Games

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Have you ever played Tic-Tac-Toe? Did you win? Did you know that each Tic-Tac-Toe game will always end in a tie unless one player makes a mistake?

There are nine squares on a Tic-Tac-Toe game board, so the first player has nine choices for the first move. The second player has eight choices for the second move (one square is taken), which makes a total of 72 possible arrangements after the first two turns. After five turns, there are 15,120 possible arrangements.

Someone could possibly win on the sixth turn if the other player played really badly, so the nine possible winning positions must be subtracted from the total, leaving 60,471 arrangements after only six moves. No one can keep track of that many combinations, but that is not necessary. The successful Tic-Tac-Toe strategy is simply to block the other player's moves while hoping the other player makes a mistake.

A good chess player must use a sequence of opening moves that will yield the best possible position. In chess, the player in control of the white pieces (White) always moves first. Since only the eight pawns and two knights can move on the first move, there are twelve possible first moves for White. (The two knights can each move to two different positions).

There are also twelve different opening moves for the player controlling the black pieces (Black), so after only two moves, there are 144 different possible arrangements of pieces on the board. Each move opens up other pieces that can move, so after only four moves, there are about 70,000 different possible arrangements of pieces on the chessboard. Not every arrangement is of equal value, but with 70,000 different positions to consider, it is difficult for even good players to keep track of all possibilities. So good chess players remember patterns of pieces and learn to recognize certain patterns that give them an advantage over their opponents.

A great deal of mathematics is therefore involved in most games. Poker players must calculate the probabilities of certain card arrangements in order to win. (Never draw to an inside straight!) Bridge players must use probability to calculate the possible arrangements of cards in their opponents' hands in order to decide which strategy to use in playing winningly.

Bridge players use mathematics in evaluating their hands. In one popular system, an ace is worth four points, a king is worth three points, a queen is worth two points, and a jack is worth only one point. Players also count distribution points. Having no cards of one suit is worth three points, a singleton is worth two points, and only two cards of a suit is worth one point. Evaluating the hand this way allows a player to determine if a hand is "biddable."

Once play starts, math is used to determine how best to play the cards. For example, in a typical hand, one side may have nine cards of the trump suit, which means the other side has four trump cards. Suppose the missing cards are the queen, 8, 6, and 3. How are those cards likely to be distributed between the opponents' two hands? One opponent could have all four trump cards, one could have three cards, and the other opponent one card, or each opponent might have two cards. The mathematics of probabilities shows that the most likely arrangement is for one opponent to have three of the missing trump cards. So a player cannot depend on capturing the missing queen by simply leading the ace and king.

Probability, Theoretical.

McGervey, John D. Probabilities in Everyday Life. Chicago: Nelson-Hall, 1986.

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