Graham’s rearrangement conjecture

Prove that for every prime p and every set of nonzero, distinct elements of \(\mathbb{Z}_p\), the elements can be rearranged so that all partial sums are distinct.

Background

Graham’s rearrangement problem asks whether every subset of nonzero elements of a prime cyclic group admits an ordering whose successive partial sums are pairwise distinct. The paper identifies this as the original conjecture motivating its asymptotic results, but does not establish the exact statement.

References

For any $p$ prime and $a_1, a_2, \dots, a_d$ non-zero distinct elements of $\mathbb{Z}p$, there exists a rearrangement of the elements as $a{i_1}, a_{i_2}, \dots, a_{i_d}$ such that all partial sums $\sum_{j=1}t{a_{i_j}$, $1\leq t \leq d$ are distinct.

— Towards Graham's rearrangement conjecture via rainbow paths  (2503.01825 - Bucić et al., 3 Mar 2025) in Conjecture 1, Section 1