Lights Out Puzzle in p Colors: Evolution of Quiet Patterns

TL;DR

This work studies the $p$-color Lights Out puzzle by introducing an evolution operator $\mathrm{Evo}_p$ that expands quiet patterns to larger grids and an almost inverse $\mathrm{Era}_p$ up to the scalar $c_p$, the central entry of the primitive pattern. The authors show $\mathrm{Evo}_p(Q)$ preserves quietness exactly when $Q$ is quiet, and they connect $c_p$ to the Hasse invariant of a related elliptic curve, yielding primes with $c_p=0$ of density zero. They further reveal that quiet patterns over $\mathbb{F}_p$ correspond to points on the elliptic curve and its isogenies, with a natural $D_8$ symmetry acting on these patterns; this provides infinite families of quiet patterns and a rich fractal evolution structure. A key result is a Mikado-style five-light minimality theorem: any finite sequence of clicks on an infinite grid leaves at least five lit squares, with exactly five occurring precisely for evolutions $\mathrm{Evo}_p^{(n)}(k_0)$, thereby unifying combinatorics, number theory, and geometry in the study of colorful Lights Out puzzles.

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.

Figures (7)

• Figure 1: The nontrivial quiet patterns for the 2-color $2 \times 3$ puzzle.
• Figure 2: The 1-step evolutions of the quiet patterns in Figure \ref{['fig:qp2x3']}
• Figure 3: The pattern after 5 steps of evolution of the leftmost $2 \times 3$ quiet pattern in Figure \ref{['fig:qp2x3']}.
• Figure 4: Two steps of the evolution of the $1 \times 1$ array (1) for $k = 3$, showing how the diamonds coming from $\mathop{\mathrm{Evo}}\nolimits_p(1)$ are superimposed in $\mathop{\mathrm{Evo}}\nolimits_p^{(2)}(1)$.
• Figure 5: The six 3-color evolutions $\mathop{\mathrm{Evo}}\nolimits_3(1), \mathop{\mathrm{Evo}}\nolimits_3^{(2)}(1), \dots, \mathop{\mathrm{Evo}}\nolimits_3^{(6)}(1)$ (left to right). Note: images are not to scale.

Theorems & Definitions (31)

• Definition 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Definition 2.4
• Corollary 2.5
• proof
• Definition 2.6
• Theorem 2.7