Tic-Tac-Toe on Designs

We consider playing the game of Tic-Tac-Toe on block designs BIBD$\left(v,k,\lambda \right)$ and transversal designs TD$\left(k,n\right)$. Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD$\left(v,3,\lambda \right)$, are a first-player win if and only if $v\ge 5$. Further, we show that for $k=2,3$, TD$\left(k,n\right)$ is a first-player win if and only if $n\ge k$. We also consider a weak version of the game, called Maker–Breaker, in which the second player wins if they can stop the first player from winning. In this case, we adapt known bounds for when either the first or second player can win on BIBD$\left(v,k,1\right)$ and TD$\left(k,n\right)$, and show that for Maker–Breaker, BIBD$\left(v,4,1\right)$ is a first-player win if and only if $v\ge 16$. We show that TD$\left(4,4\right)$ is a second-player win, and so the second player can force a draw in the regular game by playing the same strategy.