Solution
Penney's Game
Show the problem again
In Penney's Game, Player 1 picks a sequence of three coin flip outcomes (e.g. HHT). Player 2, knowing Player 1's choice, picks their own sequence of three. The player whose sequence appears first in an infinite series of fair coin flips wins. For how many of the 8 possible sequences can Player 2 always find a sequence that wins with probability greater than 1/2?
Worked solution
The answer is 8: all of them. No matter what 3-flip sequence Player 1 chooses, Player 2 can always find a sequence that wins with probability strictly greater than 1/2. The game is non-transitive: there is no single best sequence, and Player 2 always has an advantage by choosing second.
Source: Introduced by Walter Penney (1969). Statement written for AxiomIQ.