In geometric representation theory, one often wishes to describe
representations realized on spaces of invariant functions as trace functions of
equivariant perverse sheaves. In the case of principal series representations
of GL_N over a local field, there is a description of families of these
representations realized on spaces of functions on GL_N invariant under the
translation action of the Iwahori subgroup, or a suitable smaller compact open
subgroup, studied by Howe, Bushnell and Kutzko, Roche, and others.

We study in detail the so-called looping case of Mozes's game of numbers,
which concerns the (finite) orbits in the reflection representation of affine
Weyl groups situated on the boundary of the Tits cone. We give a simple proof
that all configurations in the orbit are obtainable from each other by playing
the numbers game, and give a strategy for going from one configuration to
another. The strategy gives rise to a partition of the finite Weyl group into
finitely many graded posets, one for each extending vertex of the associated
extended Dynkin diagram.