Can $n$ hyperspheres in $\mathbb{R}^{n-1}$ be placed so all $2^n$ partitions (in the Venn diagram sense) are realized?

Question

For $n=3$ this would just be a standard Venn diagram, because it would contain 8 different regions corresponding to the various combinations of intersections of sets the circles represent.

Answer 1

Consider the $n-1$ coordinate hyperplanes, together with the unit (hyper-)sphere. Take a point not on any of these $n$ surfaces, and invert with respect to it. All the hyperplanes become spheres and the sphere remains a sphere.