{"paper":{"title":"When are translations of P-positions of Wythoff's game P-positions?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aviezri S. Fraenkel, Nhan Bao Ho","submitted_at":"2014-03-11T09:21:36Z","abstract_excerpt":"We study the problem whether there exist variants of {\\sc Wythoff}'s game whose $\\P$-positions, except for a finite number, are obtained from those of {\\sc Wythoff}'s game by adding a constant $k$ to each $\\P$-position. We solve this question by introducing a class $\\{\\W_k\\}_{k \\geq 0}$ of variants of {\\sc Wythoff}'s game in which, for any fixed $k \\geq 0$, the $\\P$-positions of $\\W_k$ form the set $\\{(i,i) | 0 \\leq i < k\\}\\cup \\{(\\lfloor \\phi n \\rfloor + k, \\lfloor \\phi^2 n \\rfloor + k) | n\\ge 0\\}$, where $\\phi$ is the golden ratio. We then analyze a class $\\{\\T_k\\}_{k \\geq 0}$ of variants of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.2512","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"}