Let A, B, and C be spheres that are tangent pairwise and whose points of tangency are distinct. Let {D₁, D₂, ..., D_n} be a set of spheres each of which is tangent to spheres A, B, and C. For i = 1 to n, D_i is externally tangent to D_i+1 (where D_n+1 = D₁).

What is the value of n?

If two of the spheres A,B,C are internal to the third the situation is as:

http://mathworld.wolfram.com/Hexlet.html

And the solution is n=6

If A,B,C are are externally tangent, I think the solution can also be 6 counting a sphere at infinity

Related is the Steiner Chain:

http://mathworld.wolfram.com/SteinerChain.html

but the number of circles is not limited.

Posted by Jer on 2005-11-04 12:47:40