[math-fun] Sphere puzzle

Define a sequence of unit spheres in 3-space as follows: Let spheres S_1, S_2, S_3, S_4 be mutually tangent. Given S_k with k >= 4, define S_(k+1) as the unique unit sphere that's tangent to S_k, S_(k-1), S_(k-2) but is unequal to S_(k-3). Puzzle: Do all spheres S_k have disjoint interiors? Prove your answer. Please submit answers to me rather than math-fun: dasimov at earthlink.net . I'll post them to math-fun in a couple of days. --Dan