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. “I think it’s the search for patterns and structure that is responsible for this.”
He would go on to write numerous papers, some with Chung, on the mathematics of juggling. But back in 1971, decades before he made that connection explicit, he posed a question that some mathematicians now say might have been inspired by juggling, too.
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—which would ruin the act.
If the numbers are all positive, then the answer to Graham’s question is obviously yes: The sums will always grow larger as you add more numbers. Similarly, if there are both positive and negative numbers in the mix, the answer is also known to be yes. But what if the numbers live in a finite world—like numbers wrapped around a clock, which repeat after a certain count?
The late mathematician Ronald Graham was also a skilled juggler.Photograph: Peter Vidor