puzzle: Sphere in a cube

[A_Square]

Here's a puzzle

If you have a cube of side 1 and put a sphere of radius 1 centered at the corner in 2D the center of the cube (square) is inside the sphere (circle) and in 3D it is, but what about 4D and higher?

[Keiji]

I can't say anything exact, but there will be some dimension for which the center of the hypercube is no longer inside the hypersphere.

[wendy]

A sphere in 4d of diameter 2 would exactly emclose a 4d cube (tesseract), but higher dimensions, the dimanete of a unit cube is larger than the sphere. In some higher dimension (i think, 19), the volume of the unit-edge cube exceeds that of a sphere of diameter 2.

[Tamfang]

Easy. The square of the distance between the center of the sphere and the center of the cube is n times (1/2) squared. So in 4space the cube center is on the surface, and in higher dimensions it's outside.