2.1. For what largest $k$ can we assert that in any coloring of $k$ cells in black in a white $7 \times 7$ square, there will necessarily remain a completely white $3 \times 3$ square with sides along the grid lines?
Answer: 3

![](https://cdn.mathpix.com/cropped/2024_05_06_1e0e40e2c1eb5d36a34fg-2.jpg?height=260&width=294&top_left_y=270&top_left_x=892)

Solution. Let's highlight four $3 \times 3$ squares that are adjacent to the corners of the $7 \times 7$ square. These squares do not overlap, so if no more than three cells are shaded, at least one of these $3 \times 3$ squares will remain entirely white. If, however, we shade 4 cells, marked in gray on the diagram, then no white $3 \times 3$ square will remain.