# A billiards-like dynamical system for attacking chess pieces

## TL;DR

This paper introduces a billiards-inspired dynamical system to analyze nonattacking chess piece arrangements, providing new bounds and proofs for counting functions related to two-move fairy chess pieces.

## Contribution

It applies a billiards-like model to characterize polytope vertices and bound periods of counting quasipolynomials for two-move riders, offering new insights and proofs.

## Key findings

- Proves the period of bishops' counting quasipolynomial is 2
- Provides bounds for periods of two-move rider counting quasipolynomials
- Draws parallels between billiard dynamics and chess piece configurations

## Abstract

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.