Construction of $\sigma ^+$-game solutions on a rectangular grid

Abstract

The $\sigma ^+$-game is played on a directed graph where the vertices (lights) are assigned an off or on state. The player can press any light, thereby toggling it along with its neighbors. The objective of the game is to turn all lights off given an initial configuration. Our particular interest are rectangular grid graphs with all lights initially on. We explore different classes of grid sizes and dimensions with unique solutions, and we construct the solutions to grids of size $(2^n-1)\times (2^n-1)$ with all lights initially on.