1079.41 – Word Triangle Arithmetic

In the word triangle below, find four different words of lengths $a, b, c,$ and $d$ such that $a^2 = bd$ and $ad = b^2c$ .

Solution

One answer is $a = \mathbf{SCHOOL}, b = \mathbf{OAK}, c = \mathbf{OVERCOAT}, d = \mathbf{DEMOCRATIC}$ .

There is something that can be done here to cut down on the guessing and checking.

\begin{aligned} a^2b =bd \text{ and } ad = b^2c & \leadsto & d = \frac{a^2}{b} = \frac{b^2 c}{a} \\ & \leadsto & a^3 = b^3 c \\ & \leadsto & c = \frac{a^3}{b^3}. \end{aligned}

Therefore, $c$ is a perfect cube. So it must be that $c = 8$ and $a = 2b$ . Now there are only a few possible values of $b$ to check, and $b$ determines everything else.