We can encode the existence of such a set $S$ by doing linear algebra over the binary field $\mathbb$, where $\chi(N)$ is the characteristic vector of the closed neighbourhood of $v$. Thus, we are searching for a subset $S$ of vertices such that $|S \cap N|$ is odd for all $v \in V$, where $N$ is the closed neighbourhood of $v$ (the neighbours of $v$ together with $v$ itself). Proof. Evidently, there is no point in pressing a vertex more than once, and the sequence in which we press vertices does not matter. For example, consider just a single edge where only one light is ON initially.) (Note that if not all the lights are ON initially, it will not always be possible to switch all the lights OFF. Then it is always possible to press a sequence of vertices to switch all the lights OFF. Let $G=(V,E)$ be a graph for which all the lights are initially ON. The goal of this post is to prove the following theorem. For example, there is a dedicated wikipedia page and this site (from which I borrowed the image below) even has the original user manuals. The version of this game where $G$ is the $5 \times 5$ grid was produced in the real world by Tiger Electronics, and has a bit of a cult following. The goal is to determine if it is possible to press some sequence of vertices to turn off all the lights. We are allowed to press any vertex $v$, which has the effect of switching the state of $v$ and all of the neighbours of $v$. At the beginning of the game, some subset of the vertices are ON, and the rest of the vertices are OFF. We are given a graph $G=(V,E)$, where we consider each vertex as a light. In this short post, I will discuss a cute graph theory problem called the Lights Out Game. The set-up is as follows. "Symmetric Matrices over F_2 and the Lights Out Problem".
^ a b c Solving Lights Out, Matthew Baker.Anderson and Feil found that in order for a configuration to be solvable (deriving the null vector from the original configuration) it must be orthogonal to the two vectors N 1 and N 2 below (pictured as a 5×5 array but not to be confused with matrices). Each entry is an element of Z 2, the field of integers modulo 2. The 5×5 grid of Lights Out can be represented as a 25x1 column vector with a 1 and 0 signifying a light in its on and off state respectively. In 1998, Marlow Anderson and Todd Feil used linear algebra to prove that not all configurations are solvable and also to prove that there are exactly four winning scenarios, not including redundant moves, for any solvable 5×5 problem. Secondly, in a minimal solution, each light needs to be pressed no more than once, because pressing a light twice is equivalent to not pressing it at all. Firstly, the order in which the lights are pressed does not matter, as the result will be the same. Several conclusions are used for the game's strategy. If a light is off, it must be toggled an even number of times (including none at all) for it to remain off. If a light is on, it must be toggled an odd number of times to be turned off. The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible. Pressing any of the lights will toggle it and the four adjacent lights. When the game starts, a random number or a stored pattern of these lights is switched on. The game consists of a 5 by 5 grid of lights. Lights to toggle to turn off a fully-lit 5×5 board
#Lightsout game free#
Tiger Toys also produced a cartridge version of Lights Out for its Game com handheld game console in 1997, shipped free with the console.Ī number of new puzzles similar to Lights Out have been released, such as Lights Out 2000 (5×5 with multiple colors), Lights Out Cube (six 3×3 faces with effects across edges), and Lights Out Deluxe (6×6). Another similar game was produced by Vulcan Electronics in 1983 under the name XL-25. Merlin, a similar electronic game, was released by Parker Brothers in the 1970s with similar rules on a 3 by 3 grid. The goal of the puzzle is to switch all the lights off, preferably with as few button presses as possible. Pressing any of the lights will toggle it and the adjacent lights. Lights Out is an electronic game released by Tiger Electronics in 1995. Selecting a square changes it and the surrounding squares.
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