The puzzle: You are the chair of a committee that has 8 members (including yourself). You want to hand over the chairship to one of the other members, using the following scheme:
The whole committee sits around a round table. You flip a fair coin. If it comes up heads, you pass it to the person on your right, and if it comes up tails, you hand it to your left. The person who receives the coin repeats the procedure, flipping it and passing it right or left, depending on the outcome of the flip. This process keeps going until all but one members of the committee have had the coin come into their hands. The lone member who has not yet touched the coin is then declared the new chair.
Which person is more likely to become the new chair, using this scheme: the person to your right, the person to your left, or the person sitting directly across from you?
A solution: Although it seems unlikely at first glance, it turns out that each of the three people are equally likely to become the new chair. Moreover, if there are n people sitting around the table, for any n (odd or even) then this process is equally likely to select any of the n-1 canditates as the next chair.
Rather than presenting a solution here, I'll just direct you to a blog post that I have written on the problem, that gives some history as well as a solution.