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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 Hales and Jewett.  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.

The pairing strategies of the 9-in-a-row game