Notes
four squares overlapping a rectangle solution

Solution to the Four Squares Overlapping a Rectangle Puzzle

Solution by Tessellation

The squares define a grid which tessellates the plane. The right-hand side of the rectangle stretches between two grid points, while the left-hand side passes through one.

As can be seen in the above diagram, the unshaded area to the lower right of the rectangle is equivalent in area to two small squares, while the small triangle at the top matches the green triangle that is outside the rectangle. On the left-hand side, the triangle in the bottom left matches the triangle marked with the orange edge at the top right, meaning that the unshaded regions on the left have area equal to half of two squares. So the unshaded regions have a total area of $3$ meaning that the rectangle has a total area of $7$.