Winning Lights Out with Fibonacci

TL;DR

This paper explores a generalized Fibonacci-based strategy for solving a cylindrical Lights Out game with multiple light states, analyzing how the game size affects the ability to turn off all lights.

Contribution

It introduces a novel Fibonacci connection to the Lights Out game with multiple states and cylindrical topology, providing new insights into game solvability.

Findings

Fibonacci numbers mod k determine game solvability

The number of rows needed depends on Fibonacci sequence properties

Extended Lights Out game with multiple states analyzed

Abstract

Lights Out is a single-player electronic handheld game from the 1990s that features a 5 by 5 grid of light-up buttons. The game begins with some lights on and others off. The goal is to turn off all lights but pressing a button changes its state and changes the states of the buttons above and below and to the left and right of the button. We examine a cylindrical Lights Out game in which the left side of the board is connected to the right. Moreover, instead of just on and off we let the lights have $k$ states for $k \ge 2$. We then apply a modified light chasing strategy in which we try to systematically turn off all lights in a row by pressing the buttons in the row below. We ask if the game begins with all lights starting at the same state, how many rows must the board have in order for all lights to be turned off using this type of modified light chasing after we press the last row of lights. We connect this light chasing strategy to the Fibonacci numbers and are able to provide answer to our question by studying the Fibonacci numbers (mod $k$).