i1 #d1 #c4 ,place #e on #g4 ,!n1 #d1 7#ab71 #h1 #d1 #f1 #c1 #b1 #f1 #c1 #i1 #d1 #c1 7#ab71 #a1 #f1 #g4 ,! fir/ & second solutions produce ,diagram ,a2 ! second & ?ird produce ,diagram ,b4 ,!re are only sixte5 moves 9 e case4 ,hav+ f\nd ! fewe/ moves1 we _h to consid] h[ we 7 to make ! burd5$ man d z little "w z possible4 ,x w at once be se5 t z ! pair h to g into ! c5tre be=e sep>at+ !y mu/ take at fewe/ two moves4 ,! lab\r ( ! burd5$ man c only be r$uc$ by adopt+ ! o!r me?od ( solution1 :i*1 h[ev]1 =ces u to take ano!r move4 #djc4--,,! ,,spani% ,,dungeon4 ,illu/ration -------------------- -------------------- #a #b #c #d #aj #i #g #d #e #f #g #h #f #e #aa #h #i #aj #aa #ab #a #b #ab #ae #ac #ad #ae #ac #ad #c -------------------- -------------------- ,?is c be/ be solv$ by work+ backw>ds--t is to say1 y mu/ fir/ cat* y\r squ>e1 & !n "w back to ! orig9al position4 ,we mu/ fir/ con/ruct ^? squ>es :i* are f\nd to require ! lea/ am\nt ( readju/m5t ( ! numb]s4 ,_m ( ^! we "k _c possibly be rea*$4 ,:5 we h be=e u ! mo/ fav\rable possible arrangem5ts1 x !n becomes a "q ( c>eful analysis to discov] :i* position c be rea*$ 9 ! fewe/ moves4 ,i am afraid1 h[ev]1 x is only aft] consid]able /udy & exp]i5ce t ! solv] is able to get su* a grasp ( ! v>i\s areas ( di/urbance & me?ods ( circulation t 8 judgm5t is ( mu* value to him4 ,! second diagram is a mo/ fav\rable magic squ>e position4 ,x w be se5 t prison]s #d1 #h1 #ac1 & #ad are left 9 _! orig9al cells4 ,?is position may be rea*$ 9 z few z ?irty-sev5 moves4 ,"h are ! moves3 #ae1 #ad1 #aj1 #f1 #g1 #c1 #b1 #g1 #f1 #aa1 #c1 #b1 #g1 #f1 #aa1 #aj1 #ad1 #c1 #b1 #aa1 #aj1 #i1 #e1 #a1 #f1 #aj1 #i1 #e1 #a1 #f1 #aj1 #i1 #e1 #b1 #ab1 #ae1 #c4 ,?is %ort solution w probably surprise _m read]s :o may n f9d a way "u f sixty to a hundr$ moves4 ,! clev] prison] 0 ,no4 #f1 :o 9 ! orig9al illu/ration w be se5 ) 8 arms ext5d$ call+ out ! moves4 ,he & ,no4 #aj did mo/ ( ! "w1 ea* *ang+ 8 cell five times4 ,no4 #ab1 ! man ) ! crook$ leg1 0 lame1 & !re=e =tunately _h only to pass f 8 cell into ! next "o :5 8 "t came r\nd4 #djd4--,,! ,,sib]ian ,,dungeons4 ,illu/ration -------------------- #h #e #aj #aa #af #ac #b #c #a #ab #g #ad #i #d #ae #f -------------------- ,9 attempt+ to solve ?is puzzle x is cle>ly necess>y to seek su* magic squ>es z seem ! mo/ fav\rable = our purpose1 & !n c>efully exam9e & try !m = fewe/ moves4 ,( c\rse x at once occurs to u t if we c adopt a squ>e 9 :i* a c]ta9 numb] ( m5 ne$ n leave _! orig9al cells1 we may save moves on ! "o h&1 b we may ob/ruct our movem5ts on ! o!r4 ,= example1 a magic squ>e may be =m$ ) ! #f1 #g1 #ac1 & #af unmov$2 b 9 su* case x is obvi\s t a solution is impossible1 s9ce cells #ad & #ae c nei!r be left nor ent]$ )\t break+ ! condition ( no two m5 ev] be+ 9 ! same cell toge!r4 ,! foll[+ solution 9 f\rte5 moves 0 f\nd by ,mr4 ,g4 ,wo!rspoon3 #h-ag1 #af-ba1 #f-af1 #ad-h1 #e-ah1 #d-ad1 #c-bd1 #aa-bj1 #aj-ai1 #b-bc1 #ac-bb1 #ab-f1 #a-e1 #i-ac4 ,z ?is solution is 9 :at ,i consid] ! !oretical m9imum numb] ( moves1 ,i am confid5t t x _c be improv$ ^u1 & on ?is po9t ,mr4 ,wo!rspoon is ( ! same op9ion4 #dje4--;,,c>d ,,magic ,,squ>es4 ,arrange ! c>ds z foll[s = ! ?ree new squ>es3-- #c #b #d #f #e #g #i #h #aj #d #c #b #g #f #e #aj #i #h #b #d #c #e #g #f #h #aj #i ,?ree aces & "o t5 are n us$4 ,! summations ( ! f\r squ>es are ?us3 #i1 #ae1 #ah1 & #bg--;all diff]5t1 z requir$4 #djf4--,,! ,,eighte5 ,,dom9oes4 ,illu/ration ,! illu/ration expla9s itself4 ,x w be f\nd t ! pips 9 e column1 r[1 & long diagonal add up #ah1 z requir$4 #djg4--,,two ,,new ,,magic ,,squ>es4 ,"h are two solutions t fulfil ! conditions3-- ,illu/ration3 ,,subtract+ ,,divid+ #aa #d #ad #ac #cf #h #ed #bg #af #g #a #b #baf #ab #a #b #f #e #c #ab #f #c #d #gb #i #ai #h #ae #i #ah #bd #ajh ,! fir/1 by subtract+1 has a con/ant #h1 & ! associat$ pairs all h a diff]5ce ( #d4 ,! second squ>e1 by divid+1 has a con/ant #i1 & all ! associat$ pairs produce #c by division4 ,^! are two rem>kable & in/ructive squ>es4 #djh4--,,magic ,,squ>es ,,( ,,two ,,degrees4 ,! foll[+ is ! squ>e t ,i con/ruct$4 ,z x /&s ! con/ant is #bfj4 ,if = e numb] y sub/itute1 9 its allott$ place1 its squ>e1 !n ! con/ant w be #aa1ahj4 ,read]s c write out = !mselves ! second degree squ>e4 ,illu/ration3 #g #ec #da #bg #b #eb #dh #cj #ab #eh #ch #bd #ac #fc #ce #ag -------------------------- #ea #a #bi #dg #ed #h #bh #db #fd #ad #ah #cf #eg #aa #bc #cg -------------------------- #be #dc #ee #e #cb #df #ej #d #bb #dj #fj #aj #ai #cc #fa #ae -------------------------- #de #ca #c #di #dd #bf #f #ef #cd #bj #af #fb #ci #ba #i #ei ,! ma9 key to ! solution is ! pretty law t if eight numb]s sum to #bfj & _! squ>es to #aa1ahj1 !n ! same w happ5 9 ! case ( ! eight numb]s t are complem5t>y to #fe4 ,?us #a #ah #bc #bf #ca #dh #ef #eg #bfj1 & ! sum ( _! squ>es is #aa1ahj4 ,!re=e #fd #dg #db #ci #cd #ag #i #h 7obta9$ by subtract+ ea* ( ! above numb]s f #fe7 w sum to #bfj & _! squ>es to #aa1ahj4 ,note t 9 e "o ( ! sixte5 small] squ>es ! two diagonals sum to #fe4 ,!re are f\r columns & f\r r[s ) _! complem5t>y columns & r[s4 ,let u pick out ! numb]s f\nd 9 ! #bnd1 #a/1 #d?1 & #crd r[s & arrange !m ?us 3-- ,illu/ration3 #a #h #bh #bi #db #dg #ea #ed #b #g #bg #cj #da #dh #eb #ec #c #f #bf #ca #dd #de #di #ef #d #e #be #cb #dc #df #ej #ee ,"h ea* column conta9s f\r consecutive numb]s cyclically arrang$1 f\r runn+ 9 "o direction & f\r 9 ! o!r4 ,! numb]s 9 ! #bnd1 #e?1 #crd1 & #h? columns ( ! squ>e may be simil>ly gr\p$4 ,! great difficulty lies 9 discov]+ ! conditions gov]n+ ^! gr\ps ( numb]s1 ! pair+ ( ! complem5t>ies 9 ! squ>es ( f\r & ! =mation ( ! diagonals4 ,b :5 a correct solution is %[n1 z above1 x discloses all ! m important keys to ! my/]y4 ,i am incl9$ to ?9k ?is squ>e ( two degrees ! mo/ elegant ?+ t exi/s 9 magics4 ,i believe su* a magic squ>e _c be con/ruct$ 9 ! case ( any ord] l[] ?an #h4 #dji4--,,! ,,baskets ,,( ,,plums4 ,z ! m]*ant told 8 man to di/ribute ! cont5ts ( "o ( ! baskets ( plums among "s *ildr51 x w\ld n be p]missible to give ! complete basketful to "o *ild2 & z x 0 also direct$ t ! man 0 to give plums to e *ild1 s t ea* %\ld receive an equal numb]1 x w\ld also n be all[$ to select j z _m *ildr5 z !re 7 plums 9 a basket & give ea* *ild a s+le plum4 ,consequ5tly1 if ! numb] ( plums 9 e basket 0 a prime numb]1 !n ! man w\ld be correct 9 say+ t ! propos$ di/ribution 0 q impossible4 ,our puzzle1 !re=e1 resolves itself into =m+ a magic squ>e ) n9e diff]5t prime numb]s4 ,illu/ration ,a ,b --------------- --------------- #g #fa #dc #hc #bi #aja #gc #cg #a #hi #ga #ec #ca #ac #fg #da #aac #ei --------------- --------------- ,c ,d --------------- --------------- #ajc #gi #cg #affi #aii #abdi #g #gc #aci #fai #ajci #adei #aji #fg #dc #hbi #ahgi #dji --------------- --------------- ,9 ,diagram ,a we h a magic squ>e 9 prime numb]s1 & x is ! "o giv+ ! smalle/ con/ant sum t is possible4 ,z to ! little trap ,i m5tion$1 x is cle> t ,diagram ,a is b>r$ out by ! words e basket conta9$ plums1 = "o plum is n plums4 ,& z we 7 ref]r$ to ! baskets1 z %[n 9 ! illu/ration1 x is p]fectly evid5t1 )\t actually attempt+ to c\nt ! plums1 t !re are at any rate m ?an #g plums 9 e basket4 ,!re=e ,c is also1 /rictly speak+1 b>r$4 ,numb]s ov] #bj & "u1 say1 #bej w\ld c]ta9ly come well )9 ! range ( possibility1 & a l>ge numb] ( arrangem5ts w\ld come )9 ^! limits4 ,diagram ,b is "o ( !m4 ,( c\rse we c all[ = ! false bottoms t are s frequ5tly us$ 9 ! baskets ( fruitsell]s to make ! basket appe> to conta9 m fruit ?an x really does4 ,sev]al correspond5ts assum$ 7on :at gr\nds ,i _c ?9k7 t 9 ! case ( ?is problem ! numb]s _c be 9 consecutive ari?metical progression1 s ,i give ,diagram ,d to %[ t !y 7 mi/ak54 ,! numb]s are #aii1 #dji1 #fai1 #hbi1 #a1jci1 #a1bdi1 #a1dei1 #a1ffi1 & #a1hgi--;all primes ) a common diff]5ce ( #baj4 #daj4--,,! ,,m&>9',s ,t ,,puzzle4 ,!re are _m diff]5t ways ( arrang+ ! numb]s1 & ei!r ! #b or ! #c may be omitt$ f ! ,t enclosure4 ,! arrangem5t t ,i give is a nasik squ>e4 ,out ( ! total ( #bh1hjj nasik squ>es ( ! fif? ord] ?is is ! only "o 7) its "o reflection7 t fulfils ! ,t condition4 ,?is puzzle 0 sugge/$ to me by ,dr4 ,c4 ,planck4 ,illu/ration3 ,,! ,,m&>9',s ,t ,,puzzle4 ------------------------- #ai #bc #aa #e #g #a #aj #ag #bd #ac #bb #ad #c #f #bj #h #af #be #ab #d #ae #b #i #ah #ba ------------------------- #daa4--;,a ,,magic ,,squ>e ,,( ,,composites4 ,! problem really am\nts to f9d+ ! smalle/ prime su* t ! next high] prime %all exce$ x by #aj at lea/4 ,if we write out a little li/ ( primes1 we %all n ne$ to exce$ #aej to discov] :at we require1 = aft] #aac ! next prime is #abg4 ,we c !n =m ! squ>e 9 ! diagram1 :]e e numb] is composite4 ,?is is ! solution 9 ! smalle/ numb]s4 ,we ?us see t ! answ] is arriv$ at q easily1 9 a squ>e ( ! ?ird ord]1 by trial4 ,b ,i propose to %[ h[ we may get an answ] 7n1 x is true1 ! "o 9 smalle/ numb]s7 )\t any tables or trials1 b 9 a v direct & rapid mann]4 ,illu/ration --------------- #aba #aad #aai #aaf #aah #abj #aag #abb #aae --------------- ,fir/ write d[n any consecutive numb]s1 ! smalle/ be+ great] ?an #a--say1 #b1 #c1 #d1 #e1 #f1 #g1 #h1 #i1 #aj4 ,! only factors 9 ^! numb]s are #b1 #c1 #e1 & #g4 ,we !re=e multiply ^! f\r numb]s toge!r & add ! product1 #baj1 to ea* ( ! n9e numb]s4 ,! result is ! n9e consecutive composite numb]s1 #bab to #bbj inclusive1 ) :i* we c =m ! requir$ squ>e4 ,e numb] w necess>ily be divisible by its diff]5ce f #baj4 ,x w be v obvi\s t by ?is me?od we may f9d z _m consecutive composites z ev] we please4 ,suppose1 = example1 we wi% to =m a magic squ>e ( sixte5 su* numb]s2 !n ! numb]s #b to #ag conta9 ! factors #b1 #c1 #e1 #g1 #aa1 #ac1 & #ag1 :i*1 multipli$ toge!r1 make #eajeaj to be add$ to produce ! sixte5 numb]s #eajeab to #eajebg inclusive1 all ( :i* are composite z be=e4 ,b1 z ,i h said1 ^! are n ! answ]s 9 ! smalle/ numb]s3 = if we add #ebc to ! numb]s #a to #af1 we get sixte5 consecutive composites2 & if we add #a1cbg to ! numb]s #a to #be1 we get tw5ty-five consecutive composites1 9 ea* case ! smalle/ numb]s possible4 ,yet if we requir$ to =m a magic squ>e ( a hundr$ su* numb]s1 we %\ld f9d x a big task by means ( tables1 ?\gh by ! process ,i h %[n x is q a simple matt]4 ,ev5 to f9d ?irty-six su* numb]s y w se>* ! tables up to #aj1jjj )\t success1 & ! difficulty increases 9 an accel]at+ ratio ) ea* squ>e ( a l>g] ord]4 #dab4--,,! ,,magic ,,knight',s ,,t\r4 -------------------------------- #df #ee #dd #ai #eh #i #bb #g -------------------------------- #dc #ah #dg #ef #ba #f #ei #aj -------------------------------- #ed #de #bj #da #ab #eg #h #bc -------------------------------- #ag #db #ec #dh #e #bd #aa #fj -------------------------------- #eb #c #cb #ac #dj #fa #cd #be -------------------------------- #ca #af #di #d #cc #bh #cg #fb -------------------------------- #b #ea #ad #bi #fd #ci #bf #ce -------------------------------- #ae #cj #a #ej #bg #cf #fc #ch -------------------------------- ,"h ea* successive numb] 79 num]ical ord]7 is a knight's move f ! prec$+ numb]1 & z #fd is a knight's move f #a1 ! t\r is re-entrant4 ,all ! columns & r[s add up #bfj4 ,un=tunately1 x is n a p]fect magic squ>e1 because ! diagonals are incorrect1 "o add+ up #bfd & ! o!r #bef--requir+ only ! transf] ( #d f "o diagonal to ! o!r4 ,i ?9k ?is is ! be/ result t has ev] be5 obta9$ 7ei!r re-entrant or n71 & nobody c yet say :e!r a p]fect solution is possible or impossible4 #dac4--;,a ,,*essbo>d ,,fallacy4 ,illu/ration ,! explanation ( ?is little fallacy is z foll[s4 ,! error lies 9 assum+ t ! little triangul> piece1 m>k$ ,c1 is exactly ! same height z "o ( ! little squ>es ( ! bo>d4 ,z a matt] ( fact1 its height 7if we make ! sixty-f\r squ>es ea* a squ>e in*7 w be #aa_/g 94 ,consequ5tly ! rectangle is really #ia_/g 94 by #g 941 s t ! area is sixty-f\r squ>e in*es 9 ei!r case4 ,n[1 al?\gh ! pieces d fit toge!r exactly to =m ! p]fect rectangle1 yet ! directions ( ! horizontal l9es 9 ! pieces w n co9cide4 ,! new diagram above w make ev]y?+ q cle> to ! read]4 #dad4--,,:o ,,0 ,,fir/8 ,biggs1 :o saw ! smoke1 w\ld be fir/2 ,c>p5t]1 :o saw ! bullet /rike ! wat]1 w\ld be second2 & ,&]son1 :o he>d ! report1 w\ld be la/ ( all4 #dae4--;,a ,,wond]ful ,,village4 ,:5 ! sun is 9 ! horizon ( any place 7:e!r 9 ,japan or else:]e71 he is ! l5g? ( half ! e>?'s diamet] m di/ant f t place ?an 9 8 m]idian at noon4 ,z ! e>?'s semi-diamet] is ne>ly #d1jjj miles1 ! sun mu/ be consid]ably m ?an #c1jjj miles ne>] at noon ?an at 8 ris+1 !re be+ no valley ev5 ! hundr$? "p ( #a1jjj miles deep4 #daf4--;,a ,,cal5d> ,,puzzle4 ,! fir/ day ( a c5tury c nev] fall on a ,sunday2 nor on a ,w$nesday or a ,friday4 #dag4--,,! ,,tir+-,,irons4 ,i w give my complete work+ ( ! solution1 s t read]s may see h[ easy x is :5 y "k h[ to proce$4 ,& fir/ ( all1 z !re is an ev5 numb] ( r+s1 ,i w say t !y may all be tak5 (f 9 "o-?ird ( 7#b7n #a7 - #b7 moves2 & s9ce n 9 our case is #ad1 all ! r+s may be tak5 (f 9 #aj1ibb moves4 ,!n ,i say #aj1ibb - #i1iii #ibc1 & proce$ to f9d ! position :5 only #ibc out ( ! #aj1ibb moves rema9 to be made4 ,"h is ! curi\s me?od ( do+ ?is4 ,x is bas$ on ! b9>y scale me?od us$ by ,monsieur ,l4 ,gros1 = an acc\nt ( :i* see ,w4,w4 ,r\se ,ball's ,ma!matical ,recreations4 ,divide #ibc by #b1 & we get #dfa & ! rema9d] #a2 divide #dfa by #b1 & we get #bcj & ! rema9d] #a2 divide #bcj by #b1 & we get #aae & ! rema9d] n\ght4 ,keep on divid+ by #b 9 ?is way z long z possible1 & all ! rema9d]s w be f\nd to be #a1 #a1 #a1 #j1 #j1 #a1 #a1 #j1 #a1 #a1 ! la/ rema9d] be+ to ! left & ! fir/ rema9d] to ! "r4 ,z !re are f\rte5 r+s & only t5 figures1 we place ! diff]5ce1 9 ! =m ( f\r n\ghts1 9 brackets to ! left1 & bracket all ^? figures t repeat a figure on _! left4 ,!n we get ! foll[+ arrangem5t3 7#j #j #j #j7 #a 7#a #a7 #j 7#j7 #a 7#a7 #j #a 7#a74 ,?is is ! correct answ] to ! puzzle1 = if we n[ place r+s bel[ ! l9e to repres5t ! figures 9 brackets & r+s on ! l9e = ! o!r figures1 we get ! solution 9 ! requir$ =m1 z bel[3-- ,o ,o ,o ,,oo ------------------------- ,,oooo ,,oo ,o ,o ,o ,?is is ! exact position ( ! r+s aft] ! #i1iii? move has be5 made1 & ! read] w f9d t ! me?od %[n w solve any simil> "q1 no matt] h[ _m r+s are on ! tir+-irons4 ,b 9 work+ ! inv]se process1 :]e y are requir$ to asc]ta9 ! numb] ( moves necess>y 9 ord] to rea* a giv5 position ( ! r+s1 ! rule w require a little modification1 because x does n necess>ily foll[ t ! position is "o t is actually rea*$ 9 c\rse ( tak+ (f all ! r+s on ! irons1 z ! read] w pres5tly see4 ,i w "h /ate t :]e ! total numb] ( r+s is odd ! numb] ( moves requir$ to take !m all (f is "o-?ird ( 7#b7n #a7 - #a74 ,) n r+s 7:]e n is odd7 !re are #bn positions c\nt+ all on & all (f4 ,9 7#a_/c77b7n #a7 #b7 positions !y are all remov$4 ,! numb] ( positions n us$ is 7#a_/c77bn - #b74 ,) n r+s 7:]e n is ev57 !re are #bn positions c\nt+ all on & all (f4 ,9 7#b7n #a7 #a7 positions !y are all remov$4 ,! numb] ( positions n us$ is "h 7#a_/c77bn - #a74 ,x w be conv5i5t to tabulate a few cases4 ---------------------------------------- -- ,no4 ( ,total ,positions ,positions ,r+s4 ,positions4 us$4 n us$4 ---------------------------------------- -- #a #b #b #j #c #h #f #b #e #cb #bb #aj #g #abh #hf #db #i #eab #cdb #agj #b #d #c #a #d #af #aa #e #f #fd #dc #ba #h #bef #aga #he #aj #ajbd #fhc #cda ---------------------------------------- -- ,note fir/ t ! numb] ( positions us$ is "o m ?an ! numb] ( moves requir$ to take all ! r+s (f1 because we are includ+ all on :i* is a position b n a move4 ,!n note t ! numb] ( positions n us$ is ! same z ! numb] ( moves us$ to take (f a set t has "o r+ few]4 ,= example1 x takes #he moves to remove #g r+s1 & ! #db positions n us$ are exactly ! numb] ( moves requir$ to take (f a set ( #f r+s4 ,! fact is t if !re are #g r+s & y take (f ! fir/ #f1 & !n wi% to remove ! #g? r+1 !re is no c\rse op5 to y b to rev]se all ^? #db moves t nev] ought to h be5 made4 ,9 o!r words1 y mu/ replace all ! #g r+s on ! loop & />t afre%6 ,y ought fir/ to h tak5 (f #e r+s1 to d :i* y %\ld h tak5 (f #c r+s1 & previ\sly to t #a r+4 ,to take (f #f y fir/ remove #b & !n #d r+s4 #dah4--,,su* ,a ,,gett+ ,,up/airs4 ,numb] ! treads 9 regul> ord] upw>ds1 #a to #h4 ,!n proce$ z foll[s3 #a 7/ep back to floor71 #a1 #b1 #c 7#b71 #c1 #d1 #e 7#d71 #e1 #f1 #g 7#f71 #g1 #h1 l&+ 7#h71 l&+4 ,! /eps 9 brackets are tak5 9 a backw>d direction4 ,x w ?us be se5 t by return+ to ! floor aft] ! fir/ /ep1 & !n always go+ ?ree /eps =w>d = "o /ep backw>d1 we p]=m ! requir$ feat 9 n9ete5 /eps4 #dai4--,,! ,,five ,,p5nies4 ,illu/ration ,fir/ lay ?ree ( ! p5nies 9 ! way %[n 9 ,fig4 #a4 ,n[ hold ! rema9+ two p5nies 9 ! position %[n 9 ,fig4 #b1 s t !y t\* "o ano!r at ! top1 & at ! base are 9 contact ) ! ?ree horizontally plac$ co9s4 ,!n ! five p5nies w be equidi/ant1 = e p5ny w t\* e o!r p5ny4 #dbj4--,,! ,,indu/ri\s ,,bookworm4 ,! ha/y read] w assume t ! bookworm1 9 bor+ f ! fir/ to ! la/ page ( a book 9 ?ree volumes1 /&+ 9 _! prop] ord] on ! %elves1 has to g ?r\gh all ?ree volumes & f\r cov]s4 ,?is1 9 our case1 w\ld mean a di/ance ( #i 941 :i* is a long way f ! correct answ]4 ,y w f9d1 on exam9+ any ?ree consecutive volumes on y\r %elves1 t ! fir/ page ( ,vol4 ,i4 & ! la/ page ( ,vol4 ,,iii4 are actually ! pages t are ne>e/ to ,vol4 ,,ii41 s t ! worm w\ld only h to p5etrate f\r cov]s 7toge!r1 947 & ! leaves 9 ! second volume 7#c 9471 or a di/ance ( #c in*es1 9 ord] to tunnel f ! fir/ page to ! la/4 #dba4--;,a ,,*a9 ,,puzzle4 ,to op5 & rejo9 a l9k co/s ?reep5ce4 ,!re=e to jo9 ! n9e pieces into an endless *a9 w\ld co/ #bs4 #c;d41 :]eas a new *a9 w\ld co/ #bs4 #b;d4 ,b if we break up ! piece ( eight l9ks1 ^! eight w jo9 toge!r ! rema9+ eight pieces at a co/ ( #bs4 ,b !re is a subtle way ( ev5 improv+ on ?is4 ,break up ! two pieces conta9+ ?ree & f\r l9ks respectively1 & ^! sev5 w jo9 toge!r ! rema9+ sev5 pieces at a co/ ( only #as4 #i;d4 #dbb4--,,! ,,sabba? ,,puzzle4 ,! way ! au?or ( ! old pos] propos$ to solve ! difficulty 0 z foll[s3 ,f ! ,jew's abode let ! ,*ri/ian & ! ,turk set out on a t\r r\nd ! globe1 ! ,*ri/ian go+ due ea/ & ! ,turk due we/4 ,read]s ( ,edg> ,allan ,poe's /ory1 ,?ree ,sundays 9 a ,week1 or ( ,jules ,v]ne's ,r\nd ! ,_w 9 ,eighty ,days1 w "k t su* a proce$+ w result 9 ! ,*ri/ian's ga9+ a day & 9 ! ,turk's los+ a day1 s t :5 !y meet aga9 at ! h\se ( ! ,jew _! reckon+ w agree ) 81 & all ?ree may keep _! ,sabba? on ! same day4 ,! correctness ( ?is answ]1 ( c\rse1 dep5ds on ! popul> notion z to ! def9ition ( a day--! av]age duration betwe5 successive sun-rises4 ,x is an old quibble1 & q s\nd en\gh = puzzle purposes4 ,/rictly speak+1 ! two travell]s ought to *ange _! reckon+s on pass+ ! #ahj? m]idian2 o!rwise we h to admit t at ! ,nor? or ,s\? ,pole !re w\ld only be "o ,sabba? 9 sev5 ye>s4 #dbc4--,,! ,,ruby ,,broo*4 ,9 ?is case we 7 %[n a sket* ( ! broo* exactly z x appe>$ aft] ! f\r rubies _h be5 /ol5 f x4 ,! read] 0 ask$ to %[ ! positions f :i* ! /ones may h be5 tak52 = x is n possible to %[ precisely h[ ! gems 7 orig9ally plac$1 because !re are _m su* ways4 ,b an important po9t 0 ! /atem5t by ,lady ,littlewood's bro!r3 ,i "k ! broo* well4 ,x orig9ally conta9$ =ty-five /ones1 & !re are n[ only =ty-"o4 ,somebody has /ol5 f\r rubies1 & !n reset z small a numb] z possible 9 su* a way t !re %all always be eight /ones 9 any ( ! directions y h m5tion$4 ,illu/ration ,! diagram %[s ! arrangem5t be=e ! robb]y4 ,x w be se5 t x 0 only necess>y to reset "o ruby--! "o 9 ! c5tre4 ,any solution involv+ ! resett+ ( m ?an "o /one is n 9 accordance ) ! bro!r's /atem5t1 & mu/ !re=e be wrong4 ,! orig9al arrangem5t 01 ( c\rse1 a little unsymmetrical1 & = ?is reason ! broo* 0 describ$ z r ecc5tric4 #dbd4--,,! ,,dovetail$ ,,block4 ,illu/ration ,! my/]y is made cle> by ! illu/ration4 ,x w be se5 at once h[ ! two pieces slide toge!r 9 a diagonal direction4 #dbe4--;,,jack ,,& ,,! ,,bean/alk4 ,! s]i\s blund] t ! arti/ made 9 ?is draw+ 0 9 depict+ ! t5drils ( ,illu/ration ! bean climb+ spirally z at ,a above1 :]eas ! ,fr5* bean1 or sc>let runn]1 ! v>iety cle>ly select$ by ! arti/ 9 ! abs5ce ( any au?oritative in=mation on ! po9t1 always climbs z %[n at ,b4 ,v few seem to be aw>e ( ?is curi\s little fact4 ,?\gh ! bean always insi/s on a s9i/rorsal gr[?1 z ,b1 ! hop pref]s to climb 9 a dextrorsal mann]1 z ,a4 ,:y1 is "o ( ! my/]ies t ,nature has n yet unfold$4 #dbf4--,,! ,,hymn-,,bo>d ,,pos]4 ,?is puzzle is n ne>ly s easy z x looks at fir/ sight4 ,x 0 requir$ to f9d ! smalle/ possible numb] ( plates t w\ld be necess>y to =m a set = ?ree hymn-bo>ds1 ea* ( :i* w\ld %[ ! five hymns sung at any p>ticul> s]vice1 & !n to discov] ! l[e/ possible co/ = ! same4 ,! hymn-book conta9s #gjj hymns1 & !re=e no high] numb] ?an #gjj c\ld possibly be ne$$4 ,n[1 z we are requir$ to use e legitimate & practical me?od ( economy1 x %\ld at once occur to u t ! plates mu/ be pa9t$ on bo? sides2 inde$1 ?is is su* a common practice 9 cases ( ?is k9d t x w\ld readily occur to mo/ solv]s4 ,we %\ld also rememb] t "s ( ! figures may possibly be rev]s$ to =m o!r figures2 b z we 7 giv5 a sket* ( ! actual %apes ( ^! figures :5 pa9t$ on ! plates1 x w\ld be se5 t ?\gh ! #f's may be turn$ upside d[n to make #i's1 none ( ! o!r figures c be s treat$4 ,x w be f\nd t 9 ! case ( ! figures #a1 #b1 #c1 #d1 & #e1 ?irty-?ree ( ea* w be requir$ 9 ord] to provide = e possible em]g5cy2 9 ! case ( #g1 #h1 & #j1 we c only ne$ ?irty ( ea*2 :ile 9 ! case ( ! figure #f 7:i* may be rev]s$ = ! figure #i7 x is necess>y to provide exactly =ty-two4 ,x is !re=e cle> t ! total numb] ( figures necess>y is #big2 b z ! figures are pa9t$ on bo? sides ( ! plates1 only #adi su* plates are requir$4 ,at fir/ x w\ld appe> z if "o ( ! plates ne$ only h a numb] on "o side1 ! o!r side be+ left blank4 ,b "h we come to a r subtle po9t 9 ! problem4 ,read]s may h rem>k$ t 9 real life x is sometimes *eap] :5 mak+ a pur*ase to buy m articles ?an we require1 on ! pr9ciple ( a r$uction on tak+ a quantity3 we get m articles & we pay less4 ,?us1 if we want to buy t5 apples1 & ! price ask$ is a p5ny ea* if b\ght s+ly1 or n9ep5ce a doz51 we %\ld bo? save a p5ny & get two apples m ?an we want$ by buy+ ! full twelve4 ,9 ! same way1 s9ce !re is a regul> scale ( r$uction = plates pa9t$ alike1 we actually save by hav+ two figures pa9t$ on t odd plate4 ,suppos+1 = example1 t we h ?irty plates pa9t$ alike ) #e on "o side & #f on ! o!r4 ,! rate w\ld be #d;d41 & ! co/ #aas4 #aj;d4 ,b if ! odd plate )1 say1 only a #e on "o side ( x h a #f pa9t$ on ! o!r side1 we get ?irty-"o plates at ! r$uc$ rate ( #d;d41 ?us sav+ a f>?+ on ea* ( ! previ\s ?irty1 & r$uc+ ! co/ ( ! la/ "o f #as4 to #d;d4 ,b ev5 aft] ^! po9ts are all se5 !re comes 9 a new difficulty3 = al?\gh x w be f\nd t all ! #h's may be on ! backs ( ! #g's1 we _c h all ! #b's on ! backs ( ! #a's1 nor all ! #d on ! backs ( ! #c's1 etc4 ,!re is a great dang]1 9 our attempts to get z _m z possible pa9t$ alike1 ( our s adju/+ ! figures t "s p>ticul> comb9ation ( hymns _c be repres5t$4 ,"h is ! solution ( ! difficulty t 0 s5t to ! vic> ( ,*umpley ,/4 ,w9ifr$4 ,:]e ! sign ,x is plac$ betwe5 two figures1 x implies t "o ( ^! figures is on "o side ( ! plate & ! o!r on ! o!r side4 d4 s4 d4 #ca plates pa9t$ #e ,x #i #d #j #aa #g #cj #g ,x #h #d #j #aa #aj #ba #a ,x #b #g #j #ab #c #ba #c ,x #j #g #j #ab #c #ab #a ,x #c #i #j #i #c #ab #b ,x #d #i #j #i #c #ab #i ,x #d #i #j #i #c #h #d ,x #j #aj #j #f #aj #a #e ,x #d #ab #j #a #j #a #e ,x #j #ab #j #a #j #adi plates #f;d4 ea* #c #ad #f ---------- #g #ai #a ,( c\rse1 if we c\ld increase ! numb] ( plates1 we might get ! pa9t+ done = no?+1 b su* a cont+5cy is prev5t$ by ! condition t ! fewe/ possible plates mu/ be provid$4 ,?is puzzle appe>$ 9 ,tit-,bits1 & ! foll[+ rem>ks1 made by me 9 ! issue = #aa? ,decemb] #ahig1 may be ( int]e/4 ,! ,hymn-,bo>d ,pos] seems to h creat$ extraord9>y int]e/4 ,! imm5se numb] ( attempts at its solution s5t to me f all p>ts ( ! ,unit$ ,k+dom & f sev]al ,cont95tal c\ntries %[ a v k9d disposition among/ our read]s to help ! wor?y vic> ( ,*umpley ,/4 ,w9ifr$ ov] 8 p>o*ial difficulty4 ,e conceivable e/imate1 f a few %ill+s up to z high a sum z #a1cdg1 #ajs41 seems to h come to h&4 ,b ! a/oni%+ "p ( x is t1 aft] go+ c>efully ?r\gh ! trem5d\s pile ( correspond5ce1 ,i f9d t only "o competitor has succe$$ 9 ma9ta9+ ! reputation ( ! ,tit-,bits solv]s = _! capacity to solve any?+1 & 8 solution is sub/antially ! same z ! "o giv5 above1 ! co/ be+ id5tical4 ,"s ( 8 figures are diff]5tly comb9$1 b 8 gr\p+ ( ! plates1 z %[n 9 ! fir/ column1 is exactly ! same4 ,?\gh a l>ge majority ( competitors cle>ly hit ^u all ! ess5tial po9ts ( ! puzzle1 !y completely collaps$ 9 ! actual arrangem5t ( ! figures4 ,accord+ to _! me?ods1 "s possible selection ( hymns1 su* z #aaa1 #aab1 #aba1 #abb1baa1 _c be set up4 ,a few correspond5ts sugge/$ t x might be possible s to pa9t ! #g's t upside d[n !y w\ld appe> z #b's or #d's2 b ?is w\ld1 ( c\rse1 be b>r$ out by ! fact t a repres5tation ( ! actual figures to be us$ 0 giv54 #dbg4--,,pheasant-,,%oot+4 ,! ari?metic ( ?is puzzle is v easy inde$4 ,!re 7 cle>ly #bd pheasants at ! />t4 ,( ^! #af 7 %ot dead1 #a 0 w\nd$ 9 ! w+1 & #g got away4 ,! read] may h conclud$ t ! answ] is1 !re=e1 t sev5 rema9$4 ,b z !y flew away x is cle>ly absurd to say t !y rema9$4 ,_h !y done s !y w\ld c]ta9ly h be5 kill$4 ,mu/ we !n conclude t ! #ag t 7 %ot rema9$1 because ! o!rs flew away8 ,no2 because ! "q 0 n h[ _m rema9$8 b h[ _m /ill rema9$8 ,n[ ! poor bird t 0 w\nd$ 9 ! w+1 ?\gh unable to fly1 0 v active 9 its pa9ful /ruggles to run away4 ,! answ] is1 !re=e1 t ! #af birds t 7 %ot dead /ill rema9$1 or rema9$ /ill4 #dbh4--,,! ,,g>d5] ,,& ,,! ,,cook4 ,nobody succe$$ 9 solv+ ! puzzle1 s ,i _h to let ! cat out ( ! bag--an op]ation t 0 dimly =e%ad[$ by ! puss 9 ! orig9al illu/ration4 ,b ,i fir/ rem9d$ ! read] t ?is puzzle appe>$ on ,april #a1 a day on :i* none ( u ev] res5ts be+ made an ,april ,fool2 ?\gh1 z ,i practically gave ! ?+ away by specially draw+ att5tion to ! fact t x 0 ,all ,fools' ,day1 x 0 q rem>kable t my correspond5ts1 )\t a s+le exception1 fell into ! trap4 ,"o l>ge body ( correspond5ts held t :at ! cook loses 9 /ride is exactly made up 9 great] spe$2 consequ5tly bo? advance at ! same rate1 & ! result mu/ be a tie4 ,b ano!r consid]able section saw t1 ?\gh ?is might be s 9 a race #bjj ft4 /raight away1 x c\ld n really be1 because !y ea* g a /at$ di/ance at e b\nd1 & z #ajj is n an exact multiple ( #c1 ! g>d5] at 8 ?irty-f\r? b\nd w g #b ft4 beyond ! m>k4 ,! g>d5] w1 !re=e1 run to a po9t #ajb ft4 /raight away & return 7#bjd ft4 9 all71 & s lose by #d ft4 ,?is po9t c]ta9ly comes into ! puzzle4 ,b ! mo/ important fact ( all is ?is1 t x s happ5s t ! g>d5] 0 a pupil f ! ,horticultural ,college = ,lady ,g>d5]s at1 if ,i rememb] aright1 ,swanley2 :ile ! cook 0 a v accompli%$ ,fr5* *ef ( ! hemale p]suasion6 ,!re=e %e 7! g>d5]7 made ?ree b\nds to 8 7! cook's7 two4 ,x w n[ be f\nd t :ile ! g>d5] is runn+ h] #bjd ft4 9 #fh b\nds ( #c ft41 ! some:at infirm old cook c only make #dea_/c ( 8 #b ft4 b\nds1 :i* equals #ij ft4 #h 94 ,! result is t ! lady g>d5] w9s ! race by #aji ft4 #d 94 at a mom5t :5 ! cook is 9 ! air1 "o-?ird ?r\gh 8 #df? b\nd4 ,! moral ( ?is puzzle is tw(old3 7#a7 ,nev] take ?+s = grant$ 9 attempt+ to solve puzzles2 7#b7 always rememb] ,all ,fools' ,day :5 x comes r\nd4 ,i 0 n writ+ ( any g>d5] & cook1 b ( a p>ticul> c\ple1 9 a race t ,i witness$4 ,! /atem5t ( ! eye-witness mu/ !re=e be accept$3 z ! read] 0 n !re1 he _c contradict x4 ,( c\rse ! in=mation suppli$ 0 insuffici5t1 b ! correct reply 03 ,assum+ ! g>d5] to be ! 'he1' ! cook w9s by #d ft42 b if ! g>d5] is ! '%e1' !n ! g>d5] w9s by #aji ft4 #d 94 ,?is w\ld h won ! prize4 ,curi\sly en\gh1 "o solit>y competitor got on to ! "r track1 b fail$ to foll[ x up4 ,he said3 ,is ?is a regul> ,april #a cat*1 mean+ t !y only ran #f ft4 ea*1 & consequ5tly ! race 0 unf9i%$8 ,if n1 ,i ?9k ! foll[+ mu/ be ! solution1 suppos+ ! g>d5] to be ! 'he' & ! cook ! '%e4' ,?\gh 8 solution 0 wrong ev5 9 ! case he suppos$1 yet he 0 ! only p]son :o suspect$ ! "q ( sex4 #dbi4--,,plac+ ,,halfp5nies4 ,?irte5 co9s may be plac$ z %[n on page #beb4 #dcj4--;,,f9d ,,! ,,man',s ,,wife4 ,!re is no guess+ requir$ 9 ?is puzzle4 ,x is all a "q ( elim9ation4 ,if we c pair (f any five ( ! ladies ) _! respective husb&s1 o!r ?an husb& ,no4 #aj1 !n ! rema9+ lady mu/ be ,no4 #aj's wife4 ,illu/ration3 ,,plac+ ,,halfp5nies4 ,i w %[ h[ ?is may be done4 ,no4 #h is se5 c>ry+ a lady's p>asol 9 ! same h& ) 8 walk+-/ick4 ,b e lady is provid$ ) a p>asol1 except ,no4 #c2 !re=e ,no4 #c may be safely said to be ! wife ( ,no4 #h4 ,!n ,no4 #ab is hold+ a bicycle1 & ! dress-gu>d & make disclose ! fact t x is a lady's bicycle4 ,! only lady 9 a cycl+ skirt is ,no4 #e2 !re=e we conclude t ,no4 #e is ,no4 #ab's wife4 ,next1 ! man ,no4 #f has a dog1 & lady ,no4 #aa is se5 c>ry+ a dog *a94 ,s we may safely pair ,no4 #f ) ,no4 #aa4 ,!n we see t man ,no4 #b is pay+ a newsboy = a pap]4 ,b we d n pay = newspap]s 9 ?is way be=e receiv+ !m1 & ! g5tleman has app>5tly n tak5 "o f ! boy4 ,b lady ,no4 #i is se5 read+ a pap]4 ,! inf]5ce is obvi\s--t %e has s5t ! boy to h] husb& = a p5ny4 ,we !re=e pair ,no4 #b ) ,no4 #i4 ,we h n[ dispos$ ( all ! ladies except ,nos4 #a & #g1 & ( all ! m5 except ,nos4 #d & #aj4 ,on look+ at ,no4 #d we f9d t he is c>ry+ a coat ov] 8 arm1 & t ! buttons are on ! left side2--n on ! "r1 z a man we>s !m4 ,s x is a lady's coat4 ,b ! coat cle>ly does n belong to ,no4 #a1 z %e is se5 to be we>+ a coat already1 :ile ,no4 #g lady is v lightly clad4 ,we !re=e pair ,no4 #g lady ) man ,no4 #d4 ,n[ ! only lady left is ,no4 #a1 & we are consequ5tly =c$ to ! conclusion t %e is ! wife ( ,no4 #aj4 ,?is is !re=e ! correct answ]4 ,,index4 ,abbot's ,puzzle1 ,!1 #bj1 #afa4 ---- ,w9d[1 ,!1 #hg1 #bac4 ,academic ,c\rtesies1 #ah1 #afj4 ,acro/ic ,puzzle1 ,an1 #hd1 #baj4 ,adam & ,eve & ! ,apples1 #ah4 ,a]oplanes1 ,! ,two1 #b1 #adh4 ,age & ,k9%ip ,puzzles1 #f4 ---- ,conc]n+ ,tommy's1 #g1 #aec4 ---- ,mamma's1 #g1 #aeb4 ---- ,mrs4 ,timpk9s's1 #g1 #aeb4 ---- ,rov]'s1 #g1 #aeb4 ,ages1 ,! ,family1 #g1 #aeb4 ---- ,_!1 #g1 #aeb4 ,alcu91 ,abbot1 #bj1 #aab4 ,almonds1 ,! ,n9e1 #fd1 #aie4 ,amazons1 ,!1 #id1 #bba4 ,&rews1 ,w4,s41 #abe4 ,apples1 ,a ,deal 91 #c1 #adi4 ---- ,buy+1 #f1 #aea4 ---- ,! ,t51 #fd1 #aie4 ,approximations 9 ,dissection1 #bh4 ,ari?metical & ,algebraical ,problems1 #a4 ---- ,v>i\s1 #ag4 ,ar?ur's ,knights1 ,k+1 #gg1 #bjc4 ,artill]ym5's ,dilemma1 #bf1 #afg4 ,asp>agus1 ,bundles (1 #adj4 ,aspects all due ,s\?1 #acg4 ,associat$ ,magic ,squ>es1 #abj4 ,axiom1 ,a ,puzzl+1 #ach4 ,ba*et de ,mziriac1 #ij1 #aji1 #aab4 ,ba*et's ,squ>e1 #ij1 #baf4 ,ball ,problem1 ,!1 #ea1 #ahc4 ,ball1 ,w4,w4 ,r\se1 #aji1 #bjd1 #bdh4 ,balls1 ,! ,glass1 #gh1 #bjd4 ,bank]'s ,puzzle1 ,!1 #be1 #afe4 ,bank ,holiday ,puzzle1 ,a1 #gc1 #bja4 ,bann] ,puzzle1 ,!1 #df1 #agi4 ---- ,/4 ,george's1 #ej1 #ahb4 ,b>rel ,puzzle1 ,!1 #aji1 #bce4 ,b>rels ( ,balsam1 ,!1 #hb1 #bjh4 ,beanfea/ ,puzzle1 ,a1 #b1 #adh4 ,beef & ,sausages1 #c1 #adi4 ,be]1 ,! ,b>rel (1 #ac1 #aee4 ,bell-ropes1 ,/eal+ !1 #di1 #aha4 ,bells1 ,! ,peal (1 #gh1 #bjd4 ,b]gholt1 ,e41 #aaf1 #aai1 #abe4 ,betsy ,ross ,puzzle1 ,!1 #dj1 #agf4 ,bicycle ,?ief1 ,!1 #f1 #aeb4 ,bi%ops--,gu>d$1 #hh1 #bad4 ---- 9 ,convocation1 #hi1 #bae4 ---- ,puzzle1 ,a ,new1 #ih1 #bbe4 ---- ,ungu>d$1 #hh1 #bad4 ,bo>d1 ,! ,*ess-1 #he4 ---- 9 ,comp>tm5ts1 ,!1 #ajb1 #bbh4 ---- ,sett+ !1 #aje1 #bca4 ,bo>ds ) ,odd ,numb] ( ,squ>es1 #hf1 #bab4 ,boat1 ,?ree ,m5 9 a1 #gh1 #bjd4 ,bookworm1 ,! ,indu/ri\s1 #adc1 #bdh4 ,boo?by1 ,guy1 #aed4 ,box1 ,! ,c>dbo>d1 #di1 #aha4 ---- ,! ,pap]1 #dj4 ,boys & ,girls1 #fg1 #aig4 ,bridges1 ,! ,monk & !1 #ge1 #bjb4 ,brig&s1 ,! ,five1 #be1 #afd4 ,brocade1 ,! ,squ>es (1 #dg1 #ahj4 ,bun ,puzzle1 ,!1 #ce1 #agj4 ,buss*op1 ,paul1 #agb4 ,buttons & ,/r+ ,me?od1 #bcj4 ,cab ,numb]s1 ,!1 #ae1 #aeg4 ,cal5d> ,puzzle1 ,a1 #adb1 #bdg4 ,cant]bury ,puzzles1 ,!1 #ad1 #bh1 #eh1 #aag1 #aba1 #aie1 #bjb1 #bje1 #bjf1 #bab1 #bac1 #bag1 #bcc4 ,c>d ,frame ,puzzle1 ,!1 #aad1 #bch4 ---- ,magic ,squ>es1 #abc1 #bdd4 ---- ,play]s1 ,a ,puzzle =1 #gh1 #bjc4 ---- ,puzzle1 ,! ,t1 #aae1 #bci4 ---- ,triangles1 #aae1 #bci4 ,c>ds1 ,! ,cross (1 #aae1 #bch4 ,c>dan1 #adb4 ,c>roll1 ,lewis1 #dc4 ,ca/le ,treasure1 ,/eal+ !1 #aac1 #bcg4 ,cats1 ! ,wiz>d's1 #db1 #agh4 ,cattle1 ,judk9s's1 #f1 #aea4 ---- ,m>ket1 ,at a1 #a1 #adh4 ,c5sus ,puzzle1 ,a1 #g1 #aeb4 ,c5tury ,puzzle1 ,!1 #af1 #aeh4 ---- ,! ,digital1 #af1 #aei4 ,*a9 ,puzzle1 ,a1 #add1 #bdi4 ---- ,! ,antiqu>y's1 #hc1 #bji4 ---- ,! ,c>dbo>d1 #dj1 #agf4 ,*ange1 ,giv+1 #d1 #aej4 ---- ,ways ( giv+1 #aea4 ,*ang+ ,places1 #aj1 #aed4 ,*annel ,isl&1 #ach4 ,*>itable ,beque/1 ,a1 #b1 #adh ,*>ity1 ,indiscrim9ate1 #b1 #adh4 ,*eckmate1 #ajg1 #bcc4 ,*eesemong]1 ,! ,ecc5tric1 #ff1 #aif4 ,*equ]$ ,bo>d ,divisions1 #he1 #baj4 ,*]ries & ,plums1 #ef1 #ahi4 ,*ess ,puzzles1 ,dynamical1 #if4 ---- ,/atical1 #hh4 ---- ,v>i\s1 #aje4 ---- ,que]1 #ajg1 #bcc4 ,*essbo>d1 ,!1 #he4 ---- ,fallacy1 ,a1 #ada1 #bdg4 ---- ,gu>d$1 #ie4 ---- ,non-attack+ ,arrangem5ts1 #if4 ---- ,problems1 #hd4 ---- ,s5t5ce1 ,!1 #hg1 #bad4 ---- ,solitaire1 #ajh1 #bcd4 ---- ,! ,*9ese1 #hg1 #bac4 ---- ,! ,cr[d$1 #ia1 #bag4 ,*e/nuts1 ,buy+1 #f1 #aeb4 ,*9ese ,money1 #d1 #aej4 ---- ,puzzle1 ,anci5t1 #ajg1 #bcc4 ---- ---- ,! ,fa%ionable1 #dc4 ,*ri/mas ,boxes1 ,!1 #d1 #aej4 ---- ,pres5t1 ,mrs4 ,smiley's1 #df1 #agi4 ---- ,pudd+1 ,!1 #dc1 #agh4 ,cig> ,puzzle1 ,!1 #aai1 #bdb4 ,circle1 ,! ,dissect$1 #fi1 #aig4 ,ci/]ns1 ,h[ to ,make1 #ed1 #ahh4 ,civil ,s]vice ,h[l]1 #aed4 ,cl>e1 ,john1 #eh4 ,clock ,=mul1 #aed4 ---- ,puzzles1 #i4 ---- ,! ,club1 #aj1 #aed4 ---- ,! ,railway ,/ation1 #aa1 #aee4 ,clocks1 ,! ,?ree1 #aa1 #aed4 ,clo!s ,l9e ,puzzle1 ,!1 #ej1 #ahb4 ,coa/1 ,r\nd !1 #fc1 #aie4 ,co9cid5ce1 ,a ,que]1 #b1 #adh4 ,co9s1 ,! ,brok51 #e1 #aej4 ---- ,! ,t51 #eg1 #aij4 ---- ,two ,anci5t1 #adj4 ,comb9ation & ,gr\p ,problems1 #gf4 ,compasses ,puzzle1 ,!1 #ec1 #ahf4 ,composite ,magic ,squ>es1 #abg1 #bdf4 ,cone ,puzzle1 ,!1 #ee1 #ahh4 ,corn1 ,reap+ !1 #bj1 #afa4 ,cornfields1 ,f>m] ,lawr5ce's1 #aja1 #bbg4 ,co/]mong]'s ,puzzle1 ,!1 #f1 #aeb4 ,c\nt] ,problems1 ,mov+1 #eh4 ---- ,puzzle1 ,a ,new1 #ih1 #bbe4 ---- ,solitaire1 #ajg1 #bcd4 ,c\nt]s1 ,! ,col\r$1 #ia1 #bag4 ---- ,! ,=ty-n9e1 #ib1 #bag4 ---- ,! ,n9e1 #ad1 #aef4 ---- ,! ,t51 #ae1 #aef4 ,cresc5t ,puzzle1 ,!1 #eb1 #ahd4 ,cresc5ts ( ,byzantium1 ,! ,five1 #ib1 #bai4 ,cricket ,mat*1 ,! ,village1 #aaf1 #bci4 ---- ,sl[1 #aaf1 #bci4 ,cross & ,triangle1 #ce1 #afi4 ---- ( ,c>ds1 #aae1 #bch4 ---- ,! ,fold$1 #ce1 #afi4 ---- ,! ,s\!rn1 #ic1 #bbj4 ,crosses1 ,c\nt]1 #ha1 #bjg4 ---- f ,"o1 ,two1 #ce1 #afh4 ---- ---- ,?ree1 #afi4 ,cross+ ,riv] ,problems1 #aab4 ,crusad]1 ,!1 #ajf1 #bcb4 ,cubes1 ,sums (1 #afe4 ,cu%ion ,cov]s1 ,!1 #df1 #agi4 ,cutt+-out ,puzzle1 ,a1 #cg1 #agb4 ,cycli/s' ,fea/1 ,!1 #b1 #adh4 ,dairyman1 ,! ,hone/1 #aaj1 #bce4 ,def9ition1 ,a ,"q (1 #bc1 #afc4 ,de ,font5ey1 #aab4 ,deifi$ ,puzzle1 ,!1 #gd1 #bjb4 ,delannoy1 #aab4 ,de ,morgan1 ,a41 #bg4 ,de ,tudor1 ,sir ,edwyn1 #ab1 #aee4 ,diabolique ,magic ,squ>es1 #abj4 ,diamond ,puzzle1 ,!1 #gd1 #bjb4 ,dice1 ,a ,trick )1 #aaf1 #bci4 ---- ,game1 ,! ,mont5egr91 #aai1 #bdb4 ---- ,numb]s1 ,!1 #ag1 #afj4 ,die1 ,pa9t+ !1 #hd1 #baj1 ,digital ,analysis1 #aeg1 #aeh4 ---- ,division1 #af1 #aeh4 ---- ,multiplication1 #ae1 #aef4 ---- ,puzzles1 #ac4 ,digits1 ,add+ !1 #af1 #aeh4 ---- & ,squ>es1 #ad1 #aee4 ---- ,odd & ,ev51 #ad1 #aef4 ,dilemma1 ,an ,amaz+1 #ajf1 #bcc4 ,diophant9e ,problem1 #afd4 ,dissection ,puzzle1 ,an ,easy1 #ce1 #agj4 ---- ,puzzles1 #bg4 ---- ---- ,v>i\s1 #ce4 ,divid+ ,magic ,squ>es1 #abd4 ,division1 ,digital1 #af1 #aeh4 ---- ,simple1 #bc1 #afc4 ,doctor's ,qu]y1 ,!1 #aji1 #bce4 ,dogs ,puzzle1 ,! ,five1 #ib1 #bah4 ,dome/ic ,economy1 #e1 #aea4 ,dom9o ,frame ,puzzle1 ,!1 #aad1 #bch4 ,dom9oes 9 ,progression1 #aad1 #bcg4 ---- ,! ,eighte51 #abc1 #bde4 ---- ,! ,fifte51 #hc1 #bji4 ---- ,! ,five1 #aad1 #bch4 ,donkey ,rid+1 #ac1 #aee4 ,dormitory ,puzzle1 ,a1 #ha1 #bjh4 ,dovetail$ ,block1 ,!1 #ade1 #bdi4 ,drayton's ,polyolbion1 #eh4 ,dungeon ,puzzle1 ,a1 #ig1 #bbd4 ,dungeons1 ,! ,sib]ian1 #abc1 #bdd4 ---- ,! ,spani%1 #abb1 #bdd4 ,dut*m5's ,wives1 ,!1 #bf1 #afg4 ,dynamical ,*ess ,puzzles1 #if4 ,e>?'s ,girdle1 ,!1 #aci4 ,educational ,times ,repr9ts1 #bjd4 ,eggs1 ,a ,deal 91 #c1 #adi4 ---- ,obta9+ !1 #adj4 ,election1 ,! ,muddlet[n1 #ai1 #afa4 ---- ,! ,p>i% ,c\ncil1 #ai1 #afa4 ,elev51 ,! ,my/ic1 #af1 #aei4 ,elopem5ts1 ,! ,f\r1 #aac1 #bcg4 ,elrick1 ,e41 #bca4 ,eng9es1 ,! ,eight1 #fa1 #aid4 ,episcopal ,visitation1 ,an1 #ih1 #bbe4 ,e/ate1 ,f>m] ,wurzel's1 #ea1 #ahd4 ,e/ates1 ,! ,york%ire1 #ea1 #ahc4 ,euclid1 #ca1 #ach4 ,eul]1 ,l41 #afe4 ,ex*ange ,puzzle1 ,!1 #ff1 #aif4 ,fallacy1 ,a ,*essbo>d1 #ada1 #bdg4 ,family ,p>ty1 ,a1 #h1 #aec4 ,f>e1 ,! ,pass5g]'s1 #ac1 #aee4 ,f>m] & 8 ,%eep1 ,!1 #bb1 #afc4 ,f5ce ,problem1 ,a1 #ba1 #afb4 ,f5ces1 ,! ,l&[n]'s1 #db1 #agh4 ,f]mat1 #afd1 #afh4 ,f9d ! ,man's ,wife1 #adg1 #bea4 ,fly on ! ,octah$ron1 ,!1 #gj1 #aih4 ,fog1 ,mr4 ,gubb9s 9 a1 #ah1 #afa4 ,football ,play]s1 ,!1 #aaf1 #bdj4 ,fraction1 ,a ,puzzl+1 #ach4 ,fractions1 ,m ,mix$1 #af1 #aei4 ,frame ,puzzle1 ,! ,c>d1 #aad1 #bch4 ---- ---- ,! ,dom9o1 #aad1 #bch4 ,frank5/e91 ,e4,n41 #bcb4 ,frnicle1 ,b41 #aai1 #afh4 ,frogs1 ,! ,educat$1 #ei1 #aid- ---- ,! ,f\r1 #ajc1 #bbi4 ---- ,! ,six1 #ei1 #aic4 ,fro/1 ,a4,h41 #abj4 ,games1 ,puzzle1 #aag4 ---- ,problems conc]n+1 #aad4 ,g>d51 ,lady ,bel9da's1 #eb1 #ahf4 ---- ,puzzle1 ,!1 #di1 #ahb4 ,g>d5] & ! ,cook1 ,!1 #adf1 #bea4 ,geometrical ,problems1 #bg4 ---- ,puzzles1 ,v>i\s1 #di4 ,george & ! ,dragon1 ,/41 #aja1 #bbg4 ,gett+ ,up/airs1 ,su* a1 #adc1 #bdh4 ,girdle1 ! ,e>?'s1 #aci4 ,goat1 ,! ,te!r$1 #ec1 #ahf4 ,gr& ,lama's ,problem1 ,!1 #hf1 #bab4 ,gras%opp] ,puzzle1 ,!1 #ei1 #aic4 ,greek ,cross ,puzzles1 #bh4 ---- ---- ,?ree f ,"o1 #afi4 ,greyh\nd ,puzzle1 ,!1 #aja1 #bbg4 ,groc] & ,drap]1 ,!1 #e1 #aea4 ,gros1 ,l41 #bdh4 ,gr\p ,problems1 ,comb9ation &1 #gf4 ,gr\ps1 ,! ,?ree1 #ad1 #aef4 ,gu>9i1 #bbi4 ,hairdress]'s ,puzzle1 ,!1 #acg4 ,halfp5nies1 ,plac+1 #adg1 #bea4 ,hampton ,c\rt ,maze solv$1 #acc4 ,hannah's ,puzzle1 #ge1 #bjb4 ,ha/+s1 ,! ,battle (1 #bc1 #afd4 ,hatfield ,maze solv$1 #acf4 ,hat ,puzzle1 ,!1 #fg1 #aif4 ,hat-peg ,puzzle1 ,!1 #ic1 #bba4 ,hats1 ,! ,wrong1 #gh1 #bjc4 ,hay1 ,! ,trusses (1 #ah1 #afa4 ,heads or ,tails1 #bb1 #afc4 ,he>?rug1 ,mrs4 ,hobson's1 #cg1 #agb4 ,helmholtz1 ,von1 #da4 ,honey1 ,! ,b>rels (1 #aaa1 #bcf4 ,honeycomb ,puzzle1 ,!1 #ge1 #bjb4 ,horse ,race ,puzzle1 ,!1 #aag1 #bdj4 ,horse%oes1 ,! ,two1 #dj1 #age4 ,h\d91 #fh4 ,hydroplane ,"q1 ,!1 #ab1 #aee4 ,hymn-bo>d ,pos]1 ,!1 #ade1 #bej4 ,icosah$ron ,puzzle1 ,!1 #gj1 #aih4 ,jack & ! ,bean/alk1 #ade1 #bdi4 ,jackson1 ,john1 #ef4 ,ja5is*1 ,c4,f4 de1 #ib4 ,jampots1 ,arrang+ !1 #fh1 #aig4 ,jeal\s ,husb&s1 ,five1 #aac1 #bcf4 ,jo9]'s ,problem1 ,!1 #cf1 #aga4 ---- ---- ,ano!r1 #cg1 #aga4 ,jolly ,gaol-,birds1 ,eight1 #abb1 #bdc4 ---- ---- ,n9e1 #abb1 #bdc4 ,j\rney1 ,! ,que5's1 #ajj1 #bbg4 ---- ,! ,rook's1 #if1 #bbd4 ,junior ,cl]ks' ,puzzle1 ,!1 #d1 #aej4 ,juv5ile ,puzzle1 ,a1 #fh1 #aig4 ,kang>oos1 ,! ,f\r1 #ajb1 #bbh4 ,kelv91 ,"l1 #da4 ,k5nel ,puzzle1 ,!1 #aje1 #bca4 ,k+ & ! ,ca/les1 ,!1 #ef1 #ahi4 ---- ,! ,=sak51 #ajf1 #bcb4 ,kite-fly+ ,puzzle1 ,a1 #ed1 #ahg4 ,knight-gu>ds1 ,!1 #ie1 #bbb4 ,knights1 ,k+ ,ar?ur's1 #gg1 #bjc4 ---- ,t\r1 ,magic1 #abg1 #bdg4 ---- ---- ,! ,cubic1 #ajc1 #bbi4 ---- ---- ,! ,f\r1 #ajc1 #bbi4 ,labosne1 ,a41 #be1 #ij1 #baf4 ,lab\r]'s ,puzzle1 ,!1 #ah1 #afj4 ,ladies' ,di>y1 #bf4 ,lagrange1 ,j4,l41 #i4 ,laisant1 ,c4,a41 #gf4 ,lamp-po/s1 ,pa9t+ !1 #ai1 #afa4 ,leap ,ye>1 #aee4 ---- ---- ,ladies1 ,!1 #ai1 #afa4 ,legacy1 ,a ,puzzl+1 #bj1 #afa4 ,legal ,difficulty1 ,a1 #bc1 #afc4 ,le ,plongeon1 ,dr41 #bi4 ,lett] ,block ,puzzle1 ,!1 #fj1 #aid4 ---- ,blocks1 ,! ,?irty-six1 #ia1 #baf4 ---- ,puzzle1 ,! ,fifte51 #gi1 #bje4 ,level ,puzzle1 ,!1 #gd1 #bjb4 ,l9oleum ,cutt+1 #dh1 #aha4 ---- ,puzzle1 ,ano!r1 #di1 #aha4 ,lion & ! ,man1 ,!1 #ig1 #bbd4 ---- ,hunt+1 #id1 #bbb4 ,lions & ,cr[ns1 #he1 #bab4 ---- ,! ,f\r1 #hh1 #bad4 ,lock]s ,puzzle1 ,!1 #ad1 #aef4 ,locomotion & ,spe$ ,puzzles1 #aa4 ,lodg+-h\se ,difficulty1 ,a1 #fa1 #aid4 ,london & ,wise1 #aca4 ,loyd1 ,sam1 #h1 #dc1 #dd1 #ih1 #add1 #bcb1 #bcc4 ,lucas1 ,ed\>d1 #af1 #gf1 #aab1 #aba4 ,lun*eons1 ,! ,city1 #gg1 #bjc4 ,mac,mahon1 ,major1 #aji4 ,magic ,knight's ,t\r1 #abg1 #bdg4 ---- ,squ>e ,problems1 #aai4 ---- ---- ,c>d1 #abc1 #bdd4 ---- ---- ( ,composites1 #abg1 #bdf4 ---- ---- ( ,primes1 #abe4 ---- ---- ( ,two ,degrees1 #abe1 #bde4 ---- ---- ,two ,new1 #abe1 #bde4 ---- ,/rips1 #aba1 #bdc4 ,magics1 ,subtract+1 ,multiply+1 & ,divid+1 #abd4 ,maid51 ,! ,langui%+1 #ig1 #bbd4 ,m&>9's ,puzzle1 ,!1 #ajc1 #bcj4 ---- ,t ,puzzle1 ,!1 #abf1 #bdf4 ,m>ket+1 ,saturday1 #bg1 #afh4 ,m>ket ,wom51 ,!1 #c1 #adi4 ,m>y & ,m>maduke1 #g1 #aeb4 ,m>y1 ,h[ ,old 01 #h1 #aec4 ,massacre ( ,innoc5ts1 #aci4 ,mat* ,my/]y1 ,a1 #aah1 #bda4 ---- ,puzzle1 ,a ,new1 #ee1 #ahh4 ,mates1 ,?irty-six1 #ajf1 #bcc4 ,mazes & h[ to ?read ,!m1 #abg4 ,measur+1 ,weigh+1 & ,pack+ ,puzzles1 #aji4 ---- ,puzzle1 ,new1 #aaj1 #bce4 ,meet+1 ,! ,suffragi/s'1 #ai1 #afa4 ,mellor1 ,w4,m4,f41 #bdb4 ,mnages1 ,problme de1 #gf4 ,m]s5ne1 ,m41 #afh4 ,mice1 ,cat*+ !1 #fe1 #aif4 ,milkmaid ,puzzle1 ,!1 #ej1 #ahc4 ,millionaire's ,p]plexity1 ,!1 #c1 #adi4 ,m9ce ,pies1 ,! ,twelve1 #eg1 #aia4 ,m9e1 ,inspect+ a1 #ga1 #aii4 ,m9]s' ,holiday1 ,!1 #bc1 #afc4 ,mis]1 ,! ,conv]t$1 #ba1 #afb4 ,mitre1 ,dissect+ a1 #ce1 #agj4 ,monad1 ,! ,great1 #ci1 #agd4 ,money1 ,a ,que] ,?+ 91 #b1 #adh4 ---- ,boxes1 ,! ,puzzl+1 #c1 #adi4 ----1 ,pocket1 #c1 #adi4 ---- ,puzzles1 #a4 ---- ,puzzle1 ,a ,new1 #b1 #adh4 ----1 ,squ>e1 #c1 #adi4 ,moni/1 ,!1 #abe4 ,monk & ! ,bridges1 ,!1 #ge1 #bjb4 ,mon/rosity1 ,!1 #ajh1 #bcd4 ,mont5egr9 ,dice ,game1 ,!1 #aai1 #bdb4 ,moreau1 #gf4 ,morris1 ,n9e ,m5's1 #eh4 ,mosaics1 ,a ,problem 91 #ij1 #bae4 ,"m & ,daught]1 #g1 #aeb4 ,motor-c> ,race1 ,!1 #aag1 #bdj4 ---- ,t\r1 ,!1 #gd1 #bja4 ---- ,g>age ,puzzle1 ,!1 #fb1 #aie4 ,motori/s1 ,a ,puzzle =1 #gc1 #bja4 ,m\se-trap ,puzzle1 ,!1 #hj1 #bjf4 ,mov+ ,c\nt] ,problems1 #eh4 ,multiplication1 ,digital1 #ae1 #aef4 ---- ,que]1 #ae1 #aeg4 ---- ,simple1 #bc1 #afc4 ,multiply+ ,magic ,squ>es1 #abd4 ,muncey1 ,j4,n41 #abe4 ,murray1 ,sir ,james1 #dd4 ,napoleon1 #dc1 #dd4 ,nasik ,magic ,squ>es1 #abj4 ,neighb\rs1 ,next-,door1 #h1 #aec4 ,newton1 ,sir ,isaac1 #ef4 ,n9e ,m5's ,morris1 #eh4 ,notation1 ,scales (1 #adi4 ,n\ghts & ,crosses1 #eh1 #aag4 ,n\velles ,annales de ,ma?matiques1 #ad4 ,numb] ,*ecks ,puzzle1 ,!1 #af1 #aeh4 ,numb]s1 ,curi\s1 #bj1 #afb4 ,nuts1 ,! ,bag (1 #h1 #aec4 ,obs]vation1 ,defective1 #d1 #aej4 ,octah$ron1 ,! ,fly on !1 #gj1 #aih4 ,oval1 ,h[ to draw an1 #ej1 #ahb4 ,ovid's ,game1 #eh4 ,pack+ 9 ,russia1 ,gold1 #aaa1 #bcf4 ---- ,puzzles1 ,measur+1 ,weigh+1 &1 #aji4 ---- ,puzzle1 ,a1 #aaa1 #bcf4 ,p&iagonal ,magic ,squ>es1 #abj4 ,papa's ,puzzle1 #ec1 #ahg4 ,pappus1 #ec4 ,p>adox ,p>ty1 ,!1 #acg4 ,p>ty1 ,a ,family1 #h1 #aec4 ,pat*work ,puzzles1 #df4 ---- ,puzzle1 ,ano!r1 #dh1 #ahj4 ---- ,! ,silk1 #cd1 #afh4 ,pati5ce1 ,/r&1 #aaf1 #bci4 ,pawns1 ,a ,puzzle )1 #id1 #bbb4 ---- ,immovable1 #ajf1 #bcc4 ---- ,! ,six1 #ajg1 #bcc4 ---- ,! ,two1 #aje1 #bca4 ,pe>ls1 ,! ,?irty-?ree1 #ah1 #afj4 ,pebble ,game1 ,!1 #aag1 #bdj4 ,p$igree1 ,a ,mix$1 #h1 #aec4 ,pellian ,equation1 #afd1 #afg4 ,p5nies1 ,! ,five1 #adc1 #bdh4 ---- ,! ,twelve1 #fe1 #aie4 ,p5sion1 ,draw+ h]1 #ab1 #aee4 ,p5tagon & ,squ>e1 ,!1 #cg1 #agb4 ---- ,draw+ a1 #cg4 ,pfeff]mann1 ,m41 #abe4 ,pheasant-,%oot+1 #adf1 #bea4 ,philadelphia ,maze solv$1 #acg4 ,pi]rot's ,puzzle1 ,!1 #ae1 #aef4 ,pigs1 ,! ,sev51 #da1 #agg4 ,planck1 ,c41 #bbj1 #bdf4 ,plane ,p>adox1 #ach4 ,plantation ,puzzle1 ,a1 #eg1 #ahi4 ---- ,! ,burmese1 #eh1 #aia4 ,plates & ,co9s1 #fe1 #aie4 ,plums1 ,! ,baskets (1 #abf1 #bde4 ,poe1 ,e4,a41 #bdi4 ,po9ts & ,l9es ,problems1 #ef4 ,po/age ,/amps1 ,! ,f\r1 #hd1 #baj4 ,po/-,(fice ,p]plexity1 ,a1 #a1 #adh4 ,potato ,puzzle1 ,!1 #da1 #agg4 ,potatoes1 ,! ,basket (1 #ac1 #aee4 ,precoci\s ,baby1 ,!1 #aci4 ,pres5ts1 ,buy+1 #b1 #adh4 ,prime ,magic ,squ>es1 #abe4 ,pr9t]'s ,error1 ,a1 #bj1 #afb4 ,prison]s1 ,ex]cise =1 #ajd1 #bcj4 ---- ,! ,t51 #fb1 #aie4 ,probabilities1 ,two ,que/ions 91 #e1 #aej4 ,problems conc]n+ ,games1 #aad4 ,puss 9 ! ,corn]1 #aah1 #bdj4 ,puzzle ,games1 #aag4 ,pyramid1 ,pa9t+ a1 #hc1 #bjh4 ,pyramids1 ,squ>e & ,triangul>1 #afg4 ,py?agoras1 #ca4 ,que51 ,!1 #abj4 ,que5s & ,bi%op ,puzzle1 #ic1 #bai4 ---- ,! ,eight1 #hi1 #bae4 ,que5's ,j\rney1 ,!1 #ajj1 #bbg4 ---- ,t\r1 ,!1 #ih1 #bbe4 ,quilt1 ,mrs4 ,p]k9s's1 #dg1 #ahj4 ,race ,puzzle1 ,! ,horse-1 #aag1 #bdj4 ---- ,! ,motor-c>1 #aag1 #bdj4 ,rackbrane's ,little ,loss1 #ba1 #afc4 ,railway ,muddle1 ,a1 #fb1 #aid4 ---- ,puzzle1 ,a1 #fa1 #aid4 ---- ,/ations1 ,! ,?ree1 #di1 #ahb4 ,rational ,amusem5t = ,w9t] ,ev5+s1 #ef4 ,rectangles1 ,c\nt+ !1 #aje1 #bcb4 ,reiss1 ,m41 #eh4 ,relation%ips1 ,que]1 #h1 #aec4 ,rev]sals1 ,a ,puzzle 91 #e1 #aea4 ,riv] ,axe1 ,cross+ !1 #aab1 #bcf4 ,riv] ,problems1 ,cross+1 #aab4 ,rook]y1 ,!1 #aje1 #bcb4 ,rook's ,j\rney1 ,!1 #if1 #bbd4 ---- ,t\r1 ,!1 #if1 #bbc4 ,rooks1 ,! ,eight1 #hh1 #bad4 ---- ,! ,two1 #aag1 #bdj4 ,r\nd ,table1 ,!1 #hj1 #bje4 ,r\te ,problems1 ,unicursal &1 #fh4 ,ruby ,broo*1 ,!1 #add1 #bdi4 ,sabba? ,puzzle1 ,!1 #add1 #bdi4 ,sailor's ,puzzle1 ,!1 #ga1 #aii4 ,sayles1 ,h4,a41 #abe4 ,s*oolboys1 ,! ,n9e1 #hj1 #bje4 ,s*oolgirls1 ,! ,fifte51 #hj1 #bjd4 ,scramble1 ,! ,great1 #ai1 #afa4 ,sculptor's ,problem1 ,!1 #bc1 #afd4 ,second ,day ( ,week1 #aci4 ,see-,saw ,puzzle1 ,!1 #bb1 #afc4 ,semi-,nasik ,magic ,squ>es1 #abj4 ,s5ior & ,junior1 #adj4 ,sev5s1 ,! ,f\r1 #ag1 #afj4 ,%>p's ,puzzle1 #bcj4 ,%eepfold1 ,!1 #eb1 #ahd4 ,%eep ,p5s1 ,! ,six1 #ee1 #ahi4 ---- ,! ,sixte51 #hj1 #bjf4 ---- ,! ,?ree1 #ib1 #bag4 ---- ,^? ,fifte51 #gg1 #bjc4 ,%opp+ ,p]plexity1 ,a1 #d1 #aej4 ,%uldham1 ,c4,d41 #abe1 #abf4 ,sib]ian ,dungeons1 ,!1 #abc1 #bdd4 ,simpleton1 ,! ,village1 #aa1 #aee4 ,skat]1 ,! ,sci5tific1 #ajj1 #bbf4 ,skeat1 ,pr(essor1 #abg4 ,solitaire1 ,c5tral1 #fc1 #aie4 ---- ,*essbo>d1 #ajh1 #bcd4 ---- ,c\nt]1 #ajg1 #bcd4 ,sons1 ,! ,f\r1 #di1 #aha4 ,spani% ,dungeons1 ,!1 #abb1 #bdd4 ---- ,mis]1 ,!1 #bd1 #afd4 ,spe$ & ,locomotion ,puzzles1 #aa4 ---- ,av]age1 #aa1 #aee4 ,spiral1 ,draw+ a1 #ej1 #ahb4 ,spot on ! ,table1 ,!1 #ag1 #afj4 ,squ>e ,numb]s1 ,*eck =1 #ac4 ---- ---- ,digital1 #af1 #aei4 ---- ( ,v5e]1 ,!1 #ci1 #age4 ---- ,puzzle1 ,an ,easy1 #ce1 #agj4 ,squ>es1 ,a ,problem 91 #bc1 #afc4 ---- ,circl+ !1 #ba1 #afb4 ---- ,diff]5ce ( ,two1 #afg4 ---- ,magic1 #aai4 ---- ,sum ( ,two1 #afe1 #age4 ---- ,! ,*ocolate1 #ce1 #agj4 ,/alemate1 #ajf1 #bcb4 ,/amp-lick+1 ,! ,g5tle ,art (1 #ia1 #bag4 ,/> ,puzzle1 ,!1 #ii1 #bbf4 ,/>s1 ,! ,eight1 #hi1 #bae4 ---- ,! ,=ty-n9e1 #ajj1 #bbf4 ,/atical ,*ess ,puzzles1 #hh4 ,/icks1 ,! ,eight1 #ec1 #ahf4 ,/onemason's ,problem1 ,!1 #be1 #afe4 ,/op-wat*1 ,!1 #aa1 #aed4 ,/r& ,magaz9e1 ,!1 #dd1 #aaf1 #bbj4 ,/r& ,pati5ce1 #aaf1 #bci4 ,/ream1 ,cross+ !1 #aab1 #bcf4 ,/rutt1 ,joseph1 #ei4 ,subtract+ ,magic ,squ>es1 #abd4 ,sultan's ,army1 ,!1 #be1 #afe4 ,supp]s1 ,! ,new ,ye>'s ,eve1 #c1 #adi4 ,surname1 ,f9d ,ada's1 #bg1 #afh4 ,swa/ika1 ,!1 #bi1 #ca1 #afi4 ,t ,c>d ,puzzle1 ,!1 #aae1 #bci4 ,table1 ,! ,r\nd1 #hj1 #bje4 ,table-top & ,/ools1 ,!1 #ch1 #agc4 ,tangram ,p>adox1 ,a1 #dc1 #agh4 ,t>get1 ,! ,cross1 #hd1 #baj4 ,t>ry1 #aab4 ,t>taglia1 #be1 #aji1 #aab4 ,tea1 ,mix+ !1 #aaa1 #bce4 ,telegraph ,po/s1 ,!1 #aci4 ,t5nis ,t\rnam5t1 ,a1 #gh1 #bjc4 ,tetrah$ron1 ,build+ !1 #hb1 #bjh4 ,?ief1 ,cat*+ !1 #ai1 #afa4 ,?rift1 ,a ,/udy 91 #be1 #aff4 ,?ompson1 ,w4,h41 #bcb4 ,ticket ,puzzle1 ,! ,excursion1 #e1 #aea4 ,"t ,puzzle1 ,a1 #aj1 #aec4 ---- ,:at 0 !1 #aj1 #aec4 ,tir+ ,irons1 ,!1 #adb1 #bdg4 ,tit-,bits1 #eh1 #gi1 #abd1 #bea4 ,torn ,numb]1 ,!1 #bj1 #afb4 ,torp$o ,practice1 #fg1 #aif4 ,t\r1 ,! ,cycli/s'1 #ga1 #aii4 ---- ,! ,gr&1 #gb1 #bjj4 ---- ,! ,que5's1 #ih1 #bbe4 ---- ,! ,rook's1 #if1 #bbc4 ,t[ns1 ,visit+ !1 #gj1 #aih4 ,tra9s1 ,! ,two1 #aa1 #aee4 ,treasure ,boxes1 ,! ,n9e1 #bd1 #afd4 ,trees1 ,! ,tw5ty-"o1 #eg1 #aij4 ,trmaux1 ,m41 #acc1 #ace4 ,triangle1 ,! ,dissect$1 #ch1 #agc4 ,triangul> ,numb]s1 #ac1 #be1 #aff4 ---- ---- ,*eck =1 #ac4 ,tr\blesome ,eight1 ,!1 #aba1 #bdb4 ,tube ,inspector's ,puzzle1 ,!1 #fi1 #aih4 ---- ,railway1 ,he>d on !1 #h1 #aec4 ,turks & ,russians1 #eh1 #aia4 ,turn+s1 ,! ,fifte51 #gj1 #aih4 ,twick5ham ,puzzle1 ,!1 #fj1 #aid4 ,two ,pieces ,problem1 ,!1 #if4 ,unclassifi$ ,puzzles1 #adb4 ,unicursal & ,r\te ,problems1 #fh4 ,union ,jack1 ,!1 #ej1 #fi1 #aig4 ,v&]monde1 ,a41 #eh1 #ajc4 ,veil1 ,"u !1 #ij1 #baf4 ,v]ne1 ,jules1 #bdi4 ,victoria ,cross ,puzzle1 ,!1 #fj1 #aid4 ,village1 ,a ,wond]ful1 #adb1 #bdg4 ,villages1 ,! ,?ree1 #ab1 #aee4 ,villas1 ,! ,eight1 #hj1 #bjf4 ,vortex ,r+s1 #dj4 ,vot]'s ,puzzle1 ,!1 #ge1 #bjb4 ,wall1 ,! ,puzzle1 #eb1 #ahd4 ,wallis1 ,j41 #adb4 ---- 7,ano!r71 #bbj4 ,walls1 ,! ,g>d51 #eb1 #ahe4 ,wap%aw's ,:>f ,my/]y1 ,!1 #aj1 #aec4 ,w> ,puzzle ,game1 ,!1 #aah1 #bdj4 ,wassail ,b[l1 ,!1 #aji1 #bce4 ,wat*1 ,a ,puzzl+1 #aj1 #aec4 ,wat]1 ,gas1 & ,electricity1 #gc1 #bjj4 ,weekly ,dispat*1 ,!1 #bh1 #abd1 #abe1 #adf1 #adh4 ,weigh+ ,puzzles1 ,measur+1 ,pack+1 &1 #aji4 ,:eels1 ,conc]n+1 #ee1 #ahh4 ,:o 0 ,fir/8 #adb1 #bdg4 ,:yte1 ,w4,t41 #adg4 ,wid['s ,legacy1 ,!1 #b1 #adh4 ,wife1 ,f9d ! ,man's1 #adg1 #bea4 ,wilk9son1 ,rev4 ,mr41 #aic4 ,wilson1 ,pr(essor1 #bi4 ,wilson's ,pos]1 #i1 #aec4 ,w9e & ,wat]1 #aaj1 #bce4 ---- ,! ,keg (1 #aaj1 #bce4 ,wo!rspoon1 ,g41 #bdd4 ,ya*t race1 ,!1 #ii1 #bbf4 ,y\?ful ,precocity1 #a1 #adh4 ,z5o1 #aci4 ,,! ,,end4