Problem 495

Since the given grid is $6 \times 6$, the size of each of the small squares is $1 \times 1$. This means that $QR = PS = 3$. Join $Q$ to $S$. [[IMAGE0]] Since $QS$ is vertical, and $QR$ and $PS$ are both horizontal, then $\angle RQS = 90^\circ$ and $\angle PSQ = 90^\circ$. We note further that $QS = 4$. Since $\triangle RQS$ is right-angled at $Q$, by the Pythagorean Theorem, $$RS^2 = QR^2 + QS^2 = 3^2 + 4^2 = 25$$ Since $RS>0$, then $RS = 5$. Similarly, $PQ = 5$. Thus, the perimeter of $PQRS$ is $PQ + QR + RS + PS = 5 + 3 + 5 + 3 =
16$.