April 23, 2010

Math for Love just made the list of 50 best blogs for math majors!

In honor of that, I’m going to point to one of the other blogs listed there, which I expected not to like, but actually found pretty fun: MAA MinuteMath.

When I see the words “minute” and “math” together, a shadow reaches into my internal landscape, as I’m made aware yet again of the cloud, the blight, that is the misguided “skill building” effort. Not that there’s anything wrong with skill building; there’s nothing wrong with locusts either—it’s just that when you see them, it’s usually the sign that there’s too many of them coming, and there won’t be a lot of other sustenance around for a while.

But the MinuteMath blog is another thing entirely: a collection of the kind of “cute,” harmless, occasionally maddeningly hard puzzles that, if you’re in the right kind of mood, give way to satisfying answers very quickly. It’s the mathematician’s equivalent of the Tuesday crossword puzzle in the Times. (These problems actually come from the AMC math contests for 8th-12th graders).

Here’s an example: What’s the probability that a randomly picked factor of 60 is less than 7?

You could do this by writing down all the factors of 60, and see how many are less than 7. But that’s boring and slow. Or, you might imagine that if you have a*b=60, then one of the numbers (say a) has to be less than the other, and in particular, it has to be less than the square root of 60 (or else a*b would be bigger than 60). And the square root of 60 is somewhere between 7 and 8 (since 7*7=49 and 8*8=64), so half of every pair is 7 or less. But 7 isn’t a factor of 60, so exactly half of its factors must be less than 7. Done!

Of course, it happens much more smoothly and satisfyingly in your own head. The writing is a little clumsy, and is mostly meant to point the right direction for you to have the idea. Still, I hope it follows, for those interested in thinking about it.

I’m not sure why puzzles are so attractive. In Ancient Puzzles, Olivastro refers to them as the detritus of intellectual exercise. Somehow, they’re the unimportant but amusing leftovers of discovery, but we get to relive the sense of discovery through them. And they seem unimportant. Even though no one will conceivably use my work on K3 surfaces in any practical way, it feels like I’m involved in an important pursuit, at least by the standard of mathematical endeavor. On the other hand, I avoided combinatorics and graph theory, fields where I’m arguably more intuitive, because they didn’t feel as “important.”

(Though probabilistic graph theory turns out to be one of the most applicable, important, and hot areas of math going right now. And arguably, I only became an algebraic geometer because my advisor played basketball with me and asked me to work with him. So perhaps I’m wrong in every respect.)

In any case, the appeal of puzzles is undeniable. If you’re a math major (my new audience), or someone with a taste for tricky, little puzzles that don’t go anywhere in particular but take some insight to solve (which is exactly what I’m in the mood for at this hour), check out the MathMinute blog.

If you feel like more accessible problems that occasionally point in interesting directions, you might take a look at Saint Ann’s problems of the week. They’re great for kids and adults, and some are great lesson builders: beginning problems leading to deeper principles and questions. Consider this problem for example. If you ask yourself one additional question—what is area of each square—and stick with it for a while, you’ll almost have to end up with the Pythagorean theorem, or at the very least it’s motivation. There’s a great, deep idea embedded in that one little problem.

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