problem of the 13 balls

mathematics

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July 14, 2026

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The "problem of the 13 balls" refers to a classic question in combinatorial geometry. It asks how many equal spheres can touch a central sphere of the same size without overlapping. While 12 spheres can be arranged to touch a central one, it was long debated whether 13 were possible. This problem, discussed by Isaac Newton, was finally proven to have 12 as the maximum in 1953. This is a type of "packing problem," which seeks to arrange geometric figures as compactly as possible without overlap. The related "Fifteen Puzzle" involves arranging numbered tiles in a grid, and its solvability is determined by the number of inversions in the tile arrangement.12

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combinatorics
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…might be possible to have 13 equal non-overlapping spheres touch another of the same size. This dilemma between 12 and 13, one of the first nontrivial problems of combinatorial geometry, was the object of discussion between Isaac Newton and David Gregory in 1694. Newton believed 12 to be the correct…
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…might be possible to have 13 equal non-overlapping spheres touch another of the same size. This dilemma between 12 and 13, one of the first nontrivial problems of combinatorial geometry, was the object of discussion between Isaac Newton and David Gregory in 1694. Newton believed 12 to be the correct…
Read More