  Given a (simple) grid polygon $P$ in a grid of equilateral triangles, Defant
and Jiradilok considered a billiards system where beams of light bounce around
inside of $P$. We study the relationship between the perimeter
$\operatorname{perim}(P)$ of $P$ and the number of different trajectories
$\operatorname{cyc}(P)$ that the billiards system has. Resolving a conjecture
of Defant and Jiradilok, we prove the sharp inequality $\operatorname{cyc}(P)
\leq (\operatorname{perim}(P) + 2)/4$ and characterize the equality cases.
