Problems

Alice and Jamie each have an identical “Lights Out” board (same size, same rules). Both boards start with all lights off, and on this board size there are no quiet plans. Alice presses a plan \(A\); Jamie presses a different plan \(B\) (not the same set of buttons). Could they end up with exactly the same final pattern of lights?

On a \(5\times5\) “Lights Out” board, it turns out that there is a simple rule to turn the whole board off regardless of which lights are on at the start:

1. Chase down. Start at the top row. For each light in that row that is turned on, press the button directly below it so that it turns off. Move to the next row and repeat. This turns off rows one by one; only the bottom row may be left with lights being on.

2. Fix the bottom. Find your bottom row in the table and press on the top row the pattern shown to the right. Then chase down again. Repeat until everything is off.

Can you explain why this works?

George is playing on the \(5\times 5\) “Lights Out" board and says that since each light can be on or off, and there are \(25\) lights on the board, the number possible light patterns that can be achieved by playing the game is \(2^{25}\). It turns out that the number is much smaller, it is \(2^{23}\). Can you explain why? You may take it as a fact that these three are the only quiet plans of the \(5\times 5\) board: