A billiards-like dynamical system for attacking chess pieces

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