A \(3\times 3\) “Lights Out" board starts with all the lights off. Explain why \(5\) is the smallest number of presses you need to turn the whole board on.
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?
Friday shows Robinson Crusoe a magic trick:
He asks Robinson to write down any 15 whole numbers of his choice on a piece of paper. Then Friday looks at the list, and is always able to pick two of the numbers so that, when one is subtracted from the other, the result is a multiple of \(13\).
Can you explain why this trick works?