# Math Problem Solved After 55 Years

> Mathematicians finally solved a 55-year-old problem about rearranging numbers to avoid repeating sums.
- Title: Math Problem Solved After 55 Years · WIRED | Shortify
- Summary: Mathematicians finally solved a 55-year-old problem about rearranging numbers to avoid repeating sums.
- Keywords: ronald graham, math conjecture, partial sums, juggling, clock arithmetic, random ordering, anti-concentration, fourier analysis, mathematical proof, graham conjecture, lis sauermann, huy tuan pham
- Source: WIRED — https://wired.com/story/a-55-year-old-math-problem-was-solved-this-time-by-humans
- Author: Shalma Wegsman
- Published: 2026-11-08T10:30:00+00:00
- Read time: 2 min
- Topics: ronald graham, math conjecture, partial sums, juggling, clock arithmetic, random ordering, anti-concentration, fourier analysis, mathematical proof, graham conjecture, lis sauermann, huy tuan pham
## Ronald Graham's Dual Life
Ronald Graham was a renowned mathematician and juggler who loved patterns and structure.

> He loved tricks
## Graham's 1971 Conjecture
Graham asked if a set of nonzero integers could always be rearranged so all partial sums are unique.
## Why the Problem Was Hard
The problem was tricky because numbers wrap around like a clock, repeating after a prime number.
## The Proof's Key Idea
Mathematicians used randomness to find patterns, avoiding zero-sum sequences that repeat sums.
## Müyesser and Pokrovskiy's Work
They tackled large sets using random ordering, fixing zero-sum sequences with spare numbers.

> Computer scientists often call this a finding the hay in the haystack problem
## Kravitz and Bedert's Approach
They solved the problem for tiny sets by showing random orderings avoid repeating sums.
## The Medium-Size Gap
No one could solve the problem for medium-sized sets, where the method didn't work.
## Sauermann and Pham's Breakthrough
They used anti-concentration and Fourier analysis to prove the conjecture for all set sizes.
## The Final Proof
Sauermann and Pham's 27-page proof showed random orderings work 90% of the time to avoid repeating sums.
## The Math Behind It
The proof works for very large prime numbers but not all cases, requiring a huge set size.
## Why It Matters
The solution shows flexibility exists even in constrained number systems, avoiding repeating sums.
## The Juggling Connection
Graham's problem was inspired by juggling, where balls must stay in the air without overlapping.
## The Proof's Impact
The solution confirms that even in limited number settings, flexible patterns always exist.
## The Math Community's Reaction
Mathematicians were surprised by how quickly Sauermann and Pham closed the remaining case.
## The Final Result
The four papers prove Graham's conjecture for all set sizes, assuming a very large prime number.
## What's Next
The problem remains unsolved for all prime numbers, but the solution works for practical juggling routines.
## Key takeaway

Mathematicians proved that any set of nonzero numbers can be rearranged to ensure all partial sums are unique, closing Ronald Graham's conjecture.