{"paper":{"title":"Slow $k$-Nim","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nhan Bao Ho, Vladimir Gurvich","submitted_at":"2015-08-24T12:24:39Z","abstract_excerpt":"Given $n$ piles of tokens and a positive integer $k \\leq n$, we study the following two impartial combinatorial games Nim$^1_{n, \\leq k}$ and Nim$^1_{n, =k}$. In the first (resp. second) game, a player, by one move, chooses at least $1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from each of these piles. For the normal and mis\\`ere version of each game we compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 = 3$. For game Nim$^1_{n, \\leq k}$ we also characterize its P-positions for the cases $n \\leq k+2$ and $n = k+3 \\leq 6$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05777","kind":"arxiv","version":1},"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"}