This is related to many of the answers already here, but a little different. When I was an undergrad, I got fixated on the following problem:

Given a deck of $n$ cards, how many perfect shuffles does it take to get back to the starting position. Does it matter if this preserves the top and bottom card (what I called an "out shuffle") or mixed them in ("an in shuffle")? This leads to the discrete logarithm problem and other stuff with cyclic groups.