A group of young mathematicians found the solution to the long unanswered conjecture, first inspired by juggling, through randomness and friendship.

Animation: Irene Pérez for Quanta Magazine

The original version of this story appeared in Quanta Magazine .

The late Ronald Graham wore two hats. He was a renowned mathematician and, at one time, president of the American Mathematical Society. He was also a serious juggler and president of the International Jugglers’ Association. “He loved tricks,” said Fan Chung, a mathematician at UC San Diego who was married to Graham. “You know, spinning a ball, spinning a coat hanger, spinning several balls together, throwing pens against the wall.”

Sometimes Graham wore both hats at once. “It’s interesting, in fact, that many mathematicians and computer scientists have an interest in juggling,” he said in a 1980 television interview .…