I just reread this problem on the Harvey Mudd Math Fun Facts website. It’s about your basic intuition in higher dimensions. I remember when I first heard it I was so shocked.
Suppose you have a square in the plane. Cut it into four quadrants, then inscribe a circle in each. Finally add a fifth circle which is just tangent to the other circles.
As you can see, it is much smaller then the other circles and the square.
You can repeat this in higher dimensions. For example, you can cut the cube into octants and inscribe eight spheres in these, then add a last final sphere, tangent to the eight inscribed spheres.
So how does the volume of this final sphere change with respect to the volume of the cube when we change dimensions? How does its size compare? It’s always smaller right? WRONG!
As you increase dimension, the final sphere gets bigger and bigger. In dimension 9 it is tangent to the original n-cube and in dimensions bigger than 9 it pushes outside the original cube. Eventually as the dimension increases, the volume of the final sphere exceeds that of the n-cube!
The details are pretty easy and can be found here:
Su, Francis E., et al. “High-Dimensional Spheres in Cubes.” Mudd Math Fun Facts. http://www.math.hmc.edu/funfacts.
There are all sorts of other cool math fun facts on this webpage. You should check it out.