A rectangle is divided into smaller rectangles, whose sides are parallel with the big rectangle. Let $x$ be how many intersections there are (a point in the interior where $4$ rectangles meet), $y$ the number of rectangles, and $z$ the number of lines in the interior of the figure.
An example would be below, where $x=2, y=7, z=4.$
Find an expression for the number of rectangles in terms of $x$ and $z.$
So far, I know that each internal line defines $2$ rectangles, and an intersection defines $4$ rectangles. I think we can proceed with Euler's Formula $(V+F=E-2)$ or $E \leq 3V-6.$ However, I'm not sure on how to find the number of edges or vertices.
How would I continue from here?