{"paper":{"title":"Tetris Hypergraphs and Combinations of Impartial Games","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":"2017-01-11T01:18:02Z","abstract_excerpt":"The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\\cH$-combinations of impartial games for any hypergraph $\\cH$. In particular, we introduce the game $NIM_\\cH$ which is the $\\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\\cH$-combination of SG decreasing games to $NIM_\\cH$. We call $\\cH$ a Tetris hypergraph if $NIM_\\cH$ is SG decreasing. We provide some necessary and some suffi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02819","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"}