Question 36
Let $n\ge3$ be an integer, and consider an $n\times n$ board divided into $n^2$ unit squares. For each $m\ge1$ there are arbitrarily many $1\times m$ rectangles (type I) and $m\times1$ rectangles (type II). Cover the board with $N$ such rectangles, without overlap or protrusion. The total number of type I rectangles must equal the total number of type II rectangles. (A $1\times1$ rectangle belongs to both types.) What is the smallest possible value of $N$?
