Here is a question that came across my desk. You may find it amusing.
We have a single long, flexible protein molecule with some number (N) of cysteines.
Any cysteine can pair with any other to form a disulfide bond. There are no
steric constraints.
We are interested in arrangements where the maximum number of such bonds
are formed, namely floor(N/2) bonds. We are not interested in the exact
shape, or even the toplogy; we are just interested in which cysteine is
bonded to which.
How many different arrangements are possible?
Remark: This is not a made-up puzzle; this question arose in the course of
serious research.
Remark: This question has a goodly amount of the "aha" property. That is, if
you approach it the right way, you can solve it in less time than it takes to
talk about it, using techniques that any high-school senior should be able to
follow. OTOH, if you don't approach it the right way, things could get ugly.
Remark: This is basically just a counting question, but it is the sort of counting
that crops up all the time in real-world physics situations, so it seems fair game
for this list.
Remark: The answer can be expressed in a simple, elegant form.
Hint: You may wish to start with the case where N is even, and deal with odd N
later.