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.

我们将最初由Arnol想到的数学台球考虑的一维离散动力系统应用于研究两步骑手（一种神仙棋子）。在此模型中，粒子沿着两个固定坡度之一的线段穿过有界凸区域。

我们应用该动力学系统来表征由无攻击棋子的计数位置引起的由内而外的多面体的顶点，并为计数拟多项式的周期提供一个边界。分析集中在区域的点上，该点在包含拐角的轨迹上或在满秩的循环上，或者在其交叉点上。

因此，我们给出一个简单的证明，即主教的准多项式计数周期为2，并提供了许多包括所有部分夜行者在内的两步骑手的准多项式计数周期的公式。我们将与数学台球理论相提并论，并提出了许多新的开放性问题。