In 1611 Johannes Kepler guessed that the way a greengrocer stacks oranges, each layer sitting in the hollows of the one below, is the tightest way to pack equal balls, filling a little over 74 per cent of the space. Every greengrocer already believed it. Proving that nothing cleverer could ever exist took until 1998, when Thomas Hales handed in a proof that leaned on enormous computer calculations. A panel of twelve referees worked on it for about four years and then reported that they were 99 per cent certain it was right, and could go no further. So Hales spent another eleven years teaching a computer to check every step, and in August 2014 it agreed. The oranges were right on the first day. The last one per cent was the part about how far you can trust someone who says they have looked everywhere.