Misère Tic-Tac-Toe on Projective Binary Steiner Triple Systems

Abstract

Imagine playing tic-tac-toe to lose. Two players, Xavier and Olivia, alternate marking squares as usual. As soon as one player owns three squares in a row, they lose. The combinatorial game “misère tic-tac-toe” generalizes this idea. The two players must first agree on a board made from points and lines, which are subsets of the points—but this need not be a traditional tic-tac-toe board. In this article, we study misère tic-tac-toe on projective binary Steiner triple systems. We provide an explicit winning strategy for the second player, Olivia. This winning strategy relies on the nested geometric structure of these systems, as well as the structure of caps within them. This article completes the final case for misère tic-tac-toe on the “geometric” Steiner triple systems, with the surprising result that the winning strategy belongs to different players on affine versus projective Steiner triple systems.