Graham’s Conjecture Proved: Clock Arithmetic Puzzle Solved

The conjecture operates within the framework of clock arithmetic, where numbers repeat after a prime number p. In this system, adding two positive integers can result in zero. Ronald Graham originally asked if any set of nonzero numbers could be rearranged such that every partial sum—the running total as you add the numbers one by one—remains unique. The difficulty of this task scales with the size of the set relative to p; as the set grows, the number of potential sum collisions increases, creating a significant computational and theoretical bottleneck.

Randomization Strategies for Large-Scale Sets

For sets containing nearly every possible number up to p, constructing a valid ordering is computationally intensive. Müyesser and Alexey Pokrovskiy of University College London determined that a purely random ordering provides a strong baseline but is rarely perfect. To resolve this, they implemented a “spare number” protocol: they isolated a small group of specially chosen numbers and randomly scrambled the remainder of the set. When the researchers encountered an interval that summed to zero—which would cause a partial sum to repeat—they inserted one of the reserved spare numbers to break the sequence and shift the sum.

This method addresses what Müyesser describes as a “finding the hay in the haystack” problem. While valid orderings are statistically likely to exist, describing them explicitly is difficult.

// Conceptual logic for avoiding zero-sum intervals in a sequence
function resolveGrahamSequence(set, p) {
    let { spares, mainSet } = splitSet(set);
    let ordering = shuffle(mainSet);
    
    for (let i = 0; i < ordering.length; i++) {
        if (isZeroSumInterval(ordering, i)) {
            let spare = spares.pop();
            ordering.splice(i, 0, spare);
            i++; // Skip the inserted element
        }
    }
    return ordering;
}

Bridging the Gap Between Small and Medium Sets

The complete proof required merging disparate combinatorial techniques. Noah Kravitz and Benjamin Bedert of the University of Oxford tackled the opposite end of the spectrum: cases where the set is tiny compared to p (e.g., 100 numbers in a group of 1 billion). They posted this proof in September 2024. Following this, Kravitz and Müyesser collaborated in August 2025 to extend the random-ordering approach to larger sets, though a "medium-size" gap remained for sets comprising roughly half of p.

The final resolution arrived in February 2026. Lisa Sauermann and Huy Tuan Pham, who had previously collaborated at Stanford University, utilized a shared chalkboard in Bonn, Germany, to bridge this remaining gap. Their work integrated the previous findings into a unified proof, ending the hiatus on the conjecture.

This progression from specific edge cases to a general solution reflects the standard lifecycle of complex algorithm development.

Disclaimer: The technical analyses and security protocols detailed in this article are for informational purposes only. Always consult with certified IT and cybersecurity professionals before altering enterprise networks or handling sensitive data.