Periodical states and marching groups in a closed owari

Owari is an old African game that consists of cyclically ordered pits that are filled with pebbles. In a sowing move all the pebbles are taken out of one pit and distributed one by one in subsequent pits. Repeated sowing will give rise to recurrent states of the owari. Bouchet studied such periodical states in an idealised setup, where there are infinitely many pits. We characterise periodical states in owaris with finitely many pits. Our result implies Bouchet's result.