Notes
subdivided hexagon solution

Solution to the Subdivided Hexagon Puzzle

Solutions using Triangle Areas

Solution One

In the above diagram, triangles $G D A$ and $G B A$ have the same base, $G A$, and the height of point $B$ above this base is half of that of $D$. So triangle $G D A$ has area twice that of $G B A$, namely $2$.

The regions $D E F G$ and $G D A$ together make up half of the hexagon and, by the above, have combined area $4$. The area of the hexagon is therefore $8$.

Solution Two

Let $x$ be the area of triangle $F B G$. As triangles $D F G$ and $B F G$ have the same base, $F G$, and the height of point $B$ above this base is half of that of $D$, the area of $F D G$ is twice of that of $F B G$, so is $2 x$.

The regions $D E F$ and $F A B$ are congruent so have the same area. The area of $D E F$ is $2 - 2 x$ and of $F A B$ is $1 + x$, so $2 - 2 x = 1 + x$. Solving this gives $x = \frac{1}{3}$ and so the area of $F A B$ is $\frac{4}{3}$.

The area of the hexagon is six times that of $F A B$, so is $8$.