Problem 1. A rectangle is divided into 7 squares (see drawing). If the perimeter of the smallest square is $2 \mathrm{~cm}$, what is the perimeter of the original rectangle.
Solution. The perimeter of the smallest square is $2 \mathrm{~cm}$, so the length of its side is $\frac{1}{2} \mathrm{~cm}$. This means the length of the side of the second, larger square is $\frac{3}{2} \mathrm{~cm}$. The length of one side of the rectangle is $\frac{7}{2}$. The sum of the lengths of two sides of the largest square is equal to the sum of the lengths of three sides of the middle square and one side of the smallest square, so the length of the side of the largest square is $\frac{7}{4} \mathrm{~cm}$. Therefore, the length of the rectangle is $\frac{14}{4} \mathrm{~cm}$, and the width is $\frac{13}{4} \mathrm{~cm}$. Thus, the perimeter of the rectangle is $\frac{54}{4} \mathrm{~cm}$.