Graham's rearrangement conjecture, a puzzle about putting numbers into a new order that Ronald Graham, who has since died, set out in 1971, has now been settled by a series of four papers from a group of young mathematicians. The last of them, written by Lisa Sauermann (University of Bonn) and Huy Tuan Pham (University of Chicago), came out in February 2026 and formally brought the problem to a close. Quanta Magazine first published this story in its original form.
Ronald Graham, who has since died, led two lives. A highly regarded figure in mathematics, he once served as head of the American Mathematical Society. He was equally committed to juggling, and headed the International Jugglers' Association. “He loved tricks,” said Fan Chung, a researcher at UC San Diego who was Graham's wife. “You know, spinning a ball, spinning a coat hanger, spinning several balls together, throwing pens against the wall.”
What Graham's rearrangement conjecture asks
Graham went on to publish many papers on the maths behind juggling, a number of them co-written with Chung. Yet the question he raised in 1971 came long before he drew that link openly, and some mathematicians today suspect that juggling may have lain behind it as well.
Take any collection of distinct whole numbers, with zero excluded. Is it always possible to put them in an order such that the running totals are all different from one another? These totals are formed by adding the first two, then the leading three, then the leading four, and so forth.
Put in terms of juggling, each ball stays aloft for its own length of time. The aim is to find a throwing order in which no two balls land at the same moment, since that would spoil the performance.
When every number is positive, Graham's question has an obvious answer of yes, because the totals only climb as more numbers are added. The answer is likewise established as a yes when the set contains a mixture of numbers above and below zero. But what happens when the numbers sit in a finite setting, such as numbers looped round a clock face, which start over after a fixed count?
Graham was curious whether the answer would remain yes, and he suspected it would. He reasoned that, even when the constraints are severe, there is frequently sufficient freedom to construct certain patterns or structures. A valid Sudoku grid or Latin square (a puzzle of a similar kind) can usually be found despite all their rules. Noga Alon, a Princeton University specialist in this field, said: “It fits nicely in all these questions about designs and about very symmetric structures,” Yet Graham's hunch went unproven for decades.
How young mathematicians finally solved it
Then, not long ago, a group of young mathematicians took up the challenge. Their proof ran across four papers and drew on several branches of mathematics, and it finally settled Graham's rearrangement conjecture. It was the last of the papers, written by Sauermann and Pham, that brought the problem to a close.
One idea ran through all the papers: randomness can bring order out of apparent disorder. In Alon's words, “It's the power of collaboration, the power of the young generation, the power of probabilistic methods” that cracked this long-standing puzzle.
Alp Müyesser, who works in mathematics at Oxford University, is usually drawn to issues that give way only when two elements are combined: a process driven by chance and some additional component. In 2022, while still a PhD student, he cracked a puzzle of precisely this sort. Not long afterwards, he came across Graham's conjecture and realised that the proof he had just completed might be useful for it as well.
Clock arithmetic and the size of the set
The conjecture lives in what mathematicians call clock arithmetic. Begin with the whole numbers laid out along a line, then bend that line round into a circle like a clock face, so that the values restart once they hit a prime, p. Take p to be 7. Here 0, 7, 14 and every other multiple of 7 count as the same thing, so two positive numbers, 3 and 4 for instance, can be added to make zero.
Graham's question was this: take any collection of nonzero numbers from this number line (whatever the value of p). Is it always possible to arrange them in an order such that the running totals you obtain, the partial sums, are all distinct?
How hard the problem is hinges on the size of your set relative to p. Choose more numbers, and you face a greater number of sums to keep under control; choose fewer, and you have less room to shuffle them around. Each of these situations calls for its own line of attack.
Large sets: finding the needle in a haystack
Müyesser and Alexey Pokrovskiy of University College London, who once supervised him, took on the situation where the set contains nearly every integer below p. Building a valid ordering for sets of that size can be enormously difficult. Even so, it emerged that random shuffling can carry you a long distance.
”In computer science this is frequently described as a 'needle in a haystack' issue,” Müyesser said. You may know that many good orderings exist, yet finding one is the hard part. “If you do it randomly, it's likely going to work, but it's hard to explicitly describe what the solution is supposed to look like.”
The task for Müyesser and Pokrovskiy was to guarantee that no stretch of consecutive numbers, wherever it sat in the ordering, summed to zero. If one did, adding it to the preceding partial sum would simply produce that same sum again.
If the figures are simply put in a random order, a few awkward stretches may well turn up. Müyesser and Pokrovskiy tackled this by first putting aside a few carefully chosen values from the set, and then randomly shuffling all the rest. They then inspected the shuffled sequence for faults.
Whenever a stretch added up to zero, one of the reserved values could be slotted in to break it up. Their result went online in 2022, though it sat inside a paper chiefly devoted to using the same technique on a wider question.
The small-set case and an unlikely team
Some two years on, Noah Kravitz of Oxford, who had not heard of the solution by Müyesser and Pokrovskiy, came across Graham's conjecture while browsing a web-based collection of open questions in mathematics. “I saw there was an open issue, and I was like, it's embarrassing for humanity that we don't know this,” Kravitz said. “This situation just had to be rectified.”
His plan was to approach the problem from the opposite direction. Working with Benjamin Bedert, also of Oxford, he looked at the situation in which the collection of numbers is very small next to p. Alon offered an illustration: 100 numbers, with p standing at 1 billion.
Kravitz and Bedert settled Graham's rearrangement conjecture for such cases, and put their proof online in September 2024. Müyesser spotted it and got in touch, passing along his own results. The three, joined by two more colleagues, then combined forces to push Müyesser's original method further.
“It was a pretty unlikely combination of people,” Kravitz said. He and Müyesser belong to two branches of combinatorics that rarely work together. “Different sections have completely different techniques,” he said.
The team's paper, which went online in August 2025, dealt with further cases in which the collection of numbers is fairly large relative to p. Yet a gap persisted between those and the cases involving tiny collections that Kravitz and Bedert had dealt with.
Nobody could work out how to tackle sets of middling size, for example those holding roughly 50 per cent of the quantity of p. “Our methods didn't work there, and there were clear reasons that they would not have worked,” Müyesser said.
Closing the gap on Graham's rearrangement conjecture
It looked as if work on Graham's rearrangement conjecture might fall quiet again for a long stretch. Then, in February 2026, something unexpected turned up online.
Lisa Sauermann and Huy Tuan Pham had been acquainted for years. Their first meeting was at Stanford University in 2015, when she was a doctoral candidate and he was an undergraduate. Today an ocean lies between the two mathematicians, with Sauermann working in Bonn, Germany, and Pham in Chicago.
A gathering in Germany in September 2025 gave them a rare chance to stand at one blackboard together again, and Pham afterwards went with Sauermann to Bonn for a brief stay. The one thing missing was a question to tackle.
Graham's conjecture featured in two conference talks, both delivered by researchers who had tried and failed to close the gap. The two were intrigued. It later emerged that Sauermann had met a similar puzzle in the International Mathematical Olympiad back in her secondary school days, one that was very nearly the same. The mathematicians giving those talks had no idea.
Her answer was correct, and by the point she left schools she had taken gold at that famous competition four times. (Chung most likely put the puzzle on that year's paper: she sat on the panel that wrote the contest questions, and she often drew on Graham's many brainteasers.)
During the three days they spent together, Sauermann and Pham figured out a way they might crack it.
Anti-concentration and the three bad events
Their approach relied on a highly demanding technique known as anti-concentration. A claim of this kind asserts that a particular event is exceptionally unlikely. But such claims are so intricate to prove that, even though Kravitz and others understood this path might succeed, “we just hadn't had the guts to actually try it,” he said.
To begin with, however, Sauermann and Pham followed the same path as those before them. They arranged their set of values in a random order and devised a method for repairing any trouble spots, that is, any run of terms whose total is zero. Whenever such a zero-total run showed up, they exchanged its final entry for a different one.
Most of the time, this method worked smoothly. Yet it could break down in three kinds of “bad events”:
- Should a zero-sum run turn up near the very end of the whole ordering, no remaining numbers would be left to exchange with it.
- Zero-sum runs might also cluster so tightly that they could not all be repaired.
- Mending one faulty run could, in turn, produce a fresh zero-sum run further along.
Sauermann and Pham aimed to demonstrate, through anti-concentration, that every one of these unfavourable outcomes was adequately improbable. If so, some way of reordering the numbers had to exist that met the conjecture's requirements.
For this, the pair turned to Fourier analysis, a branch of mathematics in which functions are expressed as combinations of simple waves. With it, they showed that when random collections of numbers are added together, no single total is unusually probable.
Using that finding, they carefully estimated how probable each unfavourable outcome was, and concluded that the overall chance of them occurring was below 100 per cent.
What the proof of Graham's rearrangement conjecture means
Only months after their stay in Germany, Sauermann, and Pham produced a 27-page paper. It showed that a suitable rearrangement can always be found. It also showed that, for a set placed in random order, the bad events can be removed through reshuffling in no more than 90% of cases, an impressively high success rate.
Mathematicians who had earlier tackled the issue did not expect the last remaining case to fall so fast. “Their approach is just completely different,” Müyesser said.
Taken together, these four studies establish Graham's rearrangement for every conjecture pieces of every size. Each of them, though, assumes that p is enormous; nobody has worked out its precise value, but picture something around 10 to the power of 100. Mathematicians are untroubled by this, because what counts is that the work happens inside clock arithmetic.
One facet of the problem is still technically open: someone could yet try to prove Graham's rearrangement conjecture for every p. And anyone hoping to turn the result into a genuine juggling routine will be disappointed: a routine matching such an enormous p would run far too long.
The proof demonstrates that even within these odd, constrained number systems, “there are some nice structures that always exist,” Alon said. Some measure of freedom is always on hand, allowing you to reorder the elements of your collection so that you never land on the same running totals twice.
“To pose a good issue is really an art,” Chung said. “I think Ron would be extremely happy to see the problem solved.”
This story was first published by Quanta Magazine, an outlet of the Simons Foundation that keeps its editorial independence, and is reproduced here with permission. Its mission is to improve how the public grasps science, by covering advances and trends across maths, the physical sciences and the life sciences.
