Related papers: Lights Out on graphs

In this paper we study a variant of the solitaire game Lights-Out, where the player's goal is to turn off a grid of lights. This variant is a two-player impartial game where the goal is to make the final valid move. This version is playable…

Lights Out! is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration…

Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set $\{0,1,2,\dots, \ell-1\}$ for $\ell \in \mathbb{N}$. The game is played by toggling vertices: when a vertex is toggled,…

Lights Out is a game played on a graph $G$ where every vertex has a light bulb that is either on or off, and pressing a vertex $v$ toggles the state of every vertex in the closed neighborhood of $v$. The goal is to find a subset of vertices…

Lights out is a game that can be played on any simple graph $G$. A configuration assigns one of the two states \emph{on} or \emph{off} to each vertex. For a given configuration, the aim of the game is to turn all vertices \emph{off} by…

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…

We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer $n$, what is the probability that a graph chosen uniformly at random from the set of graphs with $n$ vertices…

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…

We study the generalization of the game Lights Out in which the standard square grid board is replaced by a graph. We examine the probability that, when a graph is chosen uniformly at random from the set of graphs with $n$ vertices and $e$…

The Lights Out Puzzle, played on a graph $\Gamma$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements…

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…

A configuration of the lit-only $\sigma$-game on a graph $\Gamma$ is an assignment of one of two states, {\it on} or {\it off}, to each vertex of $\Gamma.$ Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allows the…

A configuration of the lit-only $\sigma$-game on a finite graph $\Gamma$ is an assignment of one of two states, on or off, to all vertices of $\Gamma.$ Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allows the…

Lights Out is a game which can be played on any graph $G$. Initially we have a configuration which assigns one of the two states on or off to each vertex. The aim of the game is to turn all vertices to off state for an initial configuration…

Let $\mathbb{F}$ be any field, we consider solving $Ax=b$ repeatedly for a matrix $A\in\mathbb{F}^{n\times n}$ of $m$ non-zero elements, and multiple different $b\in\mathbb{F}^{n}$. If we are given a zero forcing set of $A$ of size $k$, we…

We look at both the \emph{group labeling lights out game} and the \emph{neighborhood lights out game}. Our main focus is to determine necessary and sufficient conditions for when the group labeling lights out game on path graphs, cycle…

The Gale-Berlekamp Light Switching Game is played on a square board of lights. Each light has two states, either on or off. There is a switch to every row and column. Turning this switch would change the state of all the lights on that row…

A configuration of a graph is an assignment of one of two states, on or off, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an on vertex. The following…

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…

We give an elementary inductive proof of a classical result for the \emph{Lights Out problem} on graphs: from any configuration of vertices, one can reach the complementary configuration by a sequence of moves, where a move consists of…