It's easier to pack spheres in some dimensions than in others
In 1998, mathematician Thomas C. Hales made headlines by settling a nearly 400-year-old question: What is the best space-saving way to stack oranges? Johannes Kepler, the natural philosopher who first realized that planets orbit the sun in ellipses, conjectured in 1611 that fruit sellers already had it right: The best packing is the familiar pyramidal arrangement seen in markets all over the world. Despite the simplicity of this proposed solution, proving Kepler's conjecture turned out to be elusive for centuries and, in the end, required the assistance of a computer. Hales' 250-page paper is so complex that referees have spent 6 years poring over its details, and although they still haven't checked every one, they recently gave the paper the thumbs-up to be published in digest form in the Annals of Mathematics.