Resolution to Sutner's Conjecture

TL;DR

This paper proves Sutner's conjecture regarding the dimension of the kernel of a matrix associated with grid graphs in Lights Out, using polynomial GCD techniques and identities relating different grid sizes.

Contribution

We confirm Sutner's conjecture on the recursive relation of kernel dimensions for grid graphs and determine the exact values of the parameter  for different grid sizes.

Findings

Confirmed Sutner's conjecture for all grid sizes.

Derived explicit conditions for  to be 0 or 2.

Connected polynomial GCD properties to graph kernel dimensions.

Abstract

Consider a game played on a simple graph $G = (V,E)$ where each vertex consists of a clickable light. Clicking any vertex $v$ toggles the on/off state of $v$ and its neighbors. One wins the game by finding a sequence of clicks that turns off all the lights. When $G$ is a $5 \times 5$ grid, this game was commercially available from Tiger Electronics as Lights Out. Sutner was one of the first to study these games mathematically. He found that when $d(G) = \text{dim}(\text{ker}(A + I))$ over the field $GF(2)$, where $A$ is the adjacency matrix of $G$, is 0 all initial configurations are solvable. When investigating $n \times n$ grid graphs, Sutner conjectured that $d_{2n+1} = 2d_{n} + \delta_{n}, \delta_{n} \in \{0,2\}, \delta_{2n+1} = \delta_{n}$, where $d_n = d(G)$ for $G$ an $n \times n$ grid graph. We resolve this conjecture in the affirmative. We use results from Sutner that give $d_n$ as the GCD of two polynomials in the ring $\mathbb{Z}_2[x]$. We then apply identities from Hunziker, Machiavelo, and Park that relate the polynomials of $(2n+1) \times (2n+1)$ grids and $n \times n$ grids. Finally, we use a result from Ore about the GCD of two products. Together these results allow us to prove Sutner's conjecture. We then go further and show for exactly which values of $n$ $\delta_n$ is 0 or 2.