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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:
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.
But now comes the second part of the story.
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:
Let me know - there's a comment box below.
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