Mathematicians Solve 55-Year-Old Graham Conjecture on Number Rearrangement
A collaborative effort by young researchers finally proves a long-standing mathematical puzzle.
1The Graham Conjecture Origins
In 1971, mathematician Ronald Graham proposed a puzzle involving the rearrangement of integers. He asked if any set of nonzero numbers could be reordered so that all partial sums remain unique. While simple for positive integers, the problem becomes complex when applied to finite clock arithmetic, where numbers repeat after prime intervals.
2Juggling and Mathematical Patterns
Graham, a renowned mathematician and skilled juggler, often explored the intersection of these two fields. He believed that the search for structure in juggling mirrored the search for patterns in mathematics. This specific conjecture was likely inspired by the mechanics of juggling, where balls must not crash down on the same beat.
3The Challenge of Clock Arithmetic
The conjecture operates within clock arithmetic, where numbers wrap around a prime number p. In this system, adding two positive numbers can result in zero, complicating the sequence. The difficulty of finding a valid arrangement depends on the set size relative to p, requiring sophisticated methods to ensure no partial sums repeat.
4Early Breakthroughs by Müyesser
Alp Müyesser of Oxford University began addressing the problem in 2022. He focused on sets containing nearly every number up to p. By using a random ordering and swapping out problematic sequences that added to zero, he demonstrated that randomness could effectively navigate the haystack of potential solutions to find valid arrangements.
5Collaborative Efforts at Oxford
Noah Kravitz and Benjamin Bedert approached the conjecture from the opposite end, focusing on tiny sets compared to p. After Müyesser shared his work, the researchers teamed up to extend the findings. Their combined efforts covered large and small sets, but a significant gap remained for medium-sized sets that defied their methods.
6Sauermann and Pham Step In
In February 2026, Lisa Sauermann and Huy Tuan Pham finalized the proof. The two mathematicians, who met at Stanford, collaborated during a visit to Germany. Sauermann had previously encountered a related problem during the International Mathematical Olympiad, providing her with unique insight into the structure of the puzzle and its potential solutions.
7The Power of Anti-Concentration
The duo employed a technically demanding method known as anti-concentration. This approach asserts that specific events have a low probability of occurring. By using Fourier analysis to rewrite functions as simple waves, they estimated the likelihood of bad events, proving that a valid, non-repeating arrangement is almost always possible to construct.
8Addressing Bad Events
The researchers identified three types of bad events that could cause a sequence to fail, such as zero-sum sequences appearing too close together. By proving these events were sufficiently unlikely, they showed that random orderings could be successfully rearranged to eliminate errors at least 90 percent of the time, solving the conjecture.
9A Four-Paper Solution
The final resolution of Graham’s conjecture spanned four distinct papers published over several years. While the proof assumes p is an extremely large prime number, it successfully closes the problem for all set sizes. This collective work highlights the power of modern probabilistic methods and the effectiveness of cross-disciplinary mathematical collaboration.
10Limitations and Practical Reality
While the proof is mathematically sound for large prime numbers, it does not provide a practical guide for real-world juggling. The prime numbers required for the proof are so vast that any corresponding juggling routine would be impossibly long. The result remains a theoretical triumph rather than a guide for physical performance.
11Legacy of the Conjecture
The resolution of this 55-year-old problem confirms that even within rigid, finite constraints, flexible structures exist. Fan Chung, Graham’s widow, noted that the late mathematician would have been pleased to see his puzzle solved. The proof stands as a testament to the enduring influence of Graham’s work on contemporary mathematical research.