How Do You Like Them Rectangles?

A FiveThirtyEight Riddler puzzle.

By Vamshi Jandhyala in mathematics Riddler

September 22, 2017

Riddler Express

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 = $44/11 = 4 \ in$

Riddler Classic

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 $xy = \frac{64}{11} \frac{11}{2} = 32 \ in.^2$