Lights Out Puzzle in p Colors: Evolution of Quiet Patterns

TL;DR

This paper explores the evolution and inverse of quiet patterns in the Lights Out puzzle across larger grids with prime number states, utilizing elliptic curves to analyze their properties and minimal click configurations.

Contribution

It introduces a method to evolve quiet patterns into larger grids for prime state counts and demonstrates the existence of inverse patterns using elliptic curves.

Findings

Evolved quiet patterns in larger grids from smaller ones for prime p.

Existence of inverse patterns for most prime p using elliptic curves.

Characterization of minimal click configurations with five nonzero states.

Abstract

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.