  We study a system of interacting urns where balls of different colour/type
compete for their survival, and annihilate upon contact. For competition
between two types, the underlying graph (finite and connected), determining the
interaction between the urns, is known to be irrelevant for the possibility of
coexistence, whereas for $K\ge3$ types the structure of the graph does affect
the possibility of coexistence. We show that when the underlying graph is a
cycle, competition between $K\ge3$ types almost surely has a single survivor,
thus establishing a conjecture of Griffiths, Janson, Morris and the first
author. Along the way, we give a detailed description of an auto-annihilative
process on the cycle, which can be perceived as an expression of the geometry
of a M\"obius strip in a discrete setting.
