Ronald Graham, a mathematician and juggler, posed a question in 1971 that may have grown out of his hobby. In the finite world of clock arithmetic, where numbers wrap around after a prime p, he asked whether any collection of distinct nonzero numbers can be rearranged so that the running sums — the first two, first three, and so on — are all different. The answer was known to be yes when the numbers are all positive or when they mix positive and negative values, but the modular case resisted proof for decades.
Now a group of young mathematicians has finally resolved it. Their work, spread across four papers, concluded with a February 2026 paper by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago. The proof draws on several areas of mathematics, but a common thread is the deliberate use of randomness to find order. Princeton mathematician Noga Alon credited the collaboration, the young researchers, and probabilistic methods for the breakthrough.
One key approach, developed by Alp Müyesser of Oxford and Alexey Pokrovskiy of University College London, handled the hardest case: sets containing almost every number up to p. Rather than trying to construct a valid ordering directly, they started with a random ordering and set aside a few chosen numbers. Whenever they found a block of numbers that summed to zero, they inserted one of the spare numbers to break the pattern. This 'random plus repair' strategy turned a seemingly rigid constraint into a manageable search.