first of all, i don't know what kind of pool you've been playing, but i can fit 15 billiard balls in a standard pool rack. second of all, assuming a somewhat smaller rack that only fits 10 balls, it is easy to fit 8 balls that do not touch. an arrangement can be found here: http://www2.stetson.edu/~efriedma/cirintri/ <http://www2.stetson.edu/~efriedma/cirintri/> third of all, the non-touching condition does not add anything new. the question "how many balls of radius r+e will fit inside a shape of size s?" is equivalent to "how many balls of radius r will fit inside a shape of size s-e?", for slightly different e. fourth of all, i don't know off the top of my head melissen's proof that n=8 is the maximum for your problem, but i can probably look it up if it is important. it likely uses a more sophisticated version of the pigeonhole principle, like most packing proofs do. erich
On Mar 4, 2016, at 7:03 AM, James Propp <jamespropp@gmail.com> wrote:
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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun