Lights Out is a grid of switches wired badly on purpose. Every press flips the square you touched and its four neighbours, so fixing one light breaks others. It feels like whack-a-mole for about two minutes, until you notice two facts that turn it into a puzzle you can solve every time.
Those two facts are the whole game, and they are also the doorway to a neat piece of mathematics: a board of light switches is a system of linear equations, and you can solve it the way you’d solve one at school — just with arithmetic where 1 + 1 = 0.
How to play
The board is a 5×5 grid (or 3×3 in Warm-up). Some squares start lit. Pressing a square toggles it and each square directly above, below, left and right of it: lit becomes dark, dark becomes lit. Diagonals are not affected. A corner press flips three squares, an edge press four, a middle press five.
There is no turn limit and no way to lose. The board is solved as soon as every square is dark. Every board here is made by starting from a dark grid and pressing random squares, so every board can be cleared. The Minimum stat tells you the fewest presses that can clear it; your challenge is to match it.
A worked example
A Warm-up board. “on” is lit; a dot is dark. Rows and columns are counted from the top left.
| c1 | c2 | c3 | |
|---|---|---|---|
| r1 | on | on | on |
| r2 | on | · | on |
| r3 | on | on | · |
Seven lights are on. The minimum is three presses.
- Press r1c1 (a corner). It flips r1c1, r1c2 and r2c1, all of which were on. Now only r1c3, r2c3, r3c1 and r3c2 are lit.
- Press r2c3 (an edge). It flips r1c3 and r2c3 off, turns r2c2 on, and turns r3c3 on. Lit: r2c2, r3c1, r3c2, r3c3 — a T shape pointing up.
- Press r3c2. It flips exactly those four. Dark board, three presses.
Now the surprise: press the same three squares in any other order on the original board and you get the same result.
Strategy
Order doesn’t matter
Each square ends up flipped once for every press that touches it. Only the count of touches matters, never when they happened. So stop thinking in sequences. A solution is a set of squares, and you can press them in whatever order is convenient.
Pressing twice is not pressing
Two presses on one square flip the same five lights twice, returning them to how they were. So in a shortest solution, every square is pressed zero times or once. On 5×5, that means there are only 2²⁵ possible plans, and a solution is a choice of on/off for each square.
Chase the lights down
A method that always works on 5×5:
- Look at row 1. For every lit square there, press the square directly below it in row 2. That clears row 1 without touching it again.
- Do the same for row 2 (pressing in row 3), then row 3, then row 4.
- Now only row 5 can have lights left. Look up its pattern (“on” = lit) and press the listed squares in row 1, then chase again from step 1. The board clears.
| Row 5 left lit (c1–c5) | Press in row 1 |
|---|---|
| on on on · · | c2 |
| · on · on · | c1, c4 |
| on · on on · | c5 |
| on · · · on | c1, c2 |
| · on on · on | c1 |
| on on · on on | c3 |
| · · on on on | c4 |
Chasing is reliable but rarely short. To get close to the minimum, look for presses you’d make twice and cross them out.
Look for shapes
A lone lit plus sign is one press. So is a T of four lit squares against an edge, or an L of three in a corner. Learning the footprint of corner, edge and middle presses lets you read small patches of the board as single moves.
Common mistakes
- Undoing with the same square. Pressing a square to “take it back” is correct — it cancels — but it costs two moves. Use Restart if you want a clean attempt at the minimum.
- Clearing one light at a time. Every press touches up to five squares; aim to flip several lit squares at once, not one.
- Chasing the wrong way. When chasing, press below a lit light, never on it.
The idea underneath
Write each square as 1 if lit and 0 if dark. A press adds 1 to five squares, where addition wraps round: 1 + 1 = 0. That tiny number system, with just 0 and 1, is called GF(2), the field with two elements. In GF(2), adding the same thing twice gives zero — which is exactly “pressing twice is not pressing” — and addition doesn’t care about order — which is exactly “order doesn’t matter”.
So a board is a list of 25 numbers, and choosing which squares to press is another list of 25 numbers, x. Each square’s final state is its starting state plus the presses that touch it. Asking for a dark board gives 25 linear equations in 25 unknowns: for every square, “the presses on me and my neighbours add up to my starting light”. That’s A·x = b, where A is the 25×25 matrix saying which presses touch which squares.
You can solve it with Gaussian elimination, the same row-reduction taught for ordinary equations, using XOR instead of subtraction. That’s what this board does behind the Minimum stat, the hint and the solution overlay.
The 5×5 matrix is not quite invertible: its rank is 23, not 25. Two consequences:
- Only 2²³ of the 2²⁵ possible boards — a quarter — can be solved at all. Random boards are usually impossible, which is why these puzzles are built by pressing from a dark board.
- Every solvable board has exactly 4 press sets that clear it. They differ by the two “quiet” patterns that flip nothing overall. The board computes all four and shows you the one with the fewest presses. The worst 5×5 boards need 15.
The 3×3 matrix is invertible, so every Warm-up board has exactly one solution.
