Problem 1. Let several non-overlapping spheres in $\mathbb{R^3}$ are given. For which $n$ it is possible that each sphere is tangent to exactly $n$ other spheres?
Consider the smallest sphere. Since the kissing number of $\mathbb{R^3}$ is 12, then $n \le 12$. Using stereographic projection of $120$-cell we obtain example of $120$ spheres with $n=12$. If you delete some spheres from this configuration, it is possible to construct examples for each $n$ except for $n=11$ (See Deleting vertices of a regular graph to obtain a regular graph , comment of @RobPratt). Does an example exist for $n=11$?
Problem 2. Let several non-overlapping unit spheres in $\mathbb{R^3}$ are given. For which $n$ it is possible that each sphere is tangent to exactly $n$ other spheres?
Considering lowest sphere we obtain $n\le 8$. Using regular polyhedra it is easy to construct examples for $n\le 5$. Is it possible to construct examples for $n=6,7,8$?
Remark. It is also interesting to consider these problems in greater dimensions. Here Chromatic number of graphs of tangent closed balls there is a discussion on chromatic number of arbitrary graphs of tangent spheres.
