From WIRED · Shalma Wegsman · · 2m
Math Problem Solved After 55 Years
Mathematicians finally solved a 55-year-old problem about rearranging numbers to avoid repeating sums.
In brief
Mathematicians finally solved a 55-year-old problem about rearranging numbers to avoid repeating sums. Mathematicians proved that any set of nonzero numbers can be rearranged to ensure all partial sums are unique, closing Ronald Graham's conjecture. Originally reported by WIRED.
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.
Here's the gist
Mathematicians proved that any set of nonzero numbers can be rearranged to ensure all partial sums are unique, closing Ronald Graham's conjecture.
