Mathematicians Harness Randomness To Crack a 55-Year-Old Conjecture

The conjecture was proposed in 1971 by Ronald Graham, a mathematician known for his work in combinatorics and juggling.

Graham, who served as president of both the American Mathematical Society and the International Jugglers’ Association, often drew inspiration from juggling to explore mathematical patterns.

His conjecture asked whether, in a finite number system (like numbers on a clock face), any set of nonzero integers could be rearranged so that all partial sums are unique.

For example, if you have numbers like 1, 2, and 3 on a clock with 7 hours (where 7 = 0), can you arrange them so that 1, 1+2=3, and 1+2+3=6 are all distinct?

This problem mirrors juggling, where balls thrown at different times must not collide mid-air.