We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. We shall consider the finite setting, i.e. when the underlying graph is finite and connected. In this case it is known that coexistence is not possible between two types. However, for competition between three or more types, the possibility of coexistence depends on the underlying graph. We prove a conjecture stating that when the underlying graph is a cycle, then the competition between three or more types has a single survivor almost surely. As part of the proof 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 Möbius strip in a discrete setting. (Joint work with Carolina Fransson.)