10, 15, fiddlesticks. Let's ask this nice problem for any triangular number T_n of balls and the associated tight exact triangle. This must be the same as asking the question unconditionally for a triangle just epsilon smaller, for small enough epsilon, as Erich has pointed out previously. For n = 2, I think the maximum is 2. ((( Or, a closely related problem without edge effects: Consider a rack in the shape of a 60-120 rhombus with the edges identified to make a 60-120 torus* that just fits n^2 balls nestled against each other just as in a tight triangular rack . . . and ask the same question: How many balls can fit without touching (i.e., in a very slightly smaller toral rack) ? The side of the rhombus would be nD, where D is a ball's diameter. I like the torus shape since for any two points x, y of it, there is an isometry of the whole thing taking x to y. And the 60-120 torus comprises, along with the square torus, the two most symmetrical tori. It is like any flat torus a Lie group, so acts on itself by translation, but also has the dihedral group of order 12 acting as isometries as well. ))) —Dan —————————————————————————————————————————————————————————————————————— * A 60-120 rhombic torus can also take the guise of a regular hexagon, with opposite edges identified in each case. (Puzzle: Show how to dissect the rhombic torus into the hexagonal one.) If the side of the rhombus is L, then the side of the equivalent regular hexagon would be L/sqrt(3).
On Mar 4, 2016, at 4:03 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 15 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).