Group Actions on Winning Games of Super Tic-Tac-Toe

TL;DR

This paper explores the symmetries of super tic-tac-toe games using group actions, revealing that the symmetry group forms a Dihedral group, which helps classify equivalent game states.

Contribution

It introduces a formal group-action framework for super tic-tac-toe, demonstrating that these symmetries form a Dihedral group structure.

Findings

Group actions induce Dihedral symmetry in super tic-tac-toe

Equivalent game states can be classified via group orbits

Symmetry analysis simplifies understanding of game strategies

Abstract

Consider a $n \times n$ tic-tac-toe board. In each field of the board, draw a smaller $n\times n$ tic-tac-toe board. Now let super tic-tac-toe (STTT) be a game where each player's move dictates which field on the larger board a player must make their next move. We will play an impartial game of STTT where each player uses X. We define a set of actions on a game board which gives rise to a group-action on the game that creates equivalent games. We will discuss how the structure of this group-action forms a Dihedral group.