Math Problems of the Month

2007-2008

The department of Mathematics and Computer Science is starting a program of posing monthly recreational problems for students and faculty. There will be cool prizes (to be announced - gift cards, pizza, burritos!) for the first solution by a student, and for the first solution by a member of the faculty. If the first solution by a student is submitted before the first solution by a faculty member, the student will get both prizes. All problems will remain open until the first solutions are received in each category. If you have any questions about a problem, or the contest guidelines, contact Alison Marr, Fumiko Futumura, or Rick Denman.

Problem of Month #2 (February – basketball season!)

At the end of the third week of basketball season, point guard Betty Sharpshooter has a free-throw shooting average below 80%. At the beginning of the seventh week of the season, her free-throw shooting average has increased to above 80%. Prove or disprove, that at some time during the intervening weeks, she must have had a free-throw shooting average of exactly 80%.

Now, what property of the percentile 80% is the key to the counterexample or proof you gave for the claim above?  That is, for which fractions strictly between 0 and 1 is there a proof of the claim above, and for which of these fractions is there a counterexample?

Problem of Month #1 (October)

After a vigorous night of trick or treating, little Billy Sweettooth wants to eat his entire hoard of candy in one sitting. In order to preserve his teeth and internal organs, his loving parents Emmy and Carl (who happen to be mathematicians), offer him the following compromise. Each time little Billy wants to eat a piece of candy, he must flip a fair coin 10 times, and if the longest run of heads that he records (under supervision) is at most 1, then he can have a piece of candy.

In each attempt, what is the probability that little Billy will win a piece of candy?

How would his odds change if the rules were changed to require the longest run of heads to be at most 2?

How would his odds change if the rules were changed to require the longest run of heads to be at most 3?

Use your answers to conjecture a general recursive formula (or recurrence relation) for counting the number of possible outcomes in which the longest run of heads has length at most k, when flipping a fair coin n times..