2.1. On 40 cells of an $8 \times 8$ chessboard, stones were placed. The product of the number of stones lying on white cells and the number of stones lying on black cells was calculated. Find the minimum possible value of this product.
Answer: 256

Solution. Let $b$ and $w=40-b$ be the number of stones on black and white cells, respectively. Without loss of generality, assume that $b \geqslant w$, so $b$ can take any integer value from 20 to 32. We are interested in the minimum value of the product $b(40-b)$. The quadratic function $x(40-x)=-x^{2}+40 x$ reaches its maximum value at $x_{b}=20$ and decreases for $x \geqslant 20$. Therefore, the answer to the problem is $f(32)=32 \cdot 8=256$.