Problem 503

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 $S$.

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 $\text{RS gt;0}$, then
$RS = 5$.

Similarly, $PQ = 5$.

Thus, the perimeter of $PQRS$ is
$PQ + QR + RS + PS = 5 + 3 + 5 + 3 = 16$.