A co-worker's son came up with this one. I was able to find the answer by a Monte Carlo simulation, but it seems it should be possible to solve it analytically too.
Place three points randomly anywhere on the perimeter of a square. All positions are equally likely, and each point is placed independently. (In particular, there's no assurance that a given point will or won't fall on the same side of the square as another point.) What is the probability that the triangle connecting the three points contains the center of the square?