This is a very famous one. 100 ants are on a stick 1m long, and they start moving (each in arbitrarily chosen, but single direction), at a speed of 1m/hour. When two ants bump into each other, each reverses it's direction.
(Picture courtesy of yaniv.leviathanonline.com)
The question is, what is the shortest time after which all ants areguaranteed to fall down?