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Penney’s Game

How to play

  1. 1

    Each player chooses a pattern of three coin tosses. H = heads. T = tails.

  2. 2

    You can choose simultaneously, then reveal your patterns. Use different patterns for the race.

  3. 3

    Toss a fair coin repeatedly. After each toss, check the latest three results. The first chosen pattern to appear wins.

Each pattern has a 1-in-8 chance in a single block of three tosses. In a continuing stream, however, some patterns are more likely to appear before others.

Winning probabilities

Each arrow points from a pattern to the one it beats. Its percentage is the chance of winning that pairwise race with a fair coin.

If the other player’s pattern is known, choose a pattern whose arrow points to it. For example, HTT → TTH means HTT wins against TTH with probability 75%.

Penney’s Game: four-pattern cycle with four outer connectionsThe central cycle: THH beats HHT with 75 percent probability; HHT beats HTT with probability two thirds; HTT beats TTH with 75 percent probability; TTH beats THH with probability two thirds. The four outer connections: THH beats HHH with 87.5 percent probability; HHT beats HTH with probability two thirds; HTT beats TTT with 87.5 percent probability; TTH beats THT with probability two thirds. Every arrow points from the favoured pattern to its opponent.THH beats HHT: 3/475%HHT beats HTT: 2/367%HTT beats TTH: 3/475%TTH beats THH: 2/367%THH beats HHH: 7/887.5%HHT beats HTH: 2/367%HTT beats TTT: 7/887.5%TTH beats THT: 2/367%HHHTHHHHTHTHHTTTTTTTHTHT
67% is rounded from 2/3. The other values are 75% = 3/4 and 87.5% = 7/8. Each percentage describes a separate matchup.

There is no single best pattern. After every toss, we check the latest three results, so consecutive windows overlap. How patterns begin and end changes which one is likely to arrive first. The cycle is non-transitive, like rock, paper, scissors.

All eight comparisons as a table
Fair-coin race, stopping when either selected pattern appears
Favoured patternOpponentChance of winning
THHHHT3/4 (75%)
HHTHTT2/3 (67%)
HTTTTH3/4 (75%)
TTHTHH2/3 (67%)
THHHHH7/8 (87.5%)
HHTHTH2/3 (67%)
HTTTTT7/8 (87.5%)
TTHTHT2/3 (67%)

Numberphile strategy (4:13) ↗ · Probability formula and strategy ↗

How do I choose a stronger response?

If the other player’s pattern is known, take the opposite of its middle symbol, then add its first two symbols. Against TTH, flip the middle T to H, then add TT: choose HTT.

This gives a better-than-even chance of winning, not a guaranteed win. With simultaneous choices, you cannot tailor your response to a pattern you have not seen. Watch the strategy at 4:13 ↗

Run the experiment

Play five rounds of 100 flips. Each round compares the first appearance of your two patterns. The player with more round wins takes the experiment.

Choose your patterns and run the experiment to see the results.

Positions count from 1. A pattern that is not found loses to one that appears; two missing patterns draw. Identical patterns draw.