Puzzles
Two players take turns removing objects from piles. One strategy always wins.
In Nim, players alternate removing any number of objects from a single pile, and whoever takes the last object wins. With several piles at once, is there a reliable way to know whether you're in a winning or losing position?
Reveal the answer
Yes: convert every pile's size to binary and XOR them all together (add each column without carrying, mod 2). If the result is anything other than zero, the position is a win for whoever moves next, and there's always a move that forces the total back to zero. Charles Bouton proved this completely in 1901, founding the mathematical theory of combinatorial games.
— Charles L. Bouton, Nim, A Game with a Complete Mathematical Theory — Annals of Mathematics, 1901