Drill a hole straight through the center of a sphere, leaving a band in the shape of a napkin ring. Suppose the height of this band is h. Now take a sphere of a different size and drill a hole through that, contriving the hole’s width so that its depth is again h. Remarkably, the two napkin rings will have the same volume.

As the sphere expands, the band must grow both wider and thinner, and it turns out that these two effects exactly cancel one another. It’s called the napkin ring problem.