This month I've been looking at the Collatz conjecture. This is the one where you start from a number, then halve it if it is even or multiply by 3 and add 1 if it is odd - it seems that you always end up reaching 1 if you do this, but it is unproven that this will happen for any starting number.
It's a really interesting and frustrating problem - great example of a problem that is easy to state but a bugger to explain. I might post more on it soon, in the meantime, here is the Wiki page, which has some useful info:
http://en.wikipedia.org/wiki/Collatz_conjecture
