I wrote about the Quantum Random Bit Generator a few weeks ago. Their signup page has a CAPTCHA -- you know, one of those things which says "to prove you're a human, read this word that's been distorted and colored funny". Except it's a math problem. And you have to know a bit of calculus to do it. The one it gave me randomly was hard enough that I didn't want to do it in my head; it was something like this one that Brad Fitzgerald mentioned. If you don't want to follow the link, it asked me to find something like
d/dx [ 7 sin 6x + 4 cos (7x + π/2) ]x=2π.
(In my defense, it's three in the morning here.) I'm not sure if it's generating these problems on the fly or if it has some repository of them it's working from; any ideas?
They're reasonable, though; they say "Note: If you do not know the answer to this question,
reload the page and you'll get another question. " Sometimes you get to differentiate something and evaluate the derivative somewhere. Sometimes you get to find the least real zero of a polynomial (of second or third degree, it looks, and the answer's always an integer). And if you refresh enough times you get something like "-4-7+6=?" Mercifully, they don't do the usual graphics tricks to distort the mathematical notation; it's just straight TeX output, I think.
Presumably in order to prove that one is human one must either be able to differentiate, or to at least recognize that one doesn't know how to differentiate. (Knowing what one does not know is often the hardest thing of all.) I suspect that most people who need really good random numbers know calculus, though.