Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture

Summary

The 55-year-old conjecture of Ronald Graham, who passed away in recent years, has finally been resolved. The conjecture asked whether any set of non-zero integers in modular arithmetic could be rearranged so that all its partial sums are distinct. A team of young mathematicians solved the problem using probabilistic methods across four papers, with the final proof published in February 2026.

Why it matters

This result validates the effectiveness of probabilistic methods in uncovering hidden structures in combinatorial problems that have stalled for half a century due to deterministic approaches. It also demonstrates how a problem rooted in juggling art and clock arithmetic connects to broader design theory issues concerning symmetric structures. This multi-author, multi-paper solution highlights mathematics' modern shift toward collaboration and distributed proof strategies.

Highlights

• Müyesser and Pokrovskiy's 'large set' strategy: randomly shuffle the majority of the main set, then add a few reserve numbers to break zero-sum intervals that might create partial sum repetitions.
• Kravitz's stated motivation—'It is a shame for humanity that we do not know this'—drove him to tackle the complementary 'small set' case independently of the Oxford team's work.
• Noga Alon highlighted 'the power of collaboration, the power of the young generation, and the power of probabilistic methods' as the combined force that made the solution possible.

Critical take

The article emphasizes Graham's identity as a juggler and positive quotes while addressing the technical mechanism only superficially. It also conceals the fact that Müyesser's 2022 paper already contained the key 'large set' argument. This positions the 2026 finale merely as a culmination rather than an independent breakthrough.