  We apply a one-dimensional discrete dynamical system originally considered by
Arnol'd reminiscent of mathematical billiards to the study of two-move riders,
a type of fairy chess piece. In this model, particles travel through a bounded
convex region along line segments of one of two fixed slopes.
  We apply this dynamical system to characterize the vertices of the inside-out
polytope arising from counting placements of nonattacking chess pieces and also
to give a bound for the period of the counting quasipolynomial. The analysis
focuses on points of the region that are on trajectories that contain a corner
or on cycles of full rank, or are crossing points thereof.
  As a consequence, we give a simple proof that the period of the bishops'
counting quasipolynomial is 2, and provide formulas bounding periods of
counting quasipolynomials for many two-move riders including all partial
nightriders. We draw parallels to the theory of mathematical billiards and pose
many new open questions.
