Four papers settle Graham's 1971 conjecture on partial sums

What happened

• Wired published on November 8, 2026, with permission from Quanta Magazine, the story of how the rearrangement conjecture mathematician Ronald Graham posed in 1971 was solved.
• The question: given a set of nonzero numbers, can they always be ordered so the partial sums (the first two, the first three, and so on) all come out different?
• If the numbers are all positive, or both positive and negative, the answer was already yes. Graham asked what happens in the finite world of “clock” numbers, which repeat after a prime p.
• The proof came in four papers. The last, by Lisa Sauermann (Bonn) and Huy Tuan Pham (Chicago), appeared in February 2026 and closed the problem.

Why it matters

The proof confirms that, even in those rigid settings, ordered structures always exist. Noga Alon, of Princeton, ties it to other questions about designs and highly symmetric structures.

The result came from different people working on parts of the problem: sets nearly as large as p, tiny sets and medium-size sets. That the proof was human, and not produced by a machine, is part of the story.

The number

55 years. That is the time between Graham’s question in 1971 and the last paper in February 2026. In between came work in 2022 (Müyesser and Pokrovskiy), September 2024 (Kravitz and Bedert) and August 2025 (an expanded team).

Context

Each case required a different technique. For sets nearly as large as p, Müyesser and Alexey Pokrovskiy started from a random ordering: they set aside some numbers, shuffled the rest, scanned for stretches adding to zero and fixed them with the reserved numbers. Noah Kravitz and Benjamin Bedert attacked small sets, for example 100 numbers with p equal to a billion.

A gap remained: medium-size sets, with about half as many numbers as p. Sauermann and Pham closed it with a technically demanding method called anti-concentration, which asserts that some event has a very low chance of happening. Mathematicians knew the route but had not dared to try it, according to Kravitz.

The proof does not settle everything: the conjecture could still be attempted for all p, and it is no use for choreographing a real juggling routine, because p would be too large. Graham was a juggler, and the analogy is that each ball stays in the air for a different time without landing on the same beat.

At Mamífero we have followed mathematical advances. An AI-assisted proof closed a 2008 problem on quantum advantage, Axiom verified the Stacks Project with agents, the parietal cortex readjusts its number code by estimated range and origami structures compute with three states.

What’s next

• Mathematicians could try to resolve the conjecture for all p, not only the cases covered.
• The anti-concentration technique remains as a tool for other combinatorics problems.
• Chung, Graham’s colleague, said he “would be extremely happy” to see the problem solved.

Bottom line

It is a story about collaboration: people from areas that rarely cross solved by parts a problem that had stood for half a century. The value goes beyond the statement, because it leaves methods of randomness and combinatorics available for other questions.

Sources

• Wired (reprinted from Quanta Magazine), “A 55-Year-Old Math Problem Was Solved—This Time by Humans”, November 8, 2026, link.
• Cover image: Marcus Jeffrey from Kenora, Canada, CC BY-SA 2.0, via Wikimedia Commons

Edited by Rodrigo Cornejo. How we select and verify: who writes these notes.