Numerous bamboo species collectively flower and seed at dramatically extended, regular intervals – some as long as 120years. These collective seed releases, termed masts’, are thought to be a strategy to overwhelm seed predators or to maximise pollination rates. But why are the intervals so long, and how did they evolve? We propose a simple mathematical model that supports their evolution as a two-step process: First, an initial phase in which a mostly annually flowering population synchronises onto a small multi-year interval. Second, a phase of successive small multiplications of the initial synchronisation interval, resulting in the extraordinary intervals seen today. A prediction of the hypothesis is that mast intervals observed today should factorise into small prime numbers. Using a historical data set of bamboo flowering observations, we find strong evidence in favour of this prediction. Our hypothesis provides the first theoretical explanation for the mechanism underlying this remarkable phenomenon.