Mozes' Game of Numbers on Directed Graphs

Abstract

In 1986, the contestants of the $27$th International Mathematical Olympiad were given a game of numbers played on a pentagon. In 1987, Mozes generalized this game to an arbitrary undirected, weighted, connected graph. The convergence properties and total number of moves of any convergent game have been resolved by Mozes using Weyl groups. Eriksson provided an alternate proof using matrix theory and graph theory. In this paper, we briefly discuss the results of Mozes and Eriksson on undirected graphs. Then we generalize this game to arbitrary directed, strongly connected graphs and investigate the convergence properties of the game of numbers.