ABSTRACT
B ritish mathematician Steve Humble and Japanese math-ematician Yutaka Nishiyama came up with an exciting cardgame. You play against an opponent using an ordinary deck of 52 cards consisting of 26 black (B) cards and 26 red (R) cards, thoroughly shuffled. Before play starts, each player chooses a three-
card-code sequence of red and black. For example, your opponent chooses BBR and you choose RBR. The cards are laid on the table, face up, one at a time. Each time that one of the two chosen sequences of red and black appears, the player of this sequence gets one point. The cards that were laid down are removed and the game continues with the remaining cards. The player who collects the most points is the winner, with a tie being declared if both players have the same number of points. Your opponent is first to choose a sequence. The 64,000 dollar question is this: how can you choose, in response to the sequence chosen by your opponent, in such a way as to give you the maximum probability of winning the game? The counter-strategy is simple and renders you a surprisingly high win probability. But let’s defer that revelation momentarily to take a look at a precursor to that game: the Penney Ante coin toss game introduced in 1969 by Walter Penney in a magazine for recreational mathematics. In this game, a fair coin is tossed for which the probability of an outcome of heads (H) and the probability of an outcome of tails (T ) is equal to 12 . The game is played by two players, 1 and 2, each of whom must choose, beforehand, a sequence of H’s and T ’s of length three. The coin is then repeatedly tossed until one of the two chosen sequences appears for the first time. And that is the end of the game. The winner is the player whose sequence appears first. Let’s say that player 1 chooses a sequence first, and shows it to player 2. Player 2 will always have an edge, because depending on which sequence player 1 chooses, a simple method of choosing an appropriate counter sequence will give player 2 a higher win probability. Player 1 chooses from among these combinations:
HHH, HHT , HTH, HTT , TTT , TTH, THT , THH,
which leaves player 2 to parry with the following countermoves
THH, THH, HHT , HHT , HTT , HTT , TTH, TTH.
