Lights Out Puzzle in p Colors: Evolution of Quiet Patterns

The Lights Out Puzzle represents a cellular automaton based on a grid of squares where clicking a square changes its state and the states of surrounding squares. A “quiet pattern” is a way to click such that in the end, no change is effected. We introduce a way to “evolve” quiet patterns in smaller grids into ones in $p$ times larger grids when the number of possible states of a square is a prime $p$. Using elliptic curves, we also find that an inverse “de-evolution” exists for most $p$. We also describe the only ways to click a grid of squares such that only 5 (the minimum) number of squares have a nonzero state.