The hypercube is the strange thing, not the red sphere. Placing the blue spheres tangent to the hypercube is an artificial construct which only “bounds” the red sphere in small dimensions. Our intuition is wrong because we think of the problem the wrong way (“the red sphere must be bounded by the box”, but there is no geometrical argument for that in n dimensions).

How does the enclosed sphere's radius changes with the number of dimensions, if the enclosing spheres are the following:

2D: 3 mutually touching 2-spheres (circles)

3D: 4 mutually touching 3-spheres (or spheres)

...

This variation of the problem doesn't rely on an artificial construct of a hypercube, I wonder if this yields a similarly unintuitive result.

If my calculations are correct, then for this variation the enclosed n-sphere's radius converges to sqrt(2)-1 from below, and remains enclosed in the bounding hyper-tetrahedron.

Very interesting, I've considered doing something similar with other regular polyhedra, like the n-simplex (the one you analyzed) and n-orthoplex.

What was the side length in your calculation? did you find an equation for the size of the center n-ball?

Buf, you may be right but I just cannot visualize it. It took me quite a while to do for the cube, imagine a tetrahedron. But you might be right.