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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. 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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. 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