A game where the first player can always win, but nobody knows the winning move
In Chomp, two players take turns 'eating' a bite out of a grid of squares, like a chocolate bar: each bite removes the chosen square plus everything below and to the right of it. Whoever is forced to eat the single poisoned square in the corner loses. Can the player who moves first always force a win?
Reveal the answer
Yes, surprisingly, but the proof doesn't tell you how. Using a 'strategy-stealing' argument, mathematicians showed the first player always has a winning strategy for any grid larger than 1x1, without ever identifying what that opening move actually is in general. The game was devised by Fred Schuh in the 1950s and independently reinvented by David Gale in 1974; Martin Gardner gave it the name 'Chomp.'