Thursday, August 18, 2005


According to this poll, math is the most hated subject in school. It's also, however, tied for the most loved subject in school. I'm in the second category, obviously (otherwise I'd be some kind of masochist, right?). I find it all fascinating. And really, the feeling you get when you work on a hard problem for a while and can't find an answer, and then suddenly it comes to you, in a flash of inspiration...nearly indescribable.

In college, my junior year, we had a problem. Let me give you a little background. There is a sequence of numbers called the triangular numbers. These are pretty easy to understand. The first one is 1, the second one is 1+2=3, the third one is 1+2+3=6, the fourth one is 1+2+3+4=10, and so on. The sequence goes 1,3,6,10,15,21,28,36,45,55,66,78,91... and so on, to infinity. It turns out that there's a very easy formula to find the "nth" number in the sequence of triangular numbers. Say you want the fifth number in the sequence. Then you have: 5*(5+1)/2. That's it. 30/2, or 15. Go look in the sequence, if you don't believe me. In general, to find the nth number, you have: n*(n+1)/2. There's a simple way to derive this formula, but I don't have the notation and you don't have the interest.

Anyway, the problem in college was this: We had to come up with a formula for the tetrahedral numbers. Just like the nth triangular number is the sum of all of the integers up to and including n, the nth tetrahedral number is the sum of all the triangular numbers up to and including the nth one. It's just like putting another layer on the problem.

Well, I struggled and struggled, and I played with things, and nothing seemed to work. I was disappointed in myself, and a little dejected. My class was the next day, and I'd have nothing to show for it. I decided to take a shower. But while I was in the shower, my mind was still working on it, and all of a sudden, out of nowhere (that I could tell), it was there! I had it! I thought on the first couple of examples, and it worked! I finished showering, went to my room, and wrote down: n*(n+1)*(n+2)/6. It's really just an extension of the formula for the triangular numbers. Look at the sequence of triangular numbers again: 1,3,6,10,15,21,28,36,45,55,66,78,91... From that, we can get that the sequence of tetrahedral numbers ought to be: 1,4,10,20,35,56,84,120,165,220,286,364,455... So we can use my formula to see if, say, the fifth one is right. It ought to be 35. So 5*6*7/6=35. It worked!

If you didn't understand any of this, that's OK. I just wanted to explain what it was like to have a real moment joy in mathematics. That moment when the formula came to me was just indescribable, it was so neat. Like for just a second, I'd been allowed to tap into some intelligence much bigger than myself, or something. Or into reservoirs of intelligence that I normally don't get to use. Something. Anyway, it was pretty cool.