# Solution to the Four Squares V Puzzle +-- {.image} [[FourSquaresV.png:pic]] > A design made from four squares. Is more of it shaded pink or purple? =-- ## Solution by [[Symmetry]] and [[Dissection]] +-- {.image} [[FourSquaresVLabelled.png:pic]] =-- With the points labelled as above, consider rotating the diagram about the centre of the purple square by $90^\circ$ clockwise. This takes the [[line segment]] $M F$ to $L E$, and the side $A D$ rotates so that it overlaps with side $D C$, with $A$ rotating to $G$. The point $N$, as the intersection of $A D$ with the line through $M$ and $F$, therefore rotates to $C$. Comparing lengths, $N F$ and $C E$ are the same length, $N K$ and $C B$ are the same length, and therefore $K F$ and $B E$ are the same length. Since angles $L \hat{F} K$ and $J \hat{E} B$ are the same, this establishes triangles $K F L$ and $B E J$ as [[congruent]] and therefore the area of quadrilateral $B J F L$ is the same as that of triangle $K E F$. This triangle is congruent to $D A G$, so the purple region $B J F L$ and has the same area as the pink region $D A G$. This also establishes $K F$ as having the same length as $D G$. Since $H B$ and $B E$ also have the same length, this means that $G C$ and $A H$ have the same length. Therefore, triangles $G C L$ and $A H I$ are [[congruent]] and so have the same area. Therefore the pink and purple regions have the same area. ## Solution by [[Invariance Principle]] Given that the solution doesn't depend on the angle of the purple square, there is a configuration of the squares that makes it clear, as below. +-- {.image} [[FourSquaresVDiagonal.png:pic]] =--