Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture
After a long hiatus, the problem, which was likely inspired by juggling, has finally been resolved by a group of young mathematicians. The post Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture first appeared on Quanta Magazine.
The conjecture was proposed in 1971 by Ronald Graham, a mathematician known for his work in combinatorics and juggling. Graham, who served as president of both the American Mathematical Society and the International Jugglers’ Association, often drew inspiration from juggling to explore mathematical patterns. His conjecture asked whether, in a finite number system (like numbers on a clock face), any set of nonzero integers could be rearranged so that all partial sums are unique. For example, if you have numbers like 1, 2, and 3 on a clock with 7 hours (where 7 = 0), can you arrange them so that 1, 1+2=3, and 1+2+3=6 are all distinct? This problem mirrors juggling, where balls thrown at different times must not collide mid-air.
Clock arithmetic (also called modular arithmetic) is a number system where numbers wrap around after reaching a certain value, called the modulus (denoted as p). For instance, in a 7-hour clock (p = 7), the numbers 0, 7, 14, etc., are equivalent to 0, while 8 is equivalent to 1. This system creates a finite loop of numbers, making it different from the infinite number line. Graham’s conjecture operates in this finite world, where the challenge is to avoid repeating partial sums when adding numbers in any order. The conjecture was unresolved for decades because the finite nature of p introduces constraints that don’t exist in infinite systems.
In 2022, mathematicians Alp Müyesser and Alexey Pokrovskiy made progress by using randomness to solve part of the conjecture. They focused on cases where the set of numbers is large compared to p. Their approach involved randomly shuffling the numbers and then fixing any problematic sequences that summed to zero. For example, if a sequence like 2, 3, and 2 (which sums to 7 = 0 in a 7-hour clock) appeared, they would swap one number to break the zero-sum. This method worked for large sets but struggled with medium-sized sets. Around the same time, Noah Kravitz and Benjamin Bedert independently solved the conjecture for small sets by considering cases where the number of elements is tiny compared to p, such as 100 numbers in a 1 billion-hour clock.
After Müyesser and Pokrovskiy’s 2022 solution and Kravitz and Bedert’s 2024 proof, a gap remained for medium-sized sets—those with roughly half as many numbers as p. Neither approach could bridge this gap, leaving the conjecture partially unresolved. Then, in September 2025, mathematicians Lisa Sauermann and Huy Tuan Pham met at a conference in Germany and decided to tackle the remaining case. Sauermann, a four-time gold medalist in the International Mathematical Olympiad, had encountered a related problem as a high school student. Pham, her former classmate, joined her in Bonn for a three-day visit to work on the conjecture.
Sauermann and Pham’s breakthrough relied on a technique called anti-concentration, which estimates the probability that a random process produces an unlikely event. They combined this with Fourier analysis, a mathematical tool that breaks down complex patterns into simple waves to study their properties. Their method involved randomly ordering the numbers and then fixing any zero-sum sequences by swapping elements. They proved that the chance of encountering a problematic sequence was less than 100%, ensuring a valid rearrangement at least 90% of the time. This approach was technically demanding and differed significantly from previous methods, surprising other mathematicians who had worked on the problem.
In February 2026, Sauermann and Pham published a 27-page proof that resolved the conjecture for all set sizes, assuming *p* is extremely large (e.g., 10^100). While the conjecture is now proven in the context of clock arithmetic, two technicalities remain: the proof doesn’t cover all possible values of *p*, and it doesn’t translate to practical applications like juggling routines due to the enormous size of *p*. The proof confirms that even in constrained systems, flexible structures exist to avoid repeating partial sums. Ronald Graham, who passed away before the solution, would likely have been pleased with the outcome, as the conjecture reflected his belief in finding order within complexity.
