A group of young mathematicians has finally found the solution to the long unanswered Graham's rearrangement conjecture. Originally inspired by the mathematical patterns found in juggling, this problem remained unsolved for over five decades. By utilizing probabilistic methods and embracing the power of collaborative friendship, these researchers bridged the gap between randomness and structured arithmetic. Their breakthrough provides a definitive answer to the 1971 question regarding partial sums in clock arithmetic, marking a historic moment for the mathematical community and proving that human intuition remains vital in the age of computation.
The mathematical foundation of the conjecture
The problem centers on the properties of partial sums within the constraints of clock arithmetic, where numbers repeat after a certain prime number, $p$. In this setting, adding certain values can result in zero, creating a significant challenge for maintaining unique sums. Ronald Graham's original question asked if, for any set of non-zero numbers chosen from such a system, an ordering exists where every cumulative sum remains unique.
To understand the complexity, consider the following conditions:
- Positive integers: If all numbers are positive, the answer is trivial because sums always increase.
- Mixed integers: If the set contains both positive and negative numbers, the answer is also known to be yes.
- Finite cyclic groups: The difficulty arises when numbers are wrapped around a clock face, where the numbers repeat after a certain count.
In this setting, 0, 7, 14, and all other multiples of 7 are equivalent—meaning that you can add two positive numbers (like 3 and 4) and get zero. The challenge of the conjecture depends heavily on the size of the set relative to $p$. The more numbers you pick, the more sums there are to manage; however, if you choose fewer numbers, there are fewer ways to rearrange them. These different scales of set size inspire different mathematical approaches.
The legacy of Ronald Graham
The late Ronald Graham was a figure who occupied two distinct worlds: he was a renowned mathematician and former president of the American Mathematical Society, but he was also a serious juggler and president of the International Jugglers’ Association. His passion for both fields was not a coincidence; he often noted that the interest mathematicians and computer scientists have in juggling stems from a shared search for patterns and structure.
Graham's mathematical work often intersected with his hobby. He wrote numerous papers, some in collaboration with his wife, mathematician Fan Chung, on the mathematics of juggling. Even his 1971 conjecture is believed by some to have been inspired by the mechanics of the craft. In juggling terms, the conjecture asks if, given a set of balls where each stays in the air for a different amount of time, one can always find an order to throw them so that two balls do not land on the same beat, which would ruin the performance.
Graham believed that even within rigid constraints, such as those found in clock arithmetic, there is enough flexibility to construct special patterns or structures. He viewed the problem as fitting into the broader mathematical questions regarding designs and highly symmetric structures. For decades, however, this intuition remained unproven.
A multi-generational breakthrough
The resolution of Graham’s rearrangement conjecture did not come from a single stroke of genius, but through a series of papers that spanned multiple fields of mathematics. The breakthrough was driven by a new generation of mathematicians, culminating in a final paper published in February 2026 by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago. This final work officially closed a problem that had remained open for 55 years.
Noga Alon, a mathematician at Princeton University, characterized the solution as a triumph of several forces: "It’s the power of collaboration, the power of the young generation, the power of probabilistic methods" that finally solved the problem. The proof process highlighted the utility of randomness in uncovering order within complex systems.
Probabilistic methods and the 'haystack' problem
One of the key drivers in the solution was the application of probabilistic methods—using randomness to find patterns. Alp Müyesser, a mathematician at the University of Oxford, noted that many problems require a combination of a random process and an additional specific ingredient. After solving a similar problem in 2022 as a graduate student, Müyesser realized his techniques could be applied to Graham's conjecture.
Müyesser and his former adviser, Alexey Pokrovskiy of University College London, focused on the case where the set includes almost every possible number up to $p$. Dealing with such large sets makes constructing a valid ordering extremely difficult. Müyesser described this as a "finding the hay in the haystack" problem: while many valid orderings likely exist, explicitly describing or finding one is a massive challenge.
To navigate this, Müyesser and Pokrovskiy utilized a strategy of partial randomness. A completely random ordering might accidentally create a sequence of numbers that adds up to zero, which would cause a repeated partial sum. To prevent this, they first set aside a few specially chosen numbers from the set and then randomly scrambled the remaining numbers, attempting to rectify the order to ensure no sequence of numbers anywhere in the ordering summed to zero.