4.1. In a $4 \times 4$ square, each of the 16 cells was painted either black or white. Then, in each of the nine $2 \times 2$ squares that can be identified within this square, the number of black cells was counted. The resulting numbers were 0, 2, 2, 3, 3, 4, 4, 4, 4. How many black cells can there be in the large square? List all possible options.
Answer: 11

Solution. Note that squares with 0 white cells and 4 white cells cannot intersect. If a square with 0 white cells is not in a corner, then it does not intersect with more than three other squares (the square can be in the center or adjacent to the middle of a side). Let the white square be in the bottom-left corner, then we have 5 squares suspected of containing 4 black cells, of which 4 are entirely black, and one has 2 or 3 black cells. Note that the squares marked in Figure 1 contain no more than 2 black cells, so exactly 2. Then the three corner squares either all contain 4 cells, but then 5 squares contain 4 black cells; or two corners contain 4, and one contains 3. In total, this results in 11 cells. Figures 2 and 3 show examples of such arrangements.

![](https://cdn.mathpix.com/cropped/2024_05_06_eb2519c7589eeda5f668g-4.jpg?height=263&width=623&top_left_y=762&top_left_x=725)