Lights Out Solver

Lights Out (or Flip Tiles game) is an electronic game composed of a grid of lighted (sometimes with bulbs) or numbered cells (originally 5x5) which consists of turning off all the lights in a grid.

At the start of the game, a pattern of cells is lit (different states). By pressing one of the boxes, it acts like a switch, it changes state (it goes from on to off, or flips 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.

There are several variations and options in Lights Out:

— Grid size: different grid formats are possible, ranging from 3x3, 4x4, 5x5, 6x6 to larger or even rectangular sizes.

— Random or set start state: lights can start in various configurations.

— Limited actions: the number of state changes allowed can be restricted to increase the difficulty.

— Timed game: A countdown can add an extra challenge to complete the game within a time limit.

— Obstacles: some variants introduce black boxes that cannot be changed state.

An effective strategy is to observe the patterns and symmetries in the grid, if there are any, then carry out symmetrical actions: bulb on the right then bulb on the left etc.

The optimal strategy, however, is to perform linear algebra mathematical calculations that allow all the grids to be mathematically resolved.

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.