  The numbers game is a one-player game played on a finite simple graph with
certain "amplitudes" assigned to its edges and with an initial assignment of
real numbers to its nodes. The moves of the game successively transform the
numbers at the nodes using the amplitudes in a certain way. Here, the edge
amplitudes will be negative integers. Combinatorial methods are used to
investigate the convergence and divergence of numbers games played on certain
such graphs. The results obtained here provide support for results in a
companion paper.
