Every impartial two-player game secretly reduces to a single pile of Nim
In games like Nim, Dots and Boxes, or many other 'impartial' games where both players have the same moves available from any position, can you always tell who's going to win, and how, just by analyzing the game's structure before anyone makes a move?
Reveal the answer
Yes. The Sprague-Grundy theorem, proved independently by Roland Sprague in 1935 and Patrick Michael Grundy in 1939, shows that every position in any impartial game is mathematically equivalent to a single pile in Nim of some exact size, called its Grundy number. Once that number is calculated, you instantly know whether the position is a win or a loss and how to play it, turning wildly different-looking games into the same underlying arithmetic.