Combinatorial Game Distributions of Steiner Systems

TL;DR

This paper explores the connection between combinatorial games and Steiner systems, constructing specific games whose P-position sets correspond to Steiner system blocks, and characterizing systems via game distributions.

Contribution

It introduces a method to construct games from Steiner systems, characterizes the hexad game among isomorphic systems, and links game distributions to properties of Steiner triple systems.

Findings

Characterized the hexad game as unique with minimal positions.

Developed a construction method for games from Steiner systems.

Linked game distribution symmetry to projective Steiner triple systems.

Abstract

The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-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, $\mathcal{D}_{\text{sh}}$. There were, however, few known games related to Steiner systems like the hexad game. For a given Steiner system, we construct a game whose $\mathscr{P}$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $\mathcal{D}_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, \ldots, 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $\mathscr{P}$-position set is its block set. From $\mathcal{D}_{\text{sh}}$, we obtain the hexad game, and this game is characterized as a 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 $\mathcal{D}$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $\mathcal{D}$. 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.