Question 31
Let a square $ABCD$ of side length $8$ cm which is divided, with lines parallel to its sides, into $64$ small squares of side $1$ cm. Color $7$ small squares black, and the other $57$ small squares are white. Suppose that there is a positive integer $k$ such that no matter which $7$ squares are black, there is a rectangle of area $k\ \text{cm}^2$ with sides parallel to the sides of $ABCD$ and all of its small squares that contains are white. Find the maximum value of $k$.