Lights Off
Rules of the Game
"Lights Out" is a one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling the on/off state of this and all four vertically and horizontally adjacent squares. In the classic "all-ones" problem, the game begins with all lights on, and the goal is to turn all the lights off. Moreover, the goal is to turn them all of in the minimum number of moves.
A variation of the game is one where there is initially a randomly chosen light pattern, and the aim is again to turn all the lamps off. Obviously,some initial states are more difficult to solve than others.
Notes: 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 conclusion are used for the game's strategy. First, the order in which the lights are pressed does not matter, as the result will be the same. Second, each light needs to be pressed no more than once, because pressing a light twice is equivalent to not pressing it at all.