Most Clicks Problem in Lights Out

TL;DR

This paper investigates the worst-case number of clicks needed to solve the Lights Out game on grid graphs, providing bounds and exact solutions for specific grid sizes and nullity conditions.

Contribution

It extends known results by establishing an upper bound for the Most Clicks Problem on certain grid sizes and solves it exactly for nullity 2 grids, proposing a conjecture for all such grids.

Findings

Upper bound for MCP on (6k-1) x (6k-1) grids

Exact solution for nullity 2 grids of these sizes

Conjecture that all nullity 2 grids are of size (6k-1) x (6k-1)

Abstract

Consider a game played on a simple graph $G = (V, E)$ where each vertex consists of a clickable light. Clicking any vertex $v$ toggles the on/off state of $v$ and its neighbors. Starting from an initial configuration of lights, one wins the game by finding a solution: a sequence of clicks that turns off all the lights. When $G$ is a $5 \times 5$ grid, this game was commercially available from Tiger Electronics as Lights Out. Restricting ourselves to solvable initial configurations, we pose a natural question about this game, the Most Clicks Problem (MCP): How many clicks does a worst-case initial configuration on $G$ require to solve? The answer to the MCP is already known for nullity 0 graphs: those on which every initial configuration is solvable. Generalizing a technique from Scherphius, we give an upper bound to the MCP for all grids of size $(6k - 1) \times (6k - 1)$. We show the value given by this upper bound exactly solves the MCP for all nullity 2 grids of this size. We conjecture that all nullity 2 grids are of size $(6k - 1) \times (6k - 1)$, which would mean we solve the MCP for all nullity 2 square grids.