A group of young mathematicians has solved a 55-year-old puzzle by treating randomness like a tool rather than a nuisance. The proof, spread across four papers, closed the door on a question first raised by Ronald Graham in 1971. The final paper, by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, appeared in February 2026.
Graham’s Question
Graham was a juggler as well as a mathematician, married to Fan Chung, a mathematician at the University of California, San Diego. He loved tricks, she said, including spinning a ball, spinning a coat hanger, and throwing pens against the wall.
In 1971, Graham posed a question about numbers arranged in a circular pattern, like hours on a clock. Start with a set of different integers, none of them zero, and arrange them so that the sum of the first two numbers, then the first three, then the first four, and so on, always produces a different total. If the numbers are all positive, the answer is obvious: the sums keep growing. If the numbers include both positive and negative values, the answer is known too. But what if the numbers repeat after a fixed count, like hours on a clock?
Graham guessed that the answer should still be yes. He thought that even with strict limits, you could usually find enough room to build a special arrangement — much like finding a working sudoku board or Latin square despite their rules. “It fits nicely in all these questions about designs and about very symmetric structures,” said Noga Alon, a mathematician at Princeton University. For decades, no one proved him right.
Müyesser’s Haystack
Alp Müyesser, a mathematician at the University of Oxford, looks for problems that need two things: a random process and something extra on top. In 2022, while still a graduate student, he solved one such problem. Then he found Graham’s conjecture and saw that his just-finished proof could help there too.
The conjecture works in the world of clock arithmetic. Place the whole numbers on a number line, then wrap the line around a clock face so that the numbers repeat after some prime number, p. For example, if p is 7, then 0, 7, 14, and all other multiples of 7 are the same. Graham asked whether you can always rearrange any set of nonzero numbers, for any p, so that the running totals are all different.
Müyesser, along with his former adviser, Alexey Pokrovskiy of University College London, took on the case where the set includes almost every possible number up to p. With sets this large, building a valid order is extremely hard. But starting with a random order brings you most of the way there.
“Computer scientists often call this a ‘finding the hay in the haystack’ problem,” Müyesser said.
You might know that many good orders exist, but actually finding one is hard. A random order is likely to work, but it is hard to describe what the solution should look like.
Müyesser and Pokrovskiy had to make sure no sequence of numbers anywhere in the order added up to zero. Otherwise, adding those numbers to the earlier total would repeat it. A completely random order might have a few of these troublesome sequences. So they set aside a few specially chosen numbers, scrambled the rest randomly, and scanned the result for problems. If they found an interval that added up to zero, they inserted one of the spare numbers to fix it. In 2022, they posted their solution, though it was buried in a paper focused on a more general problem.
Kravitz’s Stumble
Noah Kravitz of Oxford found Graham’s conjecture a couple of years later, in an online archive of unsolved problems. He saw an open question and felt embarrassed that humanity did not know the answer.
“I saw there was an open problem, and I was like, it’s embarrassing for humanity that we don’t know this,” Kravitz said.
Kravitz was unaware of Müyesser and Pokrovskiy’s solution. He worked independently.
Four Papers, One Answer
The proof spans four papers and draws on several fields of mathematics. The final paper, by Sauermann and Pham, closed the problem officially.
As Alon put it, “It’s the power of collaboration, the power of the young generation, the power of probabilistic methods” that solved the problem.
What the Proof Shows
The proof shows that randomness can be a productive strategy, not just a way to guess. Müyesser and Pokrovskiy started with a random order and used a small set of carefully chosen numbers to clean it up. Kravitz approached the same problem from a fresh angle.
The solution also confirms Graham’s original guess about the existence of flexible arrangements within rigid constraints. The mathematics of juggling, with balls staying in the air for different lengths of time, shares the same structure as the number problem. Graham himself noted in a 1980 television interview that many mathematicians and computer scientists enjoy juggling, and suggested that the search for patterns and structure explains why.
The Significance
Graham’s conjecture stood open for 55 years. The proof closes it with a method that treats randomness as a strength rather than a weakness.
The proof is a reminder that some problems yield to chance, if you know how to look at it. The four papers, and the collaboration behind them, show that young mathematicians can resolve deep questions by combining new ideas with old ones.
The proof also honors Graham’s legacy. He wore two hats, juggler and mathematician, and his question bridged the two worlds.
See the video the story is built around at Quanta Magazine.
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