Lights Out Solver

Lights Out is an electronic game composed of a grid of lighted (sometimes with bulbs) or numbered cells (originally 5 by 5).

At the start of the game, a pattern of cells is lit (different states). By pressing one of the boxes, it switches state (it goes from on to off, or from off to on or changes color), as well as the four adjacent boxes (neighboring top, right, bottom and left).

The goal of the game is to switch all the lights to the off (or on) position, preferably by pressing as few boxes as possible.

The player must think about which squares to press to turn off all the lights.

A click on the middle box gives the following result:

(the box clicked as well as the 4 adjacent cells (top, bottom, right, left) have changed state)

For a game with $ n $ states, pressing a box $ n $ times returns it to its initial state.

The principle of resolution is mathematical, by modeling the grid by a matrix $ [a_i] $ of size $ n \times m $ and $ p \geq 2 $ possible states.

By pressing a box, some other cells have their state changed or reversed (if $ p = 2 $)

The next state is then determined, for each box $ i $, by the number of times the boxes are pressed (modulo $ p $)

This state can be represented with $ m $ calculations $ a_{i1} x_1 + ... + a_{in} x_n) \mod p = 0 $ for which the value of $ x_i $ is the solution sought.