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?

I notice that the only place where spheres must be externally tangent is between D spheres.  D spheres may be internally tangent to spheres A, B, and C.

Since the points of tangency for A, B, and C are distinct, they can't all be internally tanget to each other, but it is possible for A to contain spheres B and C, while B and C are externally tangent to each other.

I believe that the problem is asking us to maximize n.

How about this solution for n=3:  the D spheres coincide with A, B, and C (assuming A, B, and C are externally tangent to one another).  But... I'm not sure whether one sphere is really considered tangent to another if they coincide.

Posted by Tristan on 2005-11-03 18:04:43