Let a sphere of radius R, shown here in green, we consider two other brown spheres of radius et which are tangent to each other and both internally tangent to the larger green sphere.

What is the maximum radius M and the minimum radius m of a 4th sphere which would be tangent to the three preceding?

Let the radius of , how many solutions to accommodate a fifth sphere tangent to the previous four?

We can continue and thus build a necklace around both brown spheres made of spheres tangents to the first three, and by requiring that each sphere more added in the necklace is tangent to the previous one . Whatever, it's quite surprising that the construction of such a necklace perfectly closes with a last-added sphere, always tangent to . Why this necklace contain always 6 spheres?