3360.21 – Rectangle Decomposition

A rectangle of width ww and height hh is decomposed into nine squares of side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 units. What are ww and hh?

Bonus points, once you know ww and hh, if you can fit the squares back together to make the rectangle.

Solution

The area of the rectangle will be

wh=12+42+72+82+92+102+142+152+182=1056\begin{align*}wh &= 1^2 + 4^2 + 7^2 \\&\quad + 8^2 + 9^2 + 10^2 \\&\quad + 14^2 + 15^2 + 18^2 \\&= 1056\end{align*}

Now 1056 factors into 33×3233 \times 32. This suggests that ww and hh are 33 and 32. In fact, if we check, and note that the rectangle must be at least 18 units wide to accommodate the largest square, no other factorization of 1056 will work.

The squares fit together as the figure below shows.