# How Do You Like Them Rectangles?

A FiveThirtyEight Riddler puzzle.
mathematics
Riddler
Published

September 22, 2017

Area puzzle

## Solution

The length of the top rectangle = $$24/4 = 6 \ in$$.

The length of the bottom rectangle = $$6+3+2=11 \ in$$.

The height of the bottom rectangle = $$44/11 = 4 \ in$$

Area puzzle

## Solution

Let $$x in$$ and $$y in$$ be the length and height of the shaded rectangle.

We have the following constraints:

$\begin{eqnarray} (14-x)y &=& 45 \\ (11-y)x &=& 32 \\ \frac{66}{11-y} &<& 14 \implies y < \frac{44}{7} \end{eqnarray}$

Subtracting (2) from (1) we have $$14y - 11x = 13$$.

Substituting $$y = \frac{13 + 11x}{14}$$ in (1), we have

$\begin{eqnarray*} 13 + 11x - 13x/14 - 11x^2/14 &=& 45 \\ \implies 11x^2 - 141x + 448 &=& 0 \\ \implies x &=& \frac{141 \pm \sqrt{141^2 - 4 \cdot 11 \cdot 448}}{2 \cdot 11} \\ \implies x = 7 \ and \ y = \frac{45}{7} \ &or& \ x = \frac{64}{11}\ and \ y = \frac{11}{2} \end{eqnarray*}$

The first solution has to be discarded because of constraint (3).

The area of the shaded region is $$xy = \frac{64}{11} \frac{11}{2} = 32 \ in.^2$$