Manon Bischoff, writing for Scientific American, traces a math problem back to a practical shipboard question. Sir Walter Raleigh wanted to know how many cannonballs were in a stack, so he asked his scientific advisor, Thomas Harriot. Harriot found a formula for a square pyramid, then wondered if that stacking was the densest way to pack spheres at all. He wrote to Johannes Kepler in 1606. Kepler’s answer, published five years later, became Kepler’s conjecture: the best packing leaves about 26 percent empty space.

Proving it took centuries. Lagrange settled the two dimensional case in 1773. Gauss handled regular lattices in 1831. In 1953 László Fejes Tóth showed the problem could be cut down to a finite number of cases using Voronoi cells, which made a computer attack possible.

Thomas Hales and his student Samuel Ferguson checked more than 5,000 sphere arrangements, about 100,000 computer problems, and submitted a 250 page proof with three gigabytes of files in 1998.…