Here is a nice math problem I encountered recently.
N painters arrive at random positions around a circular fence. Each paints the section of fence between them self and the nearest neighboring painter. As a result of this curious procedure some sections of fence are painted twice and some are not painted at all. In the limit as N goes to infinity what fraction of the fence on average will be painted (at least once)?
Actually as originally posed the fence was not circular which doesn't change the limit but is not as easy to deal with.
You can also ask the same question for the case where each painter paints the section of fence between them self and the farther of the two adjacent painters.