Resolution to Sutner's Conjecture

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.