Young Mathematicians Crack Graham’s 55-Year-Old Juggling Conjecture
A string of four papers by young researchers has settled Graham's 1971 rearrangement conjecture for large primes, using randomness instead of brute force.
A puzzle posed in 1971 by the American mathematician Ronald Graham, and possibly inspired by his love of juggling, has been settled for all sufficiently large prime numbers. The work was done by a small group of mostly young researchers who leaned on randomness rather than computing power. The final piece came in a 27-page paper by Lisa Sauermann, a professor at Germany’s University of Bonn, and Huy Tuan Pham, a mathematician at the University of Chicago. Quanta Magazine reported that the paper closes a problem that took four separate papers and several branches of mathematics to finish.
What the problem asks
The question, known as Graham’s rearrangement conjecture, sounds simple. Take a set of distinct, nonzero numbers. Can you always put them in an order where every running total (the sum of the first two, then the first three, and so on) is different from all the others?
With ordinary positive numbers the answer is obviously yes, because the totals keep growing. Mathematicians also knew the answer was yes for sets that mix positive and negative numbers. Graham’s version is harder because it uses “clock arithmetic,” also called modular arithmetic, where numbers wrap around after reaching a prime number p. On a clock with seven positions, for example, any multiple of seven counts as zero. That makes it far easier for two running totals to land in the same place.
Quanta and the Vietnamese news outlet VnExpress both describe the problem with a juggling analogy. If every ball stays in the air for a different length of time, is there always an order of throws in which no two balls come down on the same beat? Graham was a serious juggler. He served as president of the International Jugglers’ Association and later wrote papers on the mathematics of juggling, some of them with Fan Chung, a mathematician at the University of California, San Diego, who was also his wife. Quanta noted that some mathematicians think juggling may have inspired the conjecture.
How it was solved
Progress came in stages, according to Quanta’s account:
- In 2022, Alp Müyesser, then a graduate student and now at the University of Oxford, and Alexey Pokrovskiy of University College London posted a result, buried inside a paper on a more general problem, that handled sets containing nearly every possible number below p.
- In September 2024, Oxford’s Noah Kravitz and Benjamin Bedert proved the conjecture for sets that are very small compared with p.
- In August 2025, Müyesser, Kravitz and collaborators extended the result to larger sets. That left a gap in the middle.
- In February 2026, Sauermann and Pham posted their proof covering the remaining medium-sized sets.
The shared strategy starts with chance. Instead of searching through all possible orderings, the researchers shuffle the numbers at random and then repair any stretch where the running totals collide. Sauermann and Pham used tools called anti-concentration and Fourier analysis to show that each of three kinds of “bad events” is unlikely. They then showed that the combined probability of all of them stays below 100%, which means at least one valid ordering must exist. VnExpress reported the same method in its coverage.
Other researchers in the field praised the approach. Müyesser told Quanta that the pair’s method was “just completely different” from earlier attempts. Kravitz said he had considered a similar line of attack but had not had the nerve to try it. Noga Alon, a mathematician at Princeton University, described the result as showing “the power of collaboration, the power of the young generation, the power of probabilistic methods.”
A friendship behind the proof
Sauermann and Pham have known each other for about a decade. They met at Stanford University in 2015, when she was a graduate student and he was an undergraduate. Both were stars of the International Mathematical Olympiad, the world’s top competition for high school students. VnExpress reported that Sauermann won four gold medals there and Pham won two, in 2013 and 2014. According to VnExpress, Pham was born in 1996 and attended a school for gifted students in Ho Chi Minh City before going to Stanford.
The two now work on different continents. A conference in Germany in September 2025 brought them together, and Pham then made a short visit to Bonn. Quanta reported that three days of work at a chalkboard there produced the plan that became the proof.
What remains open
The result is not a complete answer to Graham’s question. All four papers assume the prime p is extremely large. Quanta described the threshold as roughly on the order of 10 to the 100th power and said no one has worked out its exact value. Smaller primes are still technically unresolved. The proof also has no practical use for jugglers, since the routines it guarantees would be far too long to perform.
Sauermann and Pham’s paper, titled “On Graham’s rearrangement conjecture,” was posted on the arXiv preprint server. Neither outlet said whether it has finished peer review. Still, researchers quoted by Quanta treated the four papers together as settling the large-prime case of a question that has stood for more than five decades.
