Notes
four squares iii solution

Solution to the Four Squares III Puzzle

Solution by Transformations

With the points labelled as in the above diagram, point $E$ moves along the line $F G$. With respect to the point $C$, $B$ can be obtained from $E$ by a rotation by $45^\circ$ and a scaling by $\frac{1}{\sqrt{2}}$. As this is a fixed transformation, $B$ also moves along a straight line. When $E$ is at $F$, $B$ is at $G$, and when $E$ is at $G$, $B$ is at $H$. Hence $B$ moves along the line $G H$. As $H$ is the midpoint of the side $A C$, the lengths of $A B$ and $B C$ are equal. Since $A B$ is the diagonal of the blue square and $B C$ the side of the green square, the green square is twice the area of the blue square (this can be seen by dissecting each square into congruent isosceles right-angled triangles; the green into $4$ and the blue into $2$).

Hence the green square has area $4$.

Solution by Invariance Principle

When the puzzle is in one of the two configurations where $E$ is at $F$ or where $E$ is at $G$, the relationship between the green and blue squares is straightforward to see.