I guess this month I am just going to embrace my mathiness.

For every chapter in the math class I teach, I assign my students a problem set. I don't ever actually look at their homework, except in passing to mark down that they've done it. I figure homework is their chance to try their hand at problems and see what they know, and it's okay if they're making mistakes at that point. So problem sets are where they show me how they are thinking and what they are understanding. I give the students a list of maybe 6 to 8 problems, usually just a little more challenging than the average homework problem, and allow them to choose four. They solve the problems, and then write up both the solutions and their solution processes (what the problem was asking for, how they solved it, why the solution makes sense). These writeups give me insight into their thinking, and also push my students to think about how to explain mathematics. After all, they are going to be teachers someday, and will have to explain mathematics to their own students.

In general, the problems I assign in the problem set are very closely related to the topic we are studying, but every once in awhile, just for fun, I throw in a random puzzle or brain teaser. On the last problem set, I gave them one of my favorites, the Hotel Room Problem. You may have heard this before:

Three friends split a $30 hotel room equally. The manager later realized he had overcharged the friends by $5. He asked his son to deliver the refund to the friends, but the son realized it would be difficult to split $5. Therefore, he gave the friends $3 and pocketed $2. Thus, it cost each of the friends $9 for the room, plus the $2 "tip" that the son took. What happened to the other dollar?

This is a puzzle I've known the solution to for such a long time that it's hard now for me to remember what it's like not to know the answer. But once upon a time it baffled me, and I've seen it baffle other very intelligent people, and I've seen very intelligent people come up with solutions that are completely wrong. I have a memory from when I was a master's student of standing in some grad student's office or other and drawing pictures on the whiteboard to try to show what happened to the missing dollar, and a few weeks later it became one of the very first questions I ever answered as a 100 Hour Board writer.

But I love this problem not because I know the solution, or because I have a couple good memories attached to it, but because I have a great aesthetic appreciation for this puzzle. As far as brain teasers go, I think it is an awfully well put together brain teaser. It's both mathematical and linguistic, and is constructed in such a way as to create a pressing, and completely baffling, problem where there really isn't one at all. On Monday one of my students came to me because she had tried and tried to solve the problem, and had eventually chosen a different one to solve instead, but was dying to know where the extra dollar went. When I saw the light come on in her eyes, I thought, This is such a great problem! and then reflected on what a great problem it was all the way out to my car, and then briefly reflected on the implications of my reflecting on the aesthetic value of a math puzzle.