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A mathematical puzzle proposed 55 years ago by mathematician Ron Graham has finally been solved through a series of recent papers by a new generation of researchers. The problem explores whether a random set of nonzero integers in clock arithmetic can always be rearranged so that every partial sum remains unique.
Origins of the Juggling Puzzle
Ron Graham, who frequently explored the mathematics of juggling alongside his wife and fellow mathematician Fan Chung, posed the question in 1971. In juggling terms, the problem asks whether balls with different air times can always be thrown in an order that prevents two balls from landing on the exact same beat. While the answer was known to be yes for positive numbers and mixed numbers, proving it within finite clock arithmetic—where numbers repeat after a certain prime number—remained an open challenge for decades.
Princeton University mathematician Noga Alon noted that the question fits well within broader studies of symmetric structures and designs. Although Graham conjectured that a valid arrangement is always possible even under such rigid constraints, mathematicians could not prove his intuition for decades.
Collaborative Proofs Across Different Set Sizes
A breakthrough came when several young mathematicians approached the problem using probabilistic methods. Yücedeniz Müyesser and his former adviser Alexey Pokrovskiy tackled very large sets of numbers near prime $p$ by starting with a random ordering and using spare numbers to fix any troublesome sequences that added up to zero. They posted their findings in 2022.
Separately, Noah Kravitz and Benjamin Bedert of Oxford University solved the conjecture for tiny sets of numbers compared to $p$, publishing their proof in September 2024. Müyesser later connected with Kravitz, and the team collaborated to expand their approach, publishing a combined paper in August 2025 that covered relatively large sets of numbers.
Closing the Final Gap
Medium-size sets—such as those containing roughly half as many numbers as $p$—still remained unaddressed until Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago tackled the remaining gap. Sauermann had previously solved a related problem during her high school years in the International Mathematical Olympiad, where she earned four gold medals.
During a three-day visit to Germany where they attended conference talks, Sauermann and Pham devised a plan utilizing anti-concentration, a demanding mathematical method stating that certain events have a very low probability of occurring. By applying Fourier analysis to estimate the likelihood of various bad events—such as zero-sum sequences occurring near the end of an arrangement or clustering too closely—they demonstrated that the total chance of failure was less than 100 percent.
In February 2026, Sauermann and Pham posted their 27-page proof online, confirming that random orderings can eliminate problematic sequences at least 90 percent of the time. Together, the four papers successfully prove Graham’s conjecture for sets of all sizes when the prime number $p$ is extremely large. Chung expressed confidence that Graham would have been thrilled to see his longstanding puzzle resolved.
