At 01:04 PM 9/26/2003, you wrote:

To see why, let's use the usual reduction of the problem to packing spherical caps into a sphere, and let's change the size of these caps to that at which there's room for just 8 mutually tangent ones, centered at the vertices of a cube. Then after starting with two touching ones of these, your method would place a third so as to touch both of them, which would prevent it from finding the correct configuration.

Thank you, that's a very clear counter-argument. My assumption was that you need to pack them as tightly as you can as you go, but that's not true. Thanks to you and the others for the insight and discussions.