In the upper echelons of mathematical research, breakthroughs rarely emerge from a single stroke of solitary genius. Instead, they often resemble a relay race—a baton passed across decades, continents, and generations of brilliant minds. Such is the story behind the resolution of Ronald Graham’s 55-year-old rearrangement conjecture. Posed initially in 1971 by the late, multifaceted mathematician and world-renowned juggler Ronald Graham, the problem lived in obscurity for half a century before a collaborative convergence of young mathematical talent finally cracked it.
The journey culminated in a series of four interconnected papers, ending with a definitive 27-page proof published in February 2026 by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago. Together, these researchers dismantled Graham’s puzzle, proving that even within the rigid, looping confines of finite modular arithmetic, unexpected flexibility and structural harmony can always be engineered through the clever application of randomness.
This comprehensive report details the history of Graham’s conjecture, the probabilistic revolution that revitalized it, the disparate camps of researchers who chipped away at its massive structural walls, and the mathematical machinery—ranging from Fourier analysis to anti-concentration techniques—that brought a 55-year-old mystery to a triumphant close.
Detailed Chronology
1971: The Genesis of the Conjecture
The roots of the problem stretch back to 1971. Ronald Graham, who split his professional identity between serving as president of the American Mathematical Society and presiding over the International Jugglers’ Association, possessed a unique perspective on patterns. Graham famously noted that the minds drawn to juggling and advanced mathematics share a common obsession: the pursuit of hidden structures and rhythmic regularities.
Decades before he and his wife, mathematician Fan Chung of UC San Diego, published formal papers on the mathematics of juggling, Graham formulated a
deceptively simple abstract question. Imagine a random set of distinct, non-zero integers. Can you always rearrange this set such that every sequential "partial sum"—the sum of the first two numbers, the first three, the first four, and so on—yields a completely distinct value?
Translated into the physical reality of juggling, Graham’s question asks whether a juggler tossing balls with varying airborne durations can always find a throwing sequence that prevents any two balls from crashing down on the exact same beat.
The Positive and Negative Realms
For purely positive numbers, the answer was trivial: the partial sums grow larger monotonically, ensuring no repetitions. Similarly, when positive and negative integers coexist on an infinite number line, the expansive space prevents collisions.
However, Graham escalated the difficulty by dragging the problem into a finite world: clock arithmetic (modular arithmetic). By wrapping the number line around a face governed by a prime number $p$, numbers begin to repeat. Within this circular universe, multiples of $p$ act as zero, and addition can yield counterintuitive results (such as $3 + 4 equiv 0$ when modulo 7). Graham hypothesized that even within these deeply constrained, repetitive environments, a valid, collision-free arrangement of partial sums could always be found. Yet, for decades, intuition remained just that—unproven and out of reach.
2022–2025: The New Generation Takes the Field
The silence surrounding Graham’s conjecture broke in the 2020s as a wave of young researchers began revisiting the problem using probabilistic methods.
In 2022, Alp Müyesser (then a graduate student at the University of Oxford, now at the same institution) and his advisor, Alexey Pokrovskiy of University College London, examined large sets comprising almost every integer up to $p$. They realized that while constructing an explicit, deterministic solution was nearly impossible, a random ordering would almost certainly work—if paired with a corrective safety net. They embedded spare numbers into a randomized sequence to iron out any "zero-sum" intervals that would otherwise ruin the partial sums.
Unaware of this work, Noah Kravitz of Oxford encountered Graham’s conjecture in an online repository in 2024. Viewing the open problem as an "embarrassment for humanity," Kravitz teamed up with Oxford colleague Benjamin Bedert to tackle the opposite end of the spectrum: tiny sets of numbers relative to a massive prime $p$.
When Müyesser and Kravitz discovered each other’s complementary research, they joined forces. Partnering with additional colleagues, they published a paper in August 2025 that expanded the scope to handle larger sets. Still, a glaring mathematical desert remained: medium-sized sets, such as those encompassing roughly half the prime $p$, resisted all existing techniques.
February 2026: The Final Piece of the Puzzle
The breakthrough for medium-sized sets arrived unexpectedly through an old academic friendship. Lisa Sauermann and Huy Tuan Pham, who first met at Stanford University in 2015, reunited at a conference in Germany in September 2025. During the event, they attended presentations detailing the roadblocks in solving Graham’s conjecture.
Unbeknownst to most, Sauermann possessed a secret weapon: as a high school student winning four gold medals at the International Mathematical Olympiad, she had successfully solved a closely related problem. (That problem had likely been crafted by Fan Chung, who frequently mined Graham’s repertoire for competition puzzles.)
Over a three-day visit to Bonn following the conference, Sauermann and Pham formulated a bold strategy. By February 2026, they posted a 27-page paper on the arXiv preprint server, officially closing the final gap and cementing a complete proof for Graham’s 55-year-old conjecture.
Supporting Context & Metrics
To appreciate the scale of this mathematical triumph, it is necessary to examine the mechanics of the proofs and the quantitative realities of the methodologies employed.
[Graham's Conjecture Life Cycle]
1971: Ronald Graham poses the modular rearrangement question.
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2022: Müyesser & Pokrovskiy introduce random ordering with correction for large sets.
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2024: Kravitz & Bedert solve the tiny-set scenario.
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2025: Combined team extends results, leaving medium-sized sets unresolved.
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Feb 2026: Sauermann & Pham deploy anti-concentration & Fourier analysis, closing the final gap.The Anatomy of Randomness and Failure Modes
The collective strategy across the four defining papers relied on a counterintuitive premise: to bring order out of chaos, you must first embrace chaos. Instead of attempting to hand-craft a rigid sequence, researchers generated random orderings of sets.
However, complete randomness is rarely flawless. In the context of modular arithmetic, a random ordering inevitably introduces three primary "bad events" that threaten to derail the partial sums:
- Late-Stage Zero-Sums: A sequence of numbers near the tail end of the arrangement sums to zero, leaving no remaining numbers downstream to execute a corrective swap.
- Dense Clustering: Multiple zero-sum sequences cluster too closely together, exhausting the system’s capacity to adjust without causing cascading errors.
- Induced Collisions: Fixing an initial problematic sequence inadvertently creates a brand-new zero-sum sequence further down the line.
The Power of Anti-Concentration and Fourier Analysis
While earlier researchers suspected that these bad events were statistically rare, they lacked the technical framework to prove it safely. Sauermann and Pham bridged this gap by utilizing anti-concentration statements—rigorous bounds proving that specific outcomes have an infinitesimally low probability of occurring.
To establish these bounds, they turned to Fourier analysis, a mathematical tool typically deployed to dissect continuous waves and signals. By translating the discrete distribution of number sums into wave functions, they proved that when adding random sets of numbers, no single sum dominates or concentrates too heavily in any localized region.
Through these precise estimations, Sauermann and Pham calculated that the total cumulative probability of encountering a fatal bad event was strictly less than 100 percent. In fact, their randomized correction protocol successfully eliminated bad configurations at least 90 percent of the time, a staggering success rate that guaranteed the existence of a valid, collision-free arrangement.
Official Statements & Expert Perspectives
The resolution of a half-century-old conjecture has sent ripples of excitement through the global combinatorial mathematics community. Leading figures have praised both the tenacity of the young researchers and the elegance of the collaborative network that made the breakthrough possible.
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On the Nature of the Problem:
"It fits nicely in all these questions about designs and about very symmetric structures… It often happens that even when dealing with rigid constraints, you can still find enough flexibility to construct special patterns or structures."
— Noga Alon, Mathematician, Princeton University -
On Collaborative Discrepancy:
"It was a pretty unlikely combination of people… [We] come from two areas of combinatorics that don’t typically collaborate. Different sections have completely different techniques."
— Noah Kravitz, University of Oxford -
On the Final Breakthrough:
"Their approach is just completely different… It’s the power of collaboration, the power of the young generation, the power of probabilistic methods that solved the problem."
— Alp Müyesser, University of Oxford (reflecting on Sauermann and Pham’s contribution) -
On Ronald Graham’s Legacy:
"To pose a good problem is really an art. I think Ron would be extremely happy to see the problem solved."
— Fan Chung, UC San Diego, mathematician and wife of the late Ronald Graham
Future Outlook & Limitations
While the publication of Sauermann and Pham’s paper in February 2026 marks the theoretical closure of Graham’s rearrangement conjecture across all set sizes, the work opens new avenues for theoretical inquiry while presenting amusing practical limitations.
The Scale of the Prime $p$
A crucial caveat of the combined four-paper proof is its reliance on the assumption that the prime modulus $p$ is extraordinarily large—imaginatively conceived as a number on the scale of $10^100$ (a one followed by a hundred zeros) or greater. To a pure mathematician, the exact magnitude of $p$ is irrelevant; the proof establishes that the structural property holds true within the framework of modular arithmetic for sufficiently large spaces.
Practical Juggling Implications
For those hoping to translate this mathematical triumph into a physical spectacle, reality provides a sobering check. Because the proof relies on a mind-bogglingly large prime $p$ to guarantee the statistical absence of collisions, any corresponding juggling routine would require a duration far beyond human physical capacity. You will not be seeing a human juggler execute a Graham-conjecture-optimized routine on stage anytime soon.
Broader Implications for Computer Science and Combinatorics
Beyond the abstract realm of clock arithmetic, the methodologies forged during this multi-year effort hold immense value for computer science. The challenge of navigating massive combinatorial search spaces—colloquially framed as "finding the hay in the haystack"—appears in fields ranging from cryptography and coding theory to network routing and artificial intelligence.
By demonstrating that probabilistic methods and anti-concentration analysis can tame rigid constraints, these young mathematicians have provided a blueprint for tackling similarly stubborn riddles across the mathematical sciences. Ronald Graham’s enduring playground of patterns, tricks, and puzzles continues to prove that even in the most chaotic arrangements of numbers, hidden harmony is waiting to be uncovered.
