Notes
overlapping hexagons solution

Solution to the Overlapping Hexagons Puzzle

Solution by Areas of Trapezium and Parallelogram

Consider the parallelogram $A B G H$ and the trapezium $B C F G$. As the hexagons are regular, $H F$ has length $10$ and $A C$ has length $20$. Let $x$ be the length of $A B$. Then $G F$ has length $10 - x$ and $B C$ has length $20 - x$. Let $h$ be the perpendicular distance between $H F$ and $A C$. The parallelogram then has area $x h$ and the trapezium has area $\frac{1}{2} h (10 - x + 20 - x) = h(15 - x)$. As these have the same area, $15 - x = x$ so $x = 7.5$.

The length of $E H$ is then $10 + x = 17.5$, so the perimeter of the design is $10 + 10 + 17.5 + 10 + 10 + 17.5 = 75$.