The 'Sack of Potatoes' Theorem
You've got a countable collection A_1, A_2, A_3, ... of potatoes. A potato is a convex body of diameter at most 1.
(That is: if p, q are points in a potato P, P also contains the entire line segment between them; furthermore, p, q are at a distance at most 1 from each other.)
Furthermore, most of these spuds are incredibly dinky: the total volume of the entire collection is at most some finite value v, say, 4 cubic inches. You'd really like to eat them all.
Trouble is, mashing or cutting them up beforehand would ruin their freshness. You want to bring them all to work whole in your lunchbox.
Prove that these spuds can be packed into a box of finite dimensions. In fact, such dimensions can be computed solely in terms of v, independent of the precise shape of the potatoes.
How to get started on a puzzle like this? One should try to find a sequence of reductions to simplify the problem. So, first, suppose we could do it for rectangular potatoes. Show how to pack a single potato of volume c into a box of volume at most a constant multiple of c, also with bounded diameter. Then, if you can pack these boxes (which have finite total volume), you've packed the potatoes they contain.
I solved the 2-D case, but haven't yet worked on 3-D or n dimensions (where it is also true). I read about this result in a charming volume: The Scottish Book, a collection of research problems (some simple, some still open) compiled by high-caliber, and very imaginative, mathematicians, including Ulam, Banach, and Mazur, who met regularly in the 'Scottish Coffee House', confusingly located in Lwow, Poland, during the '30s. Their book of problems was kept there by the management and notes the prizes--'A bottle of wine', etc.--offered for various solutions. It's definitely worth looking over, both for its diverse contents and for the image of mathematical community it offers.