Science & Tech · Science & Environment · 7 hrs ago
Mathematicians Solve Ronald Graham’s 55-Year-Old Conjecture
A group of young mathematicians solved a 55-year-old problem posed by Ronald Graham in 1971.
The conjecture asks whether a set of distinct, nonzero integers can always be rearranged so that each successive partial sum is different.
The question concerns numbers in a finite system where values wrap around and repeat, rather than ordinary positive and negative numbers.
Graham was also a skilled juggler, and the problem has a juggling analogy: arrange throws so balls do not land at the same time.
Randomness and collaboration helped produce the proof across four papers and several areas of mathematics.
The final paper, by Lisa Sauermann of the University of Bonn and Huy Tuan Pham of the University of Chicago, appeared in February 2026.
The proof closes a longstanding question about whether such rearrangements are always possible.
Mathematicians have proved Ronald Graham’s conjecture about rearranging integers so that every partial sum is different.
The conjecture was posed in 1971 and had remained unresolved for decades.
The solution came through four papers using ideas from several fields of mathematics.
The final paper, by Lisa Sauermann and Huy Tuan Pham, appeared in February 2026.
The problem may have been inspired by juggling, a subject Graham also studied.
- Who
- Young mathematicians, including Lisa Sauermann and Huy Tuan Pham, proved the conjecture.
- What
- They resolved Graham’s conjecture that a set of distinct nonzero integers can be ordered so every partial sum is different.
- When
- The conjecture was posed in 1971; the final paper appeared in February 2026.
- Where
- Not stated.
- Why
- The proof used randomness and methods from several fields of mathematics.
This story does not have two clearly opposing sides.
Ronald Graham posed the rearrangement conjecture.
Alp Müyesser solved a problem using a proof he later realized could help with Graham’s conjecture.
The final paper resolving the conjecture, by Lisa Sauermann and Huy Tuan Pham, appeared.
- Original conjecture
- Posed by Ronald Graham in 1971
- Final paper
- Published in February 2026
- Final paper authors
- Lisa Sauermann and Huy Tuan Pham
- Proof
- Spanned four papers and used ideas from various fields of mathematics
What people are saying
He loved tricks
It fits nicely in all these questions about designs and about very symmetric structures
It’s the power of collaboration, the power of the young generation, the power of probabilistic methods