Young Mathematicians Solve Ronald Graham's 1971 Number Puzzle

Ronald Graham spun coat hangers, juggled pens off walls, and served as president of both the American Mathematical Society and the International Jugglers' Association. In 1971, he posed a question about numbers that some mathematicians now believe grew out of his juggling habit, according to Quanta Magazine, as republished by Wired. Fifty-five years later, a group of young mathematicians has finally proven him right.

"He loved tricks," said Fan Chung, a mathematician at UC San Diego who was married to Graham, describing his habit of "spinning a ball, spinning a coat hanger, spinning several balls together, throwing pens against the wall," per Quanta.

What did Graham's 1971 conjecture actually ask?

Graham's question: Start with a random set of distinct nonzero integers. Can they always be rearranged so that every partial sum — the first two numbers, then the first three, then the first four, and so on — comes out different? For ordinary positive integers, the answer is trivially yes, since the running total keeps climbing. Mixing in negative numbers doesn't break it either. The hard case is numbers that live in a finite world, such as values that wrap around a clock and repeat after a fixed count. Graham conjectured the answer stayed yes even there.

How does juggling connect to the proof?

In a 1980 television interview cited by Quanta, Graham said, "I think it's the search for patterns and structure" that draws mathematicians and computer scientists to juggling. Translated into juggling terms, his conjecture asks whether a juggler can always find a throwing order so that no two balls — each staying airborne for a different length of time — land on the same beat. Graham and Chung later co-authored papers specifically on the mathematics of juggling patterns.

Who finally closed the problem, and when?

The resolution came in four papers across different fields of mathematics, culminating in a final paper by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, published in February 2026, according to Quanta. Noga Alon, a mathematician at Princeton University, told Quanta the conjecture "fits nicely in all these questions about designs and about very symmetric structures," comparing it to the near-certainty of finding a valid sudoku board or Latin square despite rigid constraints.

"It's the power of collaboration, the power of the young generation, the power of probabilistic methods," Alon said of the proof's completion, per Quanta.

Why did randomness end up being the key tool?

Across the four papers, Quanta reports, one method kept resurfacing: using probabilistic techniques — essentially harnessing randomness itself — to prove that orderly patterns must exist somewhere inside a jumble of numbers. That approach let the researchers sidestep decades of failed direct attempts and instead show that a valid rearrangement is, in a precise mathematical sense, highly likely to exist.

What to watch next in this corner of combinatorics

• Whether Sauermann and Pham's probabilistic techniques get applied to other long-standing rearrangement or design problems.
• Further commentary from Noga Alon and Fan Chung on how the four-paper proof chain holds up under peer scrutiny.
• Any follow-up work connecting the result back to Graham and Chung's original papers on juggling mathematics.

Full technical details and the complete chain of papers are available via Wired's republication of the Quanta Magazine report.

mindfulAI Keyboard — An iPhone keyboard that rewrites a draft in a tone you pick. Four free AI actions a day.