Problem 1007

[Proof] First, draw a large square $K_{0}$ of size $2^{n} \times 2^{n}$ on the grid paper, such that it contains all $n$ black cells and the total number of white cells is at least 4 times the number of black cells. It is clear that the total area of the black cells is no more than $\frac{1}{5}$ of the area of $K_{0}$. Then divide $K_{0}$ into 4 squares $K_{1}$, each containing $2^{n-1} \times 2^{n-1}$ cells. In each $K_{1}$, the total area of the black cells does not exceed $\frac{4}{5}$ of the area of $K_{1}$. If any of these 4 $K_{1}$ squares satisfy condition (2), select them; if a $K_{1}$ contains no black cells, remove it. For the $K_{1}$ squares that do not fall into these two cases, repeat the above process, i.e., divide each of them into 4, ..., until no more squares can be divided or until $2 \times 2$ squares are obtained. At this point, remove those with no black cells, and select the small squares with 1, 2, or 3 black cells.