Additive sink subtraction
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910055017742336 |
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| author | Bhagat, Anjali Larsson, Urban Manabe, Hikaru Yamashita, Takahiro |
| author_facet | Bhagat, Anjali Larsson, Urban Manabe, Hikaru Yamashita, Takahiro |
| contents | Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp~(1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in Winning Ways~(1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard {\em wall} convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Additive sink subtraction Bhagat, Anjali Larsson, Urban Manabe, Hikaru Yamashita, Takahiro Combinatorics Discrete Mathematics 91A46 Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bound on the period length. Flammenkamp~(1997) conjectures a striking classification for three-move subtraction games: non-additive rulesets exhibit linear period lengths of the form ``the sum of two moves'', where the choice of which two moves displays fractal-like behavior, while additive sets $S=\{a,b,a+b\}$ have purely periodic outcomes with linear or quadratic period lengths. Despite early attention in Winning Ways~(1982), the general additive case remains open. We introduce and analyze a dual winning convention, which we call {\sc sink subtraction}. Unlike the standard {\em wall} convention, where moves to negative positions are forbidden, the sink convention declares a player the winner upon moving to a non-positive position. We show that {\sc additive sink subtraction} admits a complete solution: the nim-sequence is purely periodic with an explicit linear or quadratic period formula, and we conjecture a duality between additive sink subtraction and classical wall subtraction. Keywords: Additive Subtraction Game, Nimber, Periodicity, Sink Convention. |
| title | Additive sink subtraction |
| topic | Combinatorics Discrete Mathematics 91A46 |
| url | https://arxiv.org/abs/2601.18715 |