Pick a sequence of three coin flips — your opponent can always beat you
Two players each choose a sequence of three coin-flip outcomes, like Heads-Heads-Tails. A coin is flipped repeatedly until one player's sequence appears, and that player wins. Counter-intuitively, whoever picks second can always choose a sequence more likely to appear first, no matter what the first player picked. How?
Reveal the answer
The second player copies the first two symbols of the first player's sequence, flips the first one, and adds it to the front. For example, if Player 1 picks Heads-Heads-Tails, Player 2 picks Tails-Heads-Heads — which beats it about 3-to-1. It works because the game is non-transitive: no single sequence beats every other, so there's always a better reply. Invented by statistician Walter Penney in 1969.
— Walter Penney, Problem 95: Penney-Ante — Journal of Recreational Mathematics, 1969