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!)

1. sentient spanner says

The important thing about understanding these counter-intuitive problems, for me, is to not misunderstand what is being asked or described. So for the napkin ring problem I initially took the cylinders to be of the same radius until I re-read what you wrote in the first paragraph.

2. kevinkirkpatrick says

Consider how very, very close any small area of the spherical-earth’s surface is to a flat surface. The napkin problem essentially amounts to: press an actual flat surface up against this very-very-nearly flat surface, and trim the gap between them to a height of 2 inches. To me, at least, my intuitive “wow” of how razor-thin such a 2-inch-tall strip of material would be pretty closely offsets my intuitive “wow” of how far around the earth’s circumference is. Conceptually I’m left multiplying some incomprehensibly small number by another incomprehensibly large number, which closely maps to my understanding of how having 2x10^23 of something the size of a water molecule gives me the volume of a small cup of water.

Similar for the rope problem: Consider that you’re charged with adding the 6 feet to this rope. You start by raising one part of it 1 foot up (pinning it in place) You continue applying the 1-foot lift&pin as you walk from the starting point, carrying the 5 feet of slack with you. For a perfectly flat surface, you will always have 5 feet of slack. And, realizing that a band of the earth’s circumference is very, very nearly a flat surface at any point, it’s intuitively clear that as you walk along raising the rope, you’ll only be “consuming” an infinitesimal amount of the remaining 5 feet of slack… The question “how far would I need to walk to consume all 5 feet of slack” is, clearly, “a very long distance”, again going back to my sense that “multiplying a very, very small number by a very, very large number can give a result that’s neither very small nor very large”

3. Rob Grigjanis says

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.

It certainly does, since the maximum thickness of the Earth-sized ring would be roughly the size of an atom or smaller! 😉

4. Arnie says

Mano:

Of course it is not really a paradox because the video explains why this must be so.

A paradox is a condition that appears to be contradictory to facts or impossible, but is real. So yes, it’s a paradox.

It is a paradox only in that it goes against our intuition

No, that’s what makes it a paradox.

5. seachange says

Somehow the language of this implies that the cylinder that goes through the larger object must also have a larger diameter. It’s not at all apparent that this is necessary. I can imagine lots and lots of possibles of orange heighted discs that are cut through the earth, each with a different diameter cylinder chopped out of them. Only one of those discs meets this not-at-all counterintuitive equality. The video supposes without explaining that of course you would pick that one, to which my response is ‘bwuh’?

So I don’t think of this as counterintuitive so much as messing with someone who isn’t fluent in a language. In this case, the language of maths and the non-fluent is me.

6. Holms says

#4
No, it really isn’t paradoxical. A paradox is when two lines of reasoning DO conflict with one another without either of them taken individually seeming to be incorrect. A paradox usually arises when a) one of them is in error, b) or whenever relativity gets involved.

But the puzzle presented above is simply a non-obvious but completely true observation, with none of the elements conflicting with anything except our lazy intuition.