{"paper":{"title":"A Subtraction Nim with a Pass","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hikaru Manabe, Ryohei Miyadera, Urban Larsson","submitted_at":"2026-05-14T03:34:08Z","abstract_excerpt":"We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classic"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The subtraction set must be exactly {2, 4n, 4n+2} for integer n ≥ 3; the reverse-mex property is stated not to hold for n=1 or n=2, and the proofs rely on this specific arithmetic form.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Subtraction Nim with moves {2,4n,4n+2} (n≥3) and its one-time-pass variant both satisfy the reverse-mex property for Grundy numbers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c87d51fd54818645248ef59be459902a7486bd9a550cf6a3e6f6990aae57f0da"},"source":{"id":"2605.14321","kind":"arxiv","version":1},"verdict":{"id":"38d70c46-cda5-4637-8360-82f6799553d9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:38:54.294616Z","strongest_claim":"We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.","one_line_summary":"Subtraction Nim with moves {2,4n,4n+2} (n≥3) and its one-time-pass variant both satisfy the reverse-mex property for Grundy numbers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The subtraction set must be exactly {2, 4n, 4n+2} for integer n ≥ 3; the reverse-mex property is stated not to hold for n=1 or n=2, and the proofs rely on this specific arithmetic form.","pith_extraction_headline":"Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged."},"references":{"count":13,"sample":[{"doi":"","year":2019,"title":"M. H. Albert, R. J. Nowakowski, and D. W olfe, Lessons In Play: An Introduction to Combi- natorial Game Theory , second edition, A K Peters/CRC Press, Boca Raton, FL, 2019","work_id":"cdaa7b1d-712f-4108-aa4f-86d4a0bced72","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Anjali Bhagat, Urban Larsson, Hikaru Manabe, Takahiro Y amashita, Additive sink subtrac- tion, Preprint arXiv:2601.18715 (2026)","work_id":"a2f53834-0dd7-413f-9efc-327901c20852","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"W. H. Chan, R. M. Low, S. C. Locke, and O.L. W ong, A map of the P-positions in ‘Nim With a Pass’ played on heap sizes of at most four, Discrete Applied Mathematics 244 (2018), 44-55","work_id":"c21bea2b-5a59-48a8-8c14-97fd04cbcb3f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1966,"title":"S. W. Golomb, A mathematical investigation of games of “t ake-away”, J. Combinatorial Theory, 1(4) (1966), 443– 458","work_id":"3b23ee8d-cfce-4c44-af8a-0da0cfc0ce20","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"D. G. Horrocks and R. J. Nowakowski, Regularity in the G–S equences of Octal Games with a Pass, Integers 3 (2003), #G1","work_id":"f6e67158-94f9-460d-ac96-6ed44c7420ff","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"b42cd6cee05376e835ae01ae111ecda5c444e8b5471c83ce607ba26a8d7ff20f","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"}