Bowl of Integers

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to the large sphere on the extremities of one of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then , 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (OEIS A046160). The sizes and positions of the first few rings of spheres are given in the table below.

1		0	0	--
2	2		0	--
3	5
4	6			0
5	9
6	11			0
7	14
8	15
9	18			0
10	21
11	23
12	27			0,
13	30
14	33
15	38			0

Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using the equation of five tangent spheres. Letting gives

For example, , , , , , and so on, giving the sequence , 2, 3, 11, 15, 27, 35, 47, 51, 63, 75, 83, ... (OEIS A046159). The sizes and positions of the first few rings of spheres are given in the table below.

1		0	--
2	2	0	--
3	3		0
4	11
5	15		0
6	27
7	35		0
8	47
9	51
10	63		0
11	75
12	83
13	99		0
14	107
15	111
16	123
17	143		0
18	147
19	155
20	171

The analogous problem of placing two circles of bend 2 inside a circle of bend and then constructing chains of mutually tangent circles was considered by B. L. Galebach and A. R. Wilks. The circle have integral bends given by , 2, 3, 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, 38, ... (OEIS A042944). Of these, the only known numbers congruent to 2, 3, 6, 11 (mod 12) missing from this sequence are 78, 159, 207, 243, 246, 342, ... (OEIS A042945), a sequence which is conjectured to be finite.

Hannachi (pers. comm., Mar. 10, 2006) found a bowl of three spheres with bend 6 and one with bend 7 inside a sphere of bend .