Generalizations and Extensions

(Berkeley Math Tournament 2014, Geometry Focus #5) In a $100$-dimensional hypercube, each edge has length $1$. The box contains $2^{100} +1$ hyperspheres with the same radius $r$. The center of one hypersphere is the center of the hypercube, and it touches all the other spheres. Each of the other hyperspheres is tangent to $100$ faces of the hypercube. Thus, the hyperspheres are tightly packed in the hypercube. Find $r$.