Asymptotically Optimal Pairing Strategy for Tic-Tac-Toe with Numerous Directions

Padmini Mukkamala, Dömötör Pálvölgyi

Abstract


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 Kruczek and Sundberg [Electronic Journal of Combinatorics 15(1):N42, 2008] who showed that such a pairing strategy exists if $m\ge 3n$. 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.


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