[math-fun] Loose packing problem

How many billiard balls can be packed into a standard billiard rack if no two balls are allowed to touch? (Of course ten will fit if they are allowed to touch.) I can fit six, but I don't see how to prove that seven is impossible. (I can prove that in a smaller rack that could accommodate six touching balls, you can't fit more than four non-touching balls; it's a nice application of the pigeonhole principle.) Is there literature on this flavor of packing problem? Note that a question about packing non-touching disks of radius r can be paraphrased as a question of the form "Is it possible, for all epsilon > 0, to pack disks of radius r-plus-epsilon ..." (where the non-touching condition is dropped). Jim Propp