What is the maximum number of spheres that can be placed in 3D such that all inter-touch?


One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the
middle, so this number must be at least 5.


[By the way, I was trying to extend the "five points in 2D cannot be inter-connected without
a crossing" limitation to 3D with a simple statement, but this was sadly the best I could
do. If anyone knows a better simple extension, please comment.]



Inter-Kissing Number for Spheres of Different Sizes
In $\mathbb R^n$, the answer is $n+2$.


You can apply an inversion which sends two of the spheres in to two parallel hyperplanes.
The rest of the spheres will have the same radii and their centers lie in a hyperplane.
Hence everything follows.


