Tic-Tac-Toe on Designs

Abstract

We consider playing the game of Tic-Tac-Toe on block designs BIBD($v, k, λ$) and transversal designs TD($k, n$). Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD($v, 3, λ$), are a first player win if and only if $v \geq 5$. Further, we show that for $k = 2, 3$, TD($k, n$) is a first player win if and only if $n \geq k$. 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($v, k, 1$) and TD($k, n$), and show that for Maker-Breaker, BIBD($v, 4, 1$) is a first player win if and only if $v \geq 16$. We show that TD($4, 4$) is a second player win, and so the second player can force a draw in the regular game by playing the same strategy.

Figures (11)

• Figure 1: A game of Tic-Tac-Toe played on a standard $3\times 3$ board. The winning hyperedges and the state of a proposed game are also given.
• Figure 2: Seven hypergraphs, with a hyperedge of cardinality $2$ indicated as a double line and of cardinality $3$ indicated as a triangle (either solid or dashed). Each vertex that is incident with one hyperedge is white, and with two hyperedges is black.
• Figure 3: Remaining vertices of the six hyperedges in which Breaker has not yet played in Case 1 of Theorem \ref{['TD(4,4)']}. These hyperedges are $\{\alpha_3, \beta_2\}$, $\{c_2, \alpha_2, \beta_2\}$, $\{c_2, \alpha_4, \beta_1\}$,$\{c_3, \alpha_3, \beta_3\}$,$\{c_4, \alpha_2, \beta_3\}$, and $\{c_4, \alpha_4, \beta_4\}$.
• Figure 4: Scissor on $X_3$, after the first three Xeno moves played on an STS$(v)$. It is possible that $A=C$, in which case we have the configuration shown on the right.
• Figure 5: The (multi)graph $G[0]$ derived from the TS$(6,2)$ of Example \ref{['ex:ts62']}, which is the $C_5$ that results from including an edge $xy$ for each block $\{0,x,y\}$ in the TS.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2: CarrollDougherty
• Theorem 1.3
• Theorem 1.4: Beck81ES
• Theorem 1.5: Beck81
• proof
• Theorem 1.6: Beck81
• Lemma 2.1
• proof
• Corollary 2.2