Michael emailed me a link to this article, which was also noted by Teresa, about a breakthrough in understanding an obscure work by Archimedes in the field of combinatorics. Both the NYT article and Teresa's observations are interesting, though I have to wonder why reporter Gina Kolata never mentioned that the science of combinatorics, being as it is a part of probability theory, was mostly advanced early on by gamblers. I always thought that was standard-issue info.
Brad DeLong has his One Hundred Interesting Math Calculations going, which you can contribute to if you want. Andrew Northrup suggests Cantor's diagonalization proof of the density of the real line, which is a lot more intuitive than it sounds, as something cool even if it's not, strictly speaking, a calculation. I second the motion, and note that there's a new book out about Cantor's life and work. I read and enjoyed a different book on the subject, and I'd recommend anyone who's remotely curious about this sort of thing to check it out.
Getting back to Brad's quest for more math problems to
tortureenlighten his children with, the original article on Archimedes mentioned Persi Diaconis, who's known for sniffing out such things. He's the one who determined that one must shuffle a deck of cards seven times in order to achieve true randomness. As a bridge player, I can tell you that I can do that in my sleep now. Pretty much any time you see Diaconis' name in the paper, you've stumbled across an interesting problem.