Let
be a positive integer. In each of the
unit squares of an
board, one of the two diagonals is drawn. The drawn diagonals divide the
board into
regions. For each
, determine the smallest and the largest possible values of
.
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%V0 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 $K$.
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