# Inter-Kissing Number for Spheres of Different Sizes

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.]

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See Ian Agol's answer to the earlier MO question "Extensions of the Koebe–Andreev–Thurston theorem to sphere packing?" He shows that $K_6$ is not realizable in $\mathbb{R}^3$, using the same argument that Anton provides for $\mathbb{R}^n$: mathoverflow.net/questions/85547/… – Joseph O'Rourke Sep 1 '12 at 18:35

In $\mathbb R^n$, the answer is $n+2$.