In the figure, $$ABCD$$ is a square. It consists of squares and rectangles of areas $$1\ \text{cm}^2$$ and $$2\ \text{cm}^2$$ as shown. The perimeter of the square $$ABCD$$ (in cm) is

Every piece of the square has an area of $$1\ \text{cm}^2$$ or $$2\ \text{cm}^2$$, so the area of the whole square must be a whole number. If the perimeter is $$P$$, the side is $$\frac{P}{4}$$ and the area is $$\frac{P^2}{16}$$. Now $$\frac{17^2}{16}$$, $$\frac{15^2}{16}$$ and $$\frac{14^2}{16}$$ are not whole numbers, while $$\frac{16^2}{16} = 16$$ is. So the side is 4 cm and the perimeter is 16 cm.