Yesterday, a plurality of the folks hanging out were math nerds. I'm not a math nerd, but I likes me some recreational math.
Graph Pebbling has some fun little problems. The idea is that you have a graph: a bunch of nodes connected by edges. There are some pebbles on some of the nodes. You can move pebbles around, but this involves attrition: you can move a pebble from node A to node B (assuming they're connected by an edge), but you have to discard another pebble from A.
So what's a fun problem based on this? Suppose you have a cube. Your goal is to end up with at least one pebble on this here corner of the cube. You can ask for some number of pebbles to be distributed amongst the cube's corners, but you can't specify how the pebbles be distributed. What's the smallest number of pebbles such that you can guarantee being able to end up with at least one at that corner?
David Moulton passed along a math joke, a proof that there's an infinite number of composite numbers:
Suppose that there's finitely many composite numbers. Now multiply them all together and don't add one to the product.