Combinatorial Game Distributions of Steiner Systems

The $P$-position sets of some combinatorial games have special combinatorial structures. For example, the $P$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, denoted by $D_{\text{sh}}$. However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose $P$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $D_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, ..., 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $P$-position set is its block set. From $D_{\text{sh}}$, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $D$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $D$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.