# 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.