Penney's Game

A non-transitive coin sequence game — no strategy is unbeatable

Choose Your Sequence (Player A)

Pick a sequence of 3 coin flips. The second player (computer) will then pick a sequence that beats yours.

?
?
?

Computer's Counter-sequence (Player B)

The optimal counter-sequence (Conway's algorithm) and win probability:

?
?
?
Player B wins
50%
Player A wins
50%

Simulation

A wins0
B wins0
B win rate—

Non-Transitivity

Like rock-paper-scissors: for every sequence, there exists a sequence that beats it. No sequence is "best"!

Conway's rule: if A = a₁a₂a₃
then B = (NOT a₂) a₁ a₂
B beats A with odds roughly 2:1

This means a clever second player can always have an advantage — a genuine mathematical paradox.

All Win Probabilities

Row=A's sequence, Col=B's sequence. Cell shows P(B wins).