7. On a $6 \times 6$ chessboard, there are 9 rooks. How many "unattacked" cells can be on the board?

Answers:

A) 9 (+) B) 8 (-) C) $6(+)$ D) 4 (+) E) 1 (+)
Solution. If the rooks are in $k$ columns and $m$ rows, then $(6-k)(6-m)$ cells are free. 9 rooks cannot fit in fewer than 2 rows and fewer than 2 columns. Moreover, it is impossible to fit them in a $4 \times 2$ rectangle. Therefore, each of the factors does not exceed 3. All other options are possible, that is, there can be $9, 6, 4, 1$, and 0 free cells.