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Imagine you have a bowling ball without a finger hole so it's smooth all around then you take a bunch of small spherical marbles all of the same size and cover them each in glue.

Then you stick each marble on the ball, covering the entire ball while keeping each marble pushed up against adjacent marbles so there is the smallest possible gap between marbles.

At the end of this process you will find that there is some small gap that is just too small to fit another marble, but which is noticeably bigger than the tiny gaps between all the other marbles.

My question is, is there some magic relationship where a certain diameter marble would fit perfectly or nearly perfectly on a certain diameter bowling ball without a finger hole?

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This is closely related to the Tammes problem, which asks (in slightly different language) how small the marbles need to be such that a given number of them will fit on your bowling ball.

Such packing problems are in general not well understood. In particular, there doesn't seem to be any good theory that will answer the problem for any given number of marbles. Most of the solutions that are known have been found by numerical experimentation and are not proved to be optimal.

A repository of best known solutions for numbers of marbles up to 130 (maintained by Neil Sloane, he of OEIS fame) can be found at http://neilsloane.com/packings/

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It appears there is a relationship with large circles being 'kissed' by smaller circles - For any large circle, small circles which are 1/12 the size of the large circle will nearly perfectly kiss the large circle. The number of small circles doing so will be 40.

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