Strong Convergence and a Game of Numbers

Abstract S. Mozes investigated a certain solitary game played on a weighted graph. Numbers are placed on the nodes of the graph, and a move consists of changing the sign of a negative number and changing the numbers on the neighboring nodes according to the weights on the edges. Mozes proved that the game has a strong convergence property when the edges have certain positive integer weights. However, his approach would give no information in the case of other weights. In this paper we first prove that strong convergence is equivalent to the fact that the game has as a certain ‘polygon property’. We can then, in a rather elementary way, characterize the assignments of weights that imply the polygon property, and hence strong convergence. Finally, we make a natural generalization of the game, where we also have weights on the nodes. The conditions for strong convergence generalize nicely to this game.