## The pairing strategies of the 9-in-a-row game.(English)Zbl 1422.91107

Summary: One of the most useful strategies for proving Breaker’s win in Maker-Breaker positional games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the $$k$$-in-a-row game, the case $$k = 9$$ is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by A. W. Hales and R. I. Jewett [Trans. Am. Math. Soc. 106, 222–229 (1963; Zbl 0113.14802)].
In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no “irregular”, non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional $$k$$-in-a-row game.

### MSC:

 91A24 Positional games (pursuit and evasion, etc.) 91A46 Combinatorial games

Zbl 0113.14802
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### References:

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