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Probabilistic methods settle Ronald Graham's 1971 conjecture on reordering running sums
Four papers, the last by Lisa Sauermann and Huy Tuan Pham in February 2026, prove Ronald Graham's 1971 conjecture on running sums. Randomness was the common tool in all four, applied to clock arithmetic, where a running total can wrap around to a value it has already hit.
The Scientist · Science desk
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What happened
- Graham asked whether any set of nonzero numbers, in arithmetic that wraps around after a prime p, can always be reordered so that every running sum is different.
- For ordinary integers the answer was already known to be yes, both for all-positive sets and for sets that mix positive and negative numbers.
- How hard the problem is depends on the set's size relative to p: more numbers mean more sums to keep apart, fewer mean fewer orderings to choose from.
- Alp Müyesser of Oxford and his former adviser Alexey Pokrovskiy of University College London handled sets that contain almost every nonzero number up to p.
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Why it matters
- constraint Because the difficulty shifts with set size, no single argument covered every case, so confirming the result means checking four separate proofs drawn from several fields.
- precedent Müyesser's 2022 argument for a different problem carried over to Graham's. The design and Latin-square questions Alon grouped this with now have a tested approach to try.
- capability Graham's hunch that rigid rules still leave room for a valid pattern is now a theorem for sums modulo any prime, citable in later work in place of an open conjecture.
Wrapping the number line into a clock removes the easy argument. Positive integers only ever push a running total upward, so no total can repeat [4]. On a clock of size 7, though, 0, 7 and 14 are the same point, and 3 plus 4 lands on zero [5]. Take the set 1, 2, 3, 4 on that clock. In that order the running totals are 1, 3, 6 and 3, because the 3 and the 4 together make a full turn. Ordered 1, 3, 2, 4, the totals are 1, 4, 6 and 3, all different [2]. Graham conjectured that an order like the second always exists [3].
Müyesser brought an existing technique to the problem. He is drawn to problems whose solutions need two ingredients: a random process and something extra [11]. After finishing one such proof in 2022, while still a graduate student, he saw that it could help with Graham's conjecture [11]. The whole proof runs across four papers and several fields of mathematics [6]. The last of them, from Sauermann at the University of Bonn and Pham at the University of Chicago, closed the problem [7].
Randomness is the common thread through all four, according to Quanta Magazine's account [8]. What solved it, said Noga Alon of Princeton University, was "the power of collaboration, the power of the young generation, the power of probabilistic methods" [9]. Graham's own reasoning was that rigid rules usually leave enough room for a special structure, the way a valid sudoku board or Latin square can usually be found despite its many constraints [13]. "It fits nicely in all these questions about designs and about very symmetric structures," Alon said [14]. Fifty-five years separate Graham's 1971 question [1] from the final paper [1].
Graham was at one time president of both the American Mathematical Society and the International Jugglers' Association [15]. In juggling terms, the question is whether balls with different airtimes can always be thrown in an order that keeps any two from landing on the same beat [16]. The theorem says such an order exists [3][6]. The thing it doesn't tell you is how to find one quickly, and a juggler needs the order itself. Whether any of the four papers yields a fast procedure is a separate matter from what Graham asked [2].
What to watch
- Whether any of the four papers gives a fast procedure for actually finding a collision-free order, since the conjecture only asks that one exist.
- Whether the methods reach clocks whose size is not prime, a case outside the conjecture as Quanta states it.
- Whether the random-process technique from Müyesser's 2022 work gets taken up on the design and Latin-square problems Alon grouped with this one.