How to prove that $5$ unit hypercubes cannot be positioned to cover a unit hypersphere?
The unit hypersphere has hypervolume $\frac{\pi^2}{2} \approx 4.93 \lt 5$ but it seems unlikely that it is possible to waste so little space. Is there any easy way to see that it is impossible? How many hypercubes are required, and can their configuration be described in a way that is easy to "visualize"?