The problem would be easy if there are only 4 spheres. You can observe that when 4 unit spheres are placed so that their centres form a regular tetrahedron of side length 1.5, there can be 15 regions. Clearly, it is the maximum possible number of partitions of any 4 arbitrary sets of points. It would be more difficult for the 7-sphere case as the topology confines us from constructing 2^7-1 = 127 partitions._________________

What is mathematics? For those who love it: Marvellous Art That Hides Exquisite Miracles And Treasures In Common Symbols.

For those who hate it: Mazy And Terrible Hoax Evoking Many Agonies Throughout Its Clumsy Statements.