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Lights Out! A game of combinatorics and linear algebra

Cassio Vieira Morais, Tiane Marcarini

Abstract

In this work, we study a triangular variant of the Lights Out game, proposed in the 2025 Capixaba Mathematics Olympiad. We present a combinatorial description of the game, formally characterize its operations, and introduce the notion of a quiet pattern, which determines which configurations admit a solution and how many solutions they possess. We then analyze the geometry of quiet patterns and describe the propagation mechanisms that generate patterns for larger board sizes. Finally, we model the problem using linear systems over the field Z2, obtaining a matrix associated with the game and a combinatorial criterion for its invertibility. This criterion shows that the game admits a solution for every configuration if and only if the number of coverings of the triangular board by 1 x 1 and 2 x 1 tiles is odd.

Lights Out! A game of combinatorics and linear algebra

Abstract

In this work, we study a triangular variant of the Lights Out game, proposed in the 2025 Capixaba Mathematics Olympiad. We present a combinatorial description of the game, formally characterize its operations, and introduce the notion of a quiet pattern, which determines which configurations admit a solution and how many solutions they possess. We then analyze the geometry of quiet patterns and describe the propagation mechanisms that generate patterns for larger board sizes. Finally, we model the problem using linear systems over the field Z2, obtaining a matrix associated with the game and a combinatorial criterion for its invertibility. This criterion shows that the game admits a solution for every configuration if and only if the number of coverings of the triangular board by 1 x 1 and 2 x 1 tiles is odd.
Paper Structure (4 sections, 3 theorems, 37 equations, 17 figures, 1 table)

This paper contains 4 sections, 3 theorems, 37 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernels
  4. Linear Algebra

Key Result

Theorem 1

If a game of size $n$ has a non-trivial kernel, then the same holds for all games of size $n+(n+2)j$, for every integer $j \geq 1$.

Figures (17)

  • Figure 1: A configuration of the game of size $n=7$.
  • Figure 2: The configuration obtained by pressing button 10 in Figure \ref{['fig:antes']}.
  • Figure 3: The kernel of a game of size $n = 2$.
  • Figure 4: A representation of an element of the kernel of a game of size $6$.
  • Figure 5: Elements of the kernel of a game of size $n=5$.
  • ...and 12 more figures

Theorems & Definitions (9)

  • Definition 1
  • proof
  • proof
  • Remark
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof