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. 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?
That’s what Graham wanted to know. He conjectured that the answer should still be yes. It often happens, he figured, that even when dealing with rigid constraints, you can still find enough flexibility to construct special patterns or structures — just as it’s usually possible to find a valid sudoku board or Latin square (another kind of puzzle) despite their many rules.
But for decades, no one could prove Graham’s intuition to be true. That changed recently, when several young mathematicians picked up the balls. In a proof that spanned four papers and various fields of mathematics, they finally resolved Graham’s rearrangement conjecture.
The final paper, by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, appeared in February 2026, officially closing the problem. Across the papers, one theme prevailed: the power of randomness to draw out patterns. As Alon put it, “It’s the power of collaboration, the power of the young generation, the power of probabilistic methods” that solved the problem.
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