Question 179
Let $N$ be a positive integer. In each of the $N^{2}$ unit squares of an $N \times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N \times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$.
Example with $N=3, K=7$
Example with $N=3, K=7$