PreprintCC BY 4.0arXiv
The Sprague--Grundy Function of Slow Exact k-Nim with k+1 Piles
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≥2, 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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BibTeX
@misc{gurvich2026spraguegrundy,
title = {{The Sprague--Grundy Function of Slow Exact $k$-Nim with $k+1$ Piles}},
author = {Vladimir Gurvich and Mariya Naumova},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.14546},
eprint = {2611.14546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.14546},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Sprague--Grundy Function of Slow Exact k-Nim with k+1 Piles
AU - Vladimir Gurvich
AU - Mariya Naumova
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.14546
UR - https://arxiv.org/abs/2611.14546
AB - 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≥2, 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.
ER -