arXiv:2611.14546v1 Announce Type: new Abstract: We give an explicit formula for the normal-play Sprague--Grundy (SG) function of slow exact $k$-Nim with $k+1$ piles. A move removes one stone from each of exactly $k$ nonempty piles. A truncation preserves every legal move sequence. The SG value depends only on the total of the truncated coordinates modulo $2k$, their number of odd entries, and the parities of the two smallest piles. For every $k\geq2$, the set of the SG values is $\{0,1,2,3\}$. We prove the formula by the mex characterization, including the reductions needed after boundary moves. We also prove a unit-shift identity and use it with a known optimal-move theorem to explain the four-value bound independently of the explicit formula.
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