Additive Subtraction Games
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915896225693696 |
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| author | Larsson, Urban Manabe, Hikaru |
| author_facet | Larsson, Urban Manabe, Hikaru |
| contents | We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the {\mathscr P}-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Miklós and Post (2024) established outcome-periodicity, but without reference to that closed formula. The {\em primitive quadratic} case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction by Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical {\mathscr P}-positions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10414 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Additive Subtraction Games Larsson, Urban Manabe, Hikaru Combinatorics Discrete Mathematics 91A46, 05A17, 11B75, 91A05 We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the {\mathscr P}-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Miklós and Post (2024) established outcome-periodicity, but without reference to that closed formula. The {\em primitive quadratic} case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction by Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical {\mathscr P}-positions. |
| title | Additive Subtraction Games |
| topic | Combinatorics Discrete Mathematics 91A46, 05A17, 11B75, 91A05 |
| url | https://arxiv.org/abs/2603.10414 |