Napkin Ring Versus Spherical Cap

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2017/09/28 - The Napkin Ring and the Spherical Cap

Many years ago I was at a semi-social gathering and a somewhat odd incident occurred. Over the course of conversation it emerged that I was studying maths, and one chap, at one point, turned to me and said: "So, do you like puzzles, then?"

Not especially on my guard I said yes, and he went on to say something like this:

  • OK, I'll give you a puzzle. The answer is a simple rational multiple of pi.
  • A chap takes a wooden sphere and drills a circular hole through the centre, and the hole turns out to be 6 inches long.
  • What volume of wood does he have left?

Many of you will recognise this as the "The Napkin Ring Problem." Much has been written about this rather magical and, to be frank, slightly wondrous result, and if you haven't met the problem then I suggest you take some time to see if you can solve it.

I'll give the solution and explain the particular technique in a follow-up post.
As he was posing the problem, the gentleman in question was pointedly removing his watch, which he then held in front of him, clearly timing me. Well, as luck would have it, the problem readily succumbs to a classic problem solving technique I'd known and practised for years, so the answer came quickly.

Very quickly.

When I gave my answer the chap in question went "Hmm," put his watch back on, stood up, finished his drink, took his jacket from the back of his chair, and left. Which I thought a little odd. I looked around and everyone else was having quiet hysterics. Colour me confused.

After a while someone took pity on me and explained. They had also timed me, and apparently I'd answered in 27 seconds. My interlocutor (love that word - shouldn't use it) had held the record at 82 seconds and had been very proud of it.


Never saw him again, so he was obviously a bit upset. But as I say, the problem falls to a standard technique that (a) I know, and (b) was lucky enough to be the first thing I tried. Still, as Tim Gowers has said, one of the tricks to appear cleverer than you really are is to be lucky and pretend you meant it all along.

Many years later I used it in a talk and mentioned that after the occasion in question I'd gone home and done the calculations properly to verify the rather lovely trick. It's a simple enough volume of revolution calculation, but I remarked that it would be nice to have a more elementary solution. After my talk someone came up to me and said "But there is an Archimedean proof of this!" And there is.

Also in the follow-up I'll give the classic answer, and the Archimedean argument to show that it works.
So your first challenge is to answer the problem, and your second challenge is to find the Archimedean argument to verify that it's true. If all this sounds a little mysterious it's because I don't want to spoil your fun if you haven't already seen all this.

A useful exercise for the (mythical) interested reader.
One of the slightly unexpected benefits of this result is that we can quickly write down the volume of a spherical cap without having to do any lengthy calculation, which is useful if you happen to want that.

But now comes the second part of the story.

Upon re-reading this it occurs to me that not everyone will know who Conway is, so here are some links:
At the same MOVES conference where I'd spent some time with Peter Winkler I also ran into John Conway, and he mentioned a different puzzle, and a meta-puzzle. I'll set you the puzzle here.

  • Consider the Earth, and imagine a straight line, a tunnel, from the North Pole to some other point on the Earth's surface.
  • Where the tunnel emerges defines a particular latitude, and that circle of latitude therefore defines a spherical cap.
  • If the tunnel is of length $l$, show that the area of the spherical cap is $\pi{l^2}$.

Well, that's not really a puzzle - it would be interesting to recast that in the form of a puzzle, but that's not what I'm looking to do here, and it's not what Conway was asking. No, what Conway said (and I paraphrase) was something like this:

  • We can all do the calculations, that's not the problem, it's not the question.
  • The question is:

Are these actually
the same problem?

So this is the meta-puzzle, the meta-question. These two puzzles, these two calculations, to some extent appear to have several similarities. They can both be solved with Archimedean-type calculations, they both have an unexpected independence from an apparently essential quantity, they both involved the sphere, with the cap playing a role.

It just feels like they could be connected ...

But I can't see it. Really, I can't. Coming from most people I would stop there and say that they are, in fact, just not connected. But it was Conway who asked the question, and his intuition can't be discarded lightly. As a result I am loathe to dismiss the question too quickly.

So let me throw it open to you:

Are these actually
the same problem?

Let me know - there's a comment box below.

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