In February, Professor Pham Tuan Huy and mathematician Lisa Sauermann uploaded a 27‑page paper called “On Graham’s rearrangement conjecture” to arXiv, finally settling a puzzle that has lingered for more than five decades.

The puzzle goes back to 1971, when Ronald Graham asked whether any set of distinct non‑zero numbers can be reordered so that every partial sum – the sum of the first k numbers for each k – is different.

The question is easy when all numbers are positive, because the partial sums then strictly increase. The real challenge appears in finite cyclic groups of prime order, where numbers wrap around and sums can repeat.

Huy and Sauermann tackled the problem with a probabilistic tool called anticoncentration. By applying Fourier analysis, they showed that the distribution of random sums spreads out enough that “bad” configurations are rare.

They calculated the probability of each undesirable outcome and proved that the total probability stays below 100 %. This bound guarantees that at least one ordering avoids all collisions, i.e., a valid rearrangement exists.

Combined with earlier partial results, their argument completes Graham’s conjecture for all sufficiently large primes. The work demonstrates how modern techniques such as Fourier‑based anticoncentration can resolve long‑standing combinatorial questions.