MathNet 2014 Question 29

Divide the plane into an infinite square grid by drawing all the lines $x = m$ and $y = n$ for integers $m$ and $n$. 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 $rx - sy$ 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 form a closed loop.



Supplemental illustration; no extra hypothesis.