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?

The statement of the problem suggests that n is not dependent on the relative sizes of the spheres A,B, and C.  If this is true then we could assume, at least at first, that A, B, and C are equal.  Then orient the 3 spheres so that their centers are in the horizontal plane.  I'm having  trouble picturing the orientation of the D spheres.  Apparently, the D's are like a string of n pearls, where the pearls can be of varying sizes.  If the pearls encircled spheres A, B, and C horizontally, then no pearl would be touching all 3 of the ABC's.  So the center of the pearls (the D spheres) must be on the vertical axis.  A linear string of pearls might work.  If n is odd, the middle pearl (n+1)/2 could be on the same plane as the ABC spheres, then a larger sphere above and below, and so on.  Except that in that case, spheres D1 and Dn wouldn't be tangent to each other.  Unless somehow spheres D1 and Dn have infinite radius and could be somehow considered to be tangent at opposite ends of the universe.  

I can picture 2 pearls, D1 and D2, tangent to each other and each tangent to the ABC spheres.  But I can't picture any more D spheres.   So my answer so far, is n=2.

Posted by Larry on 2005-11-03 17:50:57