Lights Out is parity, not luck

One move, and what it touches

Pressing a cell flips that cell and its four orthogonal neighbours. Diagonals are untouched, and cells on an edge simply have fewer neighbours to flip.

The goal on our board is to turn every light off. Nothing else about the arrangement matters, and there is no move that only affects one cell.

Two facts that shrink the problem

Both of these follow from the fact that a light has exactly two states, and both are worth more than any amount of clicking practice.

Pressing the same cell twice does nothing

Each press flips the same five cells. Flip a light twice and it is back where it started. So a solution never needs to press any cell more than once, and any sequence containing two presses of the same cell can have both removed without changing the outcome.

The order you press in never matters

Each cell's final state depends only on how many times it was flipped, and specifically on whether that number is odd or even. Addition does not care about order, so neither does the board. Press the same set of cells in any sequence you like and you land on the same result.

Chasing the lights

Here is the practical method, and it needs no theory at all to execute.

Work down the board one row at a time. If a light in the row above you is on, press the cell directly beneath it. That press turns off the light above and disturbs only the current row and the one below — rows you have not finished with anyway.

Repeat for each row in turn. By the time you have chased past the last row, rows one to four are all dark and everything left is in the bottom row. You have turned a twenty-five cell puzzle into a five cell one, using no cleverness whatsoever.

The bottom row decides everything

If the bottom row is already dark, you are finished. Usually it is not, and this is the only point in the puzzle that requires knowledge rather than procedure.

The fix is a second pass: press a specific set of cells in the top row based on what the bottom row looks like, then chase all the way down again. The second chase clears the whole board.

For the five by five board the correspondence is short enough to work out or to look up. Reading the bottom row left to right, with 1 for lit:

• 1 0 0 0 1 → press the first cell of the top row, then chase.
• 0 1 0 0 1 → press the second cell of the top row, then chase.
• 1 1 1 0 0 → press the third cell of the top row, then chase.
• 1 0 1 1 0 → press the fourth, then chase.
• 0 0 1 1 1 → press the fifth, then chase.

Any other bottom row is a combination of these: press the top-row cells for each pattern that combines to give what you see, cancelling any cell that appears twice. That cancelling is the same rule from earlier — press it twice, it never happened.

If that feels like more bookkeeping than you came for, the board has a hint button that will show you a full solution set. It marks the round as assisted rather than clean, which is only there so your own record stays meaningful.

Why every board we deal can be solved

Not every arrangement of lights on a five by five grid is solvable. Ask for a random board and a good fraction of them have no answer at all, which would be a miserable thing to hand somebody.

So we do not generate boards by randomising the lights. We start from a dark board and press a random set of cells. Whatever comes out is reachable by definition — pressing that same set again returns it to dark, since every press cancels itself.

The generator then solves the board it produced, using the parity arithmetic described above, to confirm there is an answer and to know the minimum number of presses. That number is what your click count is compared against when you finish. Beating it is not possible; matching it means you found the shortest set.

The three board sizes behave differently, and that is genuine rather than cosmetic. On some sizes every arrangement is solvable and the solution is unique; on others, including five by five, certain boards have several different minimal solutions. Finding a shorter one than last time on the same layout is a real result.