On February 17, 2026, Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago published on arXiv the missing piece needed to close Graham's rearrangement conjecture, an open problem since 1971 that was born out of an inspiration from juggling.
The question is simple to state. If you take distinct integers, all nonzero, can you always reorder them so that the sum of the first two, then the sum of the first three, and then the sum of the first four each give a different result? Ronald Graham, a mathematician and former president of the International Jugglers' Association, conjectured in 1971 that the answer remains yes even in clock arithmetic, where numbers wrap around after a prime. In juggling terms: each ball flies for a different time, and there is always a throwing order in which two do not land at the same time. According to the Wired report published on November 8, closing the problem required four papers by young mathematicians from Oxford, Bonn and Chicago. For Mexico and Latin America, the result matters because the discrete mathematics underpinning these proofs feeds applied fields, from cryptography to data analysis.
The work was divided by set size. Alp Müyesser of Oxford and Alexey Pokrovskiy of University College London solved the case of large sets with random processes in 2022. Noah Kravitz and Benjamin Bedert covered small sets in September 2024. A team led by Müyesser extended the method in 2025 and, according to the arXiv paper, Sauermann and Pham closed the intermediate case. The technique starts from a random order, sets aside some spare numbers, and uses Fourier analysis with an anticoncentration bound to correct the stretches that sum to zero. The authors showed that a random order can be repaired at least 90% of the time. Noga Alon of Princeton spoke about the power of collaboration, of the young generation and of probabilistic methods.
The proof holds for sufficiently large primes, so the case for every prime number remains technically open, and the four papers are awaiting peer review. Meanwhile, the final 27-page text is available on arXiv, with the list of open problems a little shorter.
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