  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. This game has been
studied previously by Proctor, Mozes, Bjorner, Eriksson, and Wildberger. We
show that those connected such graphs for which the numbers game meets a
certain finiteness requirement are precisely the Dynkin diagrams associated
with the finite-dimensional complex simple Lie algebras. As a consequence of
our proof we obtain the classifications of the finite-dimensional Kac-Moody
algebras and of the finite Weyl groups. We use Coxeter group theory to
establish a more general result that applies to Eriksson's E-games: an E-game
meets the finiteness requirement if and only if a naturally associated Coxeter
group is finite. To prove this and some other finiteness results we further
develop Eriksson's theory of a geometric representation of Coxeter groups and
observe some curious differences of this representation from the standard
geometric representation.
