Non-transitive probability · 1969
Penney's Game
Play Penney's Game through five levels, a timed ranked mode, a global leaderboard, and a lab with exact Conway leading-numbers probability.
It looks like a coin toss. It is a hustle, and whoever picks second owns it.
The game
Two players each name a run of heads and tails, then flip until one sequence shows up. Penney’s Game is non-transitive: for every three-flip sequence there is another that beats it, so the second player can always counter. Against ABC, play (not B) + A + B. THH beats HHH seven times out of eight; THHH beats HHHH 15 to 1.
Three modes
Campaign has five levels with 100 starting chips: Street Warm-up, The Hustle, The Counter, Four-Bit Master, and Clean Sweep, each with a goal and a lesson. Ranked, the Blind Master mode, is a timed blind-pick run: the clock starts at 8 seconds and shrinks to a 3-second floor, with streak bonuses at 5, 10, and 20 and a global leaderboard.
The Lab runs any matchup with 3 or 4 flips: pick both sequences, set the dealer to the optimal counter, and read the exact win probability computed with Conway’s leading-numbers formula alongside a simulation.
Why it matters
The game is a small, checkable lesson in why intuition about fairness fails: the order of picking is the edge. It sits next to Conway’s Game of Life and the Cellular Automata Lab as the site’s interactive mathematics corner.