  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. Combinatorial
reasoning is used to show that those connected graphs with negative integer
amplitudes for which the numbers game meets a certain finiteness requirement
are precisely the Dynkin diagrams associated with the finite-dimensional
complex simple Lie algebras. This strengthens a result originally due to the
second author. A more general result is obtained when certain real number
amplitudes are allowed. The resulting graphs are in families, each family
corresponding to a finite irreducible Coxeter group. These results are used to
demonstrate that the only generalized Cartan matrices for which there exist
finite edge-colored ranked posets enjoying a certain structure property are the
Cartan matrices for the finite-dimensional complex semisimple Lie algebras. In
this setting, classifications of the finite-dimensional Kac--Moody algebras and
of the finite Coxeter and Weyl groups are re-derived.

Combinatorics