A strange problem I overheard Bjorn Poonen discussing

I was sitting in a lounge in the MIT math department on Tuesday evening reading and waiting to meet a friend when Bjorn Poonen walked in and began to discuss a strange problem with a student. I hadn’t really realized how strange the problem was until I started thinking about it just now. Here’s how the problem goes:

Start in two dimensions. Suppose that you have a 2×2 square and that you chop it into 4 1×1 squares. Inside each of those 4 squares draw the inscribed circle, and then draw the circle at the center of the square that is tangent to each of those 4 circles. The situation looks like this picture:

 

img_1052

Now extend the idea to 3 dimensions. So, start with a 2x2x2 cube, chop it into 8 cubes, inscribe spheres in each of those cubes, and finally draw the sphere in the middle of the cube that is tangent to each of those 8 spheres. I don’t know how much it helps, but here’s basically half of the picture with our Zometool set and some tennis balls:

img_1053

Here’s the problem:

(A) As you extend this idea into higher and higher dimensions, is there a dimension in which the sphere in the middle is actually larger than each of spheres inscribed in the 1x1x1 . . . x1 n-dimensional boxes?

(B) Is there a dimension in which the sphere in the middle actually extends outside of the 2x2x2 . . . x2 n-dimensional box?

(C) Is there a dimension in which the n-dimensional volume of the center sphere is larger than the 2x2x2 . . . x2 n-dimensional box?

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