A 55-Year-Old Math Problem Was Solved by Humans
In 1971, mathematician and avid juggler Ronald Graham posed a puzzle inspired by juggling: can any set of distinct, nonzero integers be rearranged so that every partial sum is unique? While simple for ordinary positive integers, Graham conjectured that this property also holds in clock arithmetic, where numbers wrap around after a prime p. In this setting, two positive numbers can add up to zero, creating rigid constraints that prevented mathematicians from proving Graham's intuition for more than five decades.
The 55-year-old rearrangement conjecture has now been proven across four papers using probabilistic methods. Mathematicians Alp Müyesser of the University of Oxford and Alexey Pokrovskiy of University College London tackled difficult cases where sets contain nearly all possible numbers modulo p, demonstrating how random processes can reveal structured order.
The conjecture was officially resolved in February 2026 in a final paper by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago. Praised by Princeton mathematician Noga Alon as a milestone for young researchers and probabilistic techniques, the breakthrough confirms that even in strict modular arithmetic, there is always enough flexibility to order numbers without colliding sums.
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