Graham's 55-Year-Old Rearrangement Conjecture Is Finally Solved

For positive numbers the answer is obviously yes, and it is also known for mixed positive and negative numbers. Graham asked about numbers on a clock, where values repeat after a prime number p. He conjectured the answer is still yes, in line with how valid sudoku boards or Latin squares exist despite many constraints.

Randomness and an unlikely team

The difficulty depends on the size of the set relative to p. In 2022, Alp Müyesser of Oxford and Alexey Pokrovskiy of University College London handled sets containing almost every number up to p. They set a few numbers aside, scrambled the rest randomly, and used the spare numbers to repair any stretch that summed to zero.

Unaware of that work, Noah Kravitz of Oxford found the conjecture in an online archive of unsolved problems. With Benjamin Bedert, he solved the case where the set is tiny compared to p, such as 100 numbers with p at 1 billion, and posted it in September 2024. Müyesser then joined them, with two other colleagues, and their August 2025 paper covered more of the large-set cases. A gap remained for medium-size sets, which Sauermann and Pham later closed.

Noga Alon of Princeton credited "the power of collaboration, the power of the young generation, the power of probabilistic methods." Kravitz noted that the team combined people from areas of combinatorics that rarely work together.