## Math Tutoring Service

See my Mathematics Tutoring Service on Thumbtack

### The Birthday Problem

A well-known problem asks for the smallest number of people (N) who must be in a room before it is more likely than not that two share the same birthday. The answer, surprising to most people who have not heard the problem before, is N = 23.

I thought it would be interesting to modify the problem where we ask for people who share that same day of the month for their birthday. While the answer is not as surprising as the original problem, the computation is much easier. Direct computation for the first problem using factorials will result in overflow on scientific calculators such as the TI-83. Also, the answer to the day-of-month problem (N = 7) is more suitable for empirical testing in small classes. Simply ask each student for their birth day (1 - 31) and record on a large month calendar. For N = 11 the probability of a match increases to almost 88%.

The formula for the probability of one or more matches amongst a group of N people is
Prob = 1 - (31)(30)...(32 - N)/31N
= 1 - 31! /[(31 - N)! * 31N]

### Mathematics and Humor

"Time flies like an arrow. Fruit flies like a banana." -- The Flying Karamazov Brothers.

Have you ever told a joke to someone who "doesn't get it"? If you patiently explain the referents you may get them to "understand" the joke, but they will probably respond something like "So, why is that funny?"

In the simple example above, you probably found this funny if (1) you are familiar with the maxim "Time flies like an arrow", (2) your knowledge of the English language allows you to understand that "flies" can be a verb meaning "passes swiftly" or a plural noun referring to a type of insect and that "like" can mean both "as" and "enjoy" (3) your knowledge of writing style leads you to expect that when the same word appears in two successive short sentences, it will usually have the same meaning in both sentences.

I think we face the same problem when we try to teach mathematical understanding. A proof is most memorable to us when, like in getting a pun, we make a connection between two or more apparently unconnected thoughts, what is often called an "Aha!" moment. Without previous deep knowledge of the constituent thoughts, the student may be able to follow the step-by-step logic, and may be able to remember the proof for tomorrow's test, but the proof will not be memorable, and both the theorem and the proof will soon be forgotten. One implication for pedagogy is that the curriculum must be carefully planned so that, when a mathematical topic is introduced, the students will understand the constituent parts and be able to appreciate their connection. Otherwise, we are mostly wasting our time.

I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.

Then the crossed-chords theorem tells us that (c + a)(c - a) = b2, or c2 - a2 = b2.