I recently heard this puzzle from Dror Bar-Natan, and there's a nice solution using the fundamental group.

There are $n$ nails arranged in a line on a wall. Find a way of hanging a picture from these nails so that if any 1 nail is removed, then the picture will fall.

To solve it, you can first reformulate it as follows: nails correspond to punctures in the plane, and removing a nail corresponds to filling in a puncture. The fundamental group of such a space is freely generated by loops around each puncture, and filling in a puncture corresponds to quotienting by one generator. We'd like a loop that is killed in each of these quotients, and it's easy to write one down inductively using iterated commutators.