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?

Here are my thoughts on this problem.  I'm not going to prove anything rigorously, though.  I'm assuming the relative sizes of A, B and C are arbitrary, and that everything lives in Euclidean 3-space. 

I posit that for any three mutually tangent spheres, you can find two spheres tangent to all three that are themselves externally tangent.  Thus I'm willing to assume n is at least 2.  I'm not going to prove that, though.  Can anyone come up with a proof/counterexample?

For the maximum value of n, I'll start with the case where all spheres are solid, and therefore externally tangent.  Suppose A, B, and C are the same size.  To be tangent to all three, the centers of the D's must be colinear, so there cannot be more than 2 that are pairwise-tangent.

Now suppose A and B are the same size, and C is smaller.  In this case, it is possible for the D's to 'wrap all the way around' C to form a mutually tangent ring.  Now, If we consider the limiting case here (picture two planets and a golf ball), we approach two parallel planes with a sphere between.  in this case, we can find 6 D spheres, the same size as C, that form a ring around C.  Of course, this is a limiting case, so it isn't strictly allowed, and i don't know if it's really possible to find a 6-D configuration in any real case, but i posit that for A and B bigger than C, N is somewhere between 2 and 6 inclusive.

Now, what about the case where A is big, and B and C are small?  If B and C are the same size, then the D's can form a loop in the tangent plane between B and C.  I can't say for sure without actually crunching the numbers, but I'm pretty sure i could come up with an arrangement of this type that supports 5 D's (one small D directly between A, B, and C, two slightly larger on either side, and two bigger ones that meet 'behind' B and C).  more than 5, though, doesn't seem possible (sound like a challenge?  by all means, prove me wrong!).

Finally, lets relax the external tangency assumption a bit and assume that A is hollow, and B and C are each half the diameter of A, and fit snugly inside.  In this arrangement, you can fit six D's inside A, each of which has diameter 1/3 that of A, and they form a closed ring.

Therefore, I posit that n is at least 2 and at most 6, depending on the specifics of A, B, and C.

BTW: in the case of a hollow A, if we increase the size of B relative to C, the limiting case is again parallel planes with a sphere in between, which also yields n = 6.  Is this true for all cases in between?  In other words, If B and C are externally tangent and each internally tangent to A, is n always 6? 

Posted by Josh70679 on 2005-11-04 00:34:10