TSTST 2013 Problem 3

Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m, n \in \mathbb{Z}$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1 / 4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x, y)$ be a point in the interior of any white square such that $r x-s y$ is irrational. Shoot a laser out of this point with slope $r / s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will from a closed loop.