{"paper":{"title":"On the Sprague-Grundy function of Exact $k$-Nim","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Endre Boros, Kazuhisa Makino, Nhan Bao Ho, Peter Mursic, Vladimir Gurvich","submitted_at":"2015-08-18T23:28:28Z","abstract_excerpt":"Moore's generalization of the game of {\\sc Nim} is played as follows. Let $n$ and $k$ be two integers such that $1 \\leq k \\leq n$. Given $n$ piles of tokens, two players move alternately, removing tokens from at least one and at most $k$ of the piles. The player who makes the last move wins. The game was solved by Moore in 1910 and an explicit formula for its Sprague-Grundy function was given by Jenkyns and Mayberry in 1980, for the case $n = k+1$ only. We introduce another generalization of {\\sc Nim}, called {\\sc Exact $k$-Nim}, in which each move reduces exactly $k$ piles. We give an explici"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04484","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}