The late Ronald Graham wore two hats. He was a renowned mathematician, 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 the University of California, 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. "I think it's the search for patterns and structure that is responsible for this."
Back in 1971, decades before he made that connection explicit, Graham posed a question that some mathematicians now say might have been inspired by juggling. Start with a random set of different integers, not including zero. Can you always rearrange them so that if you add up the first two numbers, then the first three, then the first four, and so on, every "partial sum" turns out different? In the language of juggling, this would mean that if each ball stays in the air for a different amount of time, you can always find an order to throw them in such that two balls won't come crashing down on the same beat.
If the numbers are all positive, then the answer to Graham's question is obviously yes. Similarly, if there are both positive and negative numbers, the answer is also known to be yes. But what if the numbers live in a finite world, like numbers wrapped around a clock that repeat after a certain count? That case resisted proof for over half a century.
After a long hiatus, a group of young mathematicians has finally resolved the problem. They harnessed randomness to show that the required ordering always exists, even in the finite setting. The result confirms that the 55-year-old conjecture holds in full generality.