A paper by Professor Pham Tuan Huy and mathematician Lisa Sauermann was posted on the online repository arXiv in February under the title "On Graham's rearrangement conjecture."

The 27-page proof addresses a question posed by renowned mathematician Ronald Graham in 1971.

The question is whether, given a set of distinct nonzero numbers, they can always be rearranged so that all partial sums, the sum of the first number, the first two, the first three and so on, are different, according to Quanta Magazine.

In the language of juggling, this means asking whether, if each ball stays in the air for a different amount of time, it is always possible to find an order in which to throw the balls so that no two land on the same beat, disrupting the performance.

If all the numbers are positive, Graham’s question has an obvious "yes" answer because the partial sums keep increasing as more numbers are added. The harder problem arises in finite systems, such as those in which numbers repeat after a fixed number of steps.

The conjecture concerns subsets of the nonzero elements of a finite cyclic group of prime order.

Huy and Sauermann's paper proves the conjecture for sets up to a specified size range, and, combined with earlier results, completes the conjecture for all sufficiently large primes.

They became interested in the problem in September last year, following a conference in Germany. They began by randomly shuffling sets of numbers and developing a procedure for handling sequences whose sum is zero.

They used a highly technical method known as "anticoncentration" to show that random sums are sufficiently spread out, making undesirable configurations unlikely. Their approach draws on Fourier analysis, a field that represents functions as combinations of simple waves.

The mathematicians then estimated the probability of each undesirable outcome. By showing that the combined probability of all such events was below 100%, they established that at least one valid arrangement must exist.

The method is so complex that mathematician Noah Kravitz of the University of Oxford, who had previously solved part of the conjecture, admitted that he "hadn't had the guts to actually try it," according to Quanta.

Pham Tuan Huy in a photo by Clay Mathematics Institute.

A few months after their time in Germany, Sauermann and Huy published their 27-page proof online. Their work not only closes the remaining gap when combined with earlier results, but also shows how a random ordering can be adjusted to avoid problematic outcomes in a large proportion of cases.

The mathematicians who had previously worked on the problem were surprised to see the remaining case closed so quickly.

"Their approach is just completely different," Alp Müyesser, a mathematician at the University of Oxford, told Quanta.

Huy, born in 1996, is a professor of mathematics at the University of Chicago. His website says he will be visiting the Institute for Advanced Study in fall 2026.

Among Vietnamese mathematicians in the U.S., he is one of the few to have become a full professor at a young age.

There are three main professor ranks in the U.S., from lowest to highest: assistant professor, associate professor and full professor. Faculty members typically take years to progress through the ranks. Huy was appointed an assistant professor at Caltech in 2025 after two years as a visiting scholar.

During his time at the High School for the Gifted under Vietnam National University Ho Chi Minh City, Huy won gold medals at the International Mathematical Olympiad (IMO) twice, in 2013 and 2014.

He earned a bachelor's degree in mathematics and a master's degree in statistics from Stanford University, as well as an advanced research master's degree in mathematics from the University of Cambridge. While pursuing his PhD at Stanford, he received a postdoctoral fellowship for promising mathematicians from the Clay Mathematics Institute.

Huy's co-author, Sauermann, 34, is also a prominent IMO competitor, having won four gold medals. She is currently a professor at the Institute for Applied Mathematics at the University of Bonn in Germany. The two mathematicians have known each other for a decade, since their time studying at Stanford.

Sauermann's research group at the University of Bonn lists "On Graham’s rearrangement conjecture" among her publications.