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THE MORE THAT I PLAY THE WINDOWS PROGRAM FreeCell, the more opportunities that I see for using it to teach mathematics. (An explanation of the game of FreeCell appears on page 407.) Since many students are acquainted with FreeCell and enjoy computer games, teachers can sneak in some fairly sophisticated mathematics without great pain to the teacher or the student.
PROBABILITY
One of the first questions that occurs to students who play FreeCell is whether the degree of difficulty increases with the number of the game. This question leads naturally to a discussion of how such subjective concepts as difficulty can be measured. Students will give various suggestions on measuring the difficulty of a game. Simple measures, such as the number of aces at the top of the stacks or the closeness of the ratio of top-black to top-red cards to one-half, soon prove by experience to be inaccurate measures.
Since FreeCell includes a cumulative counter of games played and percent won, students can take random samples of games and compare the win percents. The decision about the random-selection process to choose (random-digit table, a computer statistics package, rolling dice, and so on) affords an opportunity to discuss with students how closely their random-selection process approximates a desired simple random sample (SRS).
A student might randomly select five numbers between 1 and 32 (between 1 and 1,000 for newer versions of FreeCell that have 1,000,000 possible games). If, for example, the numbers selected are 3, 10, 14, 22, and 29, the student randomly selects twenty games numbered between 2,001 and 3,000; twenty games between 9,001 and 10,000; twenty games between 13,001 and 14,000; twenty games between 21,001 and 22,000; and twenty games between 28,001 and 29,000. The student then uses the Select Game option to play those games. If the student's win percents for each of the five sets of twenty games are approximately equal, we can conclude that the games are probably close in difficulty level. The next question is, How close is close? That question is a good introduction to hypothesis testing.
Different individuals, of course, should expect different win percents. However, I have been fascinated by how rapidly my win percent approaches a constant, 75 percent. As I play thousands of...