Algumas luminesc\^encias sobre o jogo Lights Out

TL;DR

This paper explores the Lights Out game using linear algebra, providing criteria for solvability based on matrix invertibility and calculating the determinant for specific cases.

Contribution

It establishes a solvability criterion for the game on an m by n grid using linear algebra and determines the determinant in a particular case.

Findings

Solvability depends on matrix invertibility related to grid dimensions

Provided explicit conditions for game solvability based on m and n

Calculated the determinant for a specific grid configuration

Abstract

The theory behind the Lights Out game has been developed by several authors. The aim of this work is to present some results related to this game using Linear Algebra. We establish a criterion for the solubility of this game in the case of an $m$ by $n$ grid, which depends on the invertibility of a matrix, and we present the conditions for this to occur, easily verifiable from $m$ and $n$. Furthermore, we explicitly determine the value of the determinant for a particular case.