If Pascal had a Computer -
A Do-it-yourself guide to the problem of the points and the arithmetical triangle

By Thadd Selden and Bruce Torrence
Faculty Advisor : Dr. Bruce Torrence

Working in the fall of 1998 and spring of 1999, Thadd began exploring one of the oldest problems in the history of probability, now known universally as the "problem of the points". The origins of the problem come from gambling: imagine a game of chance played in rounds, where in each round a coin is tossed. A point is scored in a round if the outcome of the coin toss is favorable (say heads for player A, and tails for player B). The first player to score a predetermined number of points wins. This is essentially what happens in the World Series in baseball, where each "round" is a game rather than a coin toss, and the first team to win 4 games is the winner. The problem of the points is as follows: If each player has contributed an equal amount of money toward the stakes, and if the series of games is interrupted at a point where player A is m points shy of winning, and player B is n points shy of winning, how can the stakes be fairly divided? The problem was solved independently by Blaise Pascal and Pierre de Fermat in the year 1654. Thadd's work shows how a computer algebra system can be used to generalize the two distinct approaches used by Pascal and Fermat, and to suggest a closed form for the two solutions. The work is available in its entirety in the November 2001 issue of Math Horizons. Thadd has graduated, but is still active on many fronts. He maintains a wonderful web site at http://www.thaddeus.net.