Lights Out (game)
Lights Out is an electronic game released by Tiger Electronics in 1995. The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the adjacent lights. The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible.
A similar electronic game Merlin was released by Parker Brothers in the 1970s with similar rules on a 3 by 3 grid. Another similar game was produced by Vulcan Electronics in 1983 under the name XL-25. Tiger Toys also produced a cartridge version of Lights Out for its Game com handheld game console in 1997, shipped free with the console. A number of new puzzles similar to Lights Out have been released, such as Lights Out 2000, Lights Out Cube, and Lights Out Deluxe.
Lights Out was created by a group of people including Avi Olti, Gyora Benedek, Zvi Herman, Revital Bloomberg, Avi Weiner and Michael Ganor. The members of the group together and individually also invented several other games, such as Hidato, NimX, iTop and many more.
The game consists of a 5 by 5 grid of lights. When the game starts, a random number or a stored pattern of these lights is switched on. Pressing any of the lights will toggle it and the four adjacent lights. The goal of the puzzle is to switch all the lights off, preferably in as few button presses as possible.
If a light is on, it must be toggled an odd number of times to be turned off. If a light is off, it must be toggled an even number of times (including none at all) for it to remain off. Several conclusions are used for the game's strategy. Firstly, the order in which the lights are pressed does not matter, as the result will be the same. 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.
In a paper written by Marlow Anderson and Todd Feil, linear algebra is used 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. 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. Each entry is an element of Z2, the field of integers modulo 2. What Anderson and Feil find in their paper is 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 N1 and N2 below (pictured as a 5x5 array but not to be confused with matrices).
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