Prove the first player in Hex can always force a win — without ever showing how
In the board game Hex, two players alternate placing stones, each trying to link their two opposite sides of a rhombus-shaped board with an unbroken chain. The board can never end in a tie — someone must eventually connect their sides. Given that, can you prove which player, playing perfectly, is guaranteed to win — before a single stone is placed?
Reveal the answer
The first player can always force a win. John Nash proved it around 1949 with a 'strategy-stealing' argument: if the second player had a guaranteed winning strategy, the first player could make an arbitrary opening move, then simply adopt and follow that same strategy (an extra stone never hurts in Hex). That would make the first player a winner too — a contradiction, since only one side can have a winning strategy. The proof guarantees a winning strategy exists on every board size, yet reveals nothing about what it actually is; explicit strategies are known only for small boards.
— John Nash, Strategy-stealing argument — Devised c. 1949, unpublished at the time