Graham’s conjecture (1971): Modulo some prime p, can you find a way to order any subset of {1,...,p-1} so that its successive partial sums are all distinct?
Since an initial breakthrough in 2022, three groups of young mathematicians have proved* this for different regimes: large subsets (as a fraction of p), small subsets, and finally those of intermediate size, covering all cases.
*Like many results in combinatorics, the proofs only hold for sufficiently large p.
“It’s interesting, in fact, that many mathematicians and computer scientists have an interest in juggling,” the late mathematician Ronald Graham said in a 1980 television interview. “I think it’s the search for patterns and structure that is responsible for this.” https://www.quantamagazine.org/mathematicians-harness-randomness-to-crack-a-55-year-old-conjecture-20260928/