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Asymptotically optimal pairing strategy for tic-tac-toe with numerous directions. (English) Zbl 1202.91044
Summary: We show that there is an \(m = 2n + o(n)\), such that, in the Maker-Breaker game played on \(\mathbb Z^d\) where Maker needs to put at least \(m\) of his marks consecutively in one of \(n\) given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of K. Kruczek and E. Sundberg [Electron. J. Comb. 15, No. 1, Research Paper N42, 6 p. (2008; Zbl 1160.91008)] who showed that such a pairing strategy exists if \(m \geqslant 3n\). A simple argument shows that m has to be at least \(2n + 1\) if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.

MSC:
91A46 Combinatorial games
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