Resolution to Sutner's Conjecture
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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 × 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) = dim(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 × n grid graphs, Sutner conjectured that d_2n+1 = 2d_n + δ_n, δ_n∈{0,2}, δ_2n+1 = δ_n, where d_n = d(G) for G an n × 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 ℤ_2[x]. We then apply identities from Hunziker, Machiavelo, and Park that relate the polynomials of (2n+1) × (2n+1) grids and n × 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 and describe exactly when δ_n is 0 or 2.
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