Marcus Ranum sent along this amusing video that describes the napkin ring paradox. Basically it says that if you take any two solid spheres, however much they differ in size, and if you then remove a cylinder of material from each sphere, with the cylinder passing symmetrically through the center of the sphere such that the heights of the remaining solids (which look like napkin rings) are the same, then the volume of the two rings are identical.
Of course it is not really a paradox because the video explains why this must be so. It is a paradox only in that it goes against our intuition that a napkin ring made from a marble could have the same amount of material as a napkin ring made from an object the size of the Earth. The key constraint is that the two rings have the same height.
The counter-intuitive nature of this problem is similar to the one where you are asked to tie a string, like a belt, tightly around the equator of the Earth, assumed it to be a smooth sphere. If you now increase the length of the string by six feet, how loose would the belt get? i.e., what would be the space between the surface of the Earth and the belt?
People might guess that it would make hardly any difference because the distance around the equator is about 25,000 miles so adding six feet to that length of string should produce negligible loosening. But the answer is that the string is now about one foot above the ground. It is easier to see this than with the napkin ring.
If the radius of the Earth is R and the circumference is C (which would also be the original length of the string), then 2πR=C, where π=3.14. Now if the length of the string is increased by an amount d and if the new radius of the circle with this larger circumference is (R+h), then 2π(R+h)=C+d. Hence 2πh=d or h=d/2π. In other words, the original radius of the sphere is immaterial when it comes to how far the belt separates from the surface. (Hey, I used algebra!)