{"paper":{"title":"On the Sprague-Grundy Function of Tetris Extensions of Proper {\\sc 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, Vladimir Gurvich","submitted_at":"2015-04-27T05:00:41Z","abstract_excerpt":"Given a hypergraph $\\cH \\subseteq 2^I \\setminus \\{\\emptyset\\}$ on the ground set $I = \\{1, \\ldots, n\\}$, we assign to each $i \\in I$ a nonnegative integer $x_i$, that is a pile of $x_i$ tokens, and consider the following generalization of the classical game of {\\sc Nim}: Two players alternate turns. In a move a player chooses an arbitrary edge $H \\in \\cH$ and reduces all piles $i \\in H$. The player who is out of moves loses. We call the obtained game hypergraph {\\sc Nim}. Such a game is called proper {\\sc Nim}, when $\\cH=2^I \\setminus\\{I,\\emptyset\\}$ is the family of all proper subsets of $I$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.06926","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"}