A question posed in 1971 by the late mathematician Ronald Graham — a former president of the American Mathematical Society who was also president of the International Jugglers' Association — has finally been answered. The final piece of the proof, by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, appeared in February 2026 and spans 27 pages.
The conjecture concerns clock arithmetic. Take the whole numbers, wrap them around a clock face so they repeat after some prime number p, and pick any set of nonzero numbers. Graham asked whether you can always rearrange them so that every partial sum — the first two numbers, then the first three, then the first four, and so on — is different. Single sums are easy to control; the trouble is that a long block of numbers can add up to zero, which makes the running total repeat.
For positive sets, and for sets mixing positive and negative numbers on an infinite line, the answer was already known to be yes. The open case was the finite, wraparound setting, where Graham's intuition was that flexibility always survives rigid constraints — the same intuition that makes a valid Sudoku or Latin square usually findable.
The resolution came in four papers. Alp Müyesser (Oxford) and Alexey Pokrovskiy (UCL) handled sets that cover almost every number up to p, posting their solution in 2022 inside a broader paper. Noah Kravitz and Benjamin Bedert, both at Oxford, attacked the opposite end — tiny sets — and posted their proof in September 2024. The two groups joined forces, and their August 2025 paper extended the large-set case. But between the small and large regimes a gap remained for medium-sized sets, where roughly half the numbers up to p are in play; the earlier methods were known to fail there.
Sauermann and Pham, old friends from Stanford in 2015, found the missing approach during a three-day visit in Bonn in late 2025, after hearing two conference talks on the problem. Like their predecessors, they start with a random ordering and then repair it, swapping out numbers whenever a zero-sum block appears. Three kinds of bad event can break that repair procedure: a zero-sum block at the very end with nothing to swap in, many such blocks crowding together, and a repair that creates a new zero-sum block downstream.
The new ingredient was anti-concentration: statements that a particular sum is unlikely to appear. Using Fourier analysis, the pair showed that when random sets of numbers are added, no single sum is especially likely, then bounded the probability of each bad event and showed the total risk falls below 100%. The upshot is not just that a valid ordering exists, but that a random ordering can be repaired at least 90% of the time.
"Their approach is just completely different," Müyesser said of the final paper. Princeton's Noga Alon credited "the power of collaboration, the power of the young generation, the power of probabilistic methods."
Two caveats remain. All four papers assume p is very large — on the order of 10 to the 100th power — which is fine for the intended mathematical setting but leaves the conjecture formally open for arbitrary p. And anyone hoping to use the result to choreograph an actual juggling routine is out of luck: the required number of balls and beats would make the pattern far too long to perform. Still, as Fan Chung, Graham's wife and a mathematician at UC San Diego, put it: "To pose a good problem is really an art. I think Ron would be extremely happy to see the problem solved."
