What was the rearrangement conjecture

In 1971, Ronald Graham — one of the most prolific mathematicians of the twentieth century and, curiously, former president of the International Jugglers' Association — posed a question that was trivial to state but tenacious enough to keep the mathematical community silent for over half a century. The question: given an arbitrary set of distinct, nonzero integers, could one always reorder them so that the running cumulative sum never repeated a value?

More precisely, for any prime p and any subset S of the nonzero elements of the cyclic group ℤₚ, does there exist an ordering of the elements of S such that all partial sums s₁, s₁+s₂, s₁+s₂+s₃, etc., are distinct from each other? Graham conjectured that the answer was yes. The intuition behind it was that even under rigid modular arithmetic constraints, there should be enough flexibility to construct valid arrangements — just as it is possible to fill a sudoku board or a Latin square despite tight rules.

The problem took root in additive combinatorics, a field dealing with symmetric structures in finite groups that connects number theory, combinatorial algebra, and asymptotic analysis. Over the decades, mathematicians solved particular cases — small sets, large sets — but the gap between them remained open like a canyon no one could cross. For more than twenty years, the problem entered an almost absolute hiatus. Only in recent years did a new generation of researchers return to the balls.

The proof in four acts

What made the resolution so extraordinary was the need to combine four different approaches, each addressing a distinct regime of set size. The proof is not a single paper — it is an edifice built in four parts by at least six different authors.

The first step came with small sets. Noah Kravitz of the University of Oxford, along with Liana Bedert and David Kravitz, established that the conjecture holds for sets with up to exp((log p)^(1/4)) elements. In 2022, Alp Müyesser — then still a graduate student — realized that a proof he had just written about a different problem also applied to the intermediate case of large sets, those containing almost every possible number up to p. Together with his advisor, Alexey Pokrovskiy of University College London, Müyesser tackled that regime: starting with a random ordering already took most of the way there.

In 2025, Bedert, Bucić, Kravitz, Montgomery, and Müyesser filled the space between large and intermediate sets. The middle case remained — the one that Noga Alon of Princeton University called the most embarrassing: "Our methods didn't work there, and there were clear reasons that they would not have worked."

The breakthrough came in February 2026. Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago posted a 27-page paper on arXiv that closed the last gap. The two had been friends since 2015, when they met at Stanford — Sauermann as a graduate student and Pham as an undergraduate. When a conference in Germany in autumn 2025 gave them the chance to share a chalkboard again, they spent three days talking through the problem. By the end, they had a plan.

The central strategy revolved around a concept called "anti-concentration": rather than trying to manually construct the perfect ordering, they showed that by starting from a random ordering and applying careful probabilistic adjustments, one could avoid the problematic values — those that would make a partial sum collide with one already visited — in a sufficiently large proportion of cases to guarantee, by probabilistic counting, that at least one valid arrangement must exist.

"It's the power of collaboration, the power of the new generation, the power of probabilistic methods," Noga Alon summarized. The entire proof shows that even within the strict constraints of modular arithmetic, beautiful structures always emerge when you allow the flexibility of chance.

What the proof reveals

The rearrangement conjecture has no direct practical applications — it is not the kind of result that will show up in a computer design or a network optimization. Its value is purely structural: it answers a question about the existence of patterns in seemingly constrained systems.

But what makes Graham's case truly fascinating is the story behind it. Graham was one of the elite mathematicians who was also a professional juggler. He presided over the International Jugglers' Association and, like many jugglers, knew the language of "patterns": the hourglass-shaped diagrams that describe sequences of throws. When he formulated his conjecture, it is plausible he thought of juggling patterns — the idea that given a set of moves (or numbers), there is always a valid sequence in which each intermediate point is unique.

Clint McCullough of Georgia Southern University, one of the authors who contributed to the intermediate case, captured the spirit of the result well: "To pose a good problem is really an art. I think Ron would be extremely happy to see the problem solved."

Huy Tuan Pham, for his part, has a trajectory that might almost mirror the conjecture's theme itself. He is Vietnamese, graduated from the High School for the Gifted under Vietnam National University in Ho Chi Minh City, and won two gold medals at the International Mathematical Olympiad (2013 and 2014). Today, as a researcher at the University of Chicago, he and Sauermann proved that given a set of numbers, one can always find an ordering for them — a nearly poetic metaphor that perseverance, collaboration, and a touch of randomness can solve even the most stubborn problem.

And if you want to use this result to choreograph a real juggling routine, you are out of luck: to correspond to a prime p large enough to cover all cases of the conjecture, the sequence would be very, very long. But the proof confirms something Graham intuited in 1971: even within tight rules, there is always room for flexibility. There is always a way to arrange the pieces.

What to watch is whether the anti-concentration method — which combines random ordering with probabilistic adjustments — could be adapted to attack other additive combinatorics conjectures that still resist. Kravitz himself admitted, citing the final method, that he "hadn't had the guts to actually try it" — an acknowledgment of how deep the abyss was and how high the leap was. There are many similar problems waiting for that combination of courage, collaboration, and a little luck.

Sources: Wired, Quanta Magazine, VnExpress International

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