In my point-set topology class we used the diagonal argument as the root of a proof for the following question:

Suppose a submarine is moving in a straight line at a constant speed in the plane such that at each hour the submarine is at a lattice point. Suppose at each hour you can explode one depth charge at a lattice point that will kill the submarine if it is there. You do not know where the submarine is nor do you know where or when it started. Prove that you can explode depth charges in such a way that you will be guaranteed to eventually kill the submarine.

If there's interest, I'll post my solution. My proof actually gives an overview of what order to bomb points in, but I have no idea what it would look like if you plotted out, say, the first 100 or 1000 points. I'd be curious to see someone implement it.