Penney's game (named after its inventor Walter Penney) is a binary (head/tail) sequence
generating game between two players. At the start of the game, the two players agree on the
length of the sequences to be generated. This length is usually taken to be three, but can be
any larger number. Player A then selects a sequence of heads and tails of the required length,
and shows this sequence to player B. Player B then selects another sequence of heads and tails
of the same length. Subsequently, a fair coin is tossed until either player A's or player B's
sequence appears as a subsequence of the coin toss outcomes. The player whose sequence
appears first wins.
The interesting feature of Penney's game is that - provided sequences of at least length three
are used - the second player (B) has an edge over the starting player (A). This is because the
game is nontransitive such that for any given sequence of length three or longer one can find
another sequence that has higher likelihood of occurring first.
For the three-bit sequence game, the second player can optimise her odds by chosing
sequences according to:
1st player's choice 2nd player's choice Odds in favour of 2nd player
HHH
THH
7 to 1
HHT
THH
3 to 1
HTH
HHT
2 to 1
HTT
HHT
2 to 1
THH
TTH
2 to 1
THT
TTH
2 to 1
TTH
HTT
3 to 1
TTT
HTT
7 to 1
Notice that in each case the second player's optimal choice is to precede the first player's
sequence with the reciprocal of the second symbol, and to eliminate the last symbol.
[edit] See also

Nontransitive game
[edit] External links

Play Penney's game against the computer
[edit] References



Walter Penney, Journal of Recreational Mathematics, October 1969, p. 241.
Martin Gardner, "Time Travel and Other Mathematical Bewilderments", W. H.
Freeman, 1988.
L.J. Guibas and A.M. Odlyzko, "String Overlaps, Pattern Matching, and Nontransitive
Games", Journal of Combinatorial Theory 30 (1981), pp183-208.