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

Ronald Graham’s 1971 rearrangement conjecture asked whether any collection of nonzero integers, viewed modulo a prime p (so numbers wrap around like a clock), can be ordered so that every partial sum is distinct - equivalently, no two partial sums coincide or any contiguous block sums to zero. The difficulty depends on the set’s size relative to p: tiny sets, huge sets, and medium-sized sets pose different obstacles. After decades without a proof, a sequence of advances by several young combinatorialists resolved the conjecture through a cascade of four papers, culminating in a final joint proof by Lisa Sauermann and Huy Tuan Pham published in February 2026.

The solutions rely centrally on randomness and probabilistic tools. Alp Müyesser and Alexey Pokrovskiy handled near-complete sets by randomly ordering most elements, reserving a few “spare” elements to break any zero-sum intervals that appeared; Noah Kravitz and Benjamin Bedert treated very small sets with complementary methods. Müyesser and collaborators then extended those ideas, leaving a persistent mid-size gap that Sauermann and Pham closed using delicate anti-concentration estimates: they showed problematic zero-sum patterns are exceedingly unlikely and devised swap-and-insert fixes when they occur. Together these probabilistic constructions and fixing procedures establish that every nonzero set mod p admits an ordering with all partial sums distinct, settling Graham’s conjecture.