4. What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, for any placement (including rotations and flips) of a figure consisting of 4 cells in the shape of the letter $\Gamma$, there is at least one colored cell?
Answer: 12.

Solution: Consider a $2 \times 3$ rectangle. It is obvious that at least 2 cells need to be colored in it. Divide the $6 \times 6$ board into 6 rectangles of $2 \times 3$. In each, at least 2 cells need to be colored, so in total, at least 12 cells need to be colored. An example with 12 cells is shown on the right.

Comment: A correct example with 12 colored cells - 3 points. An estimate that fewer than 12 cells are not enough - 4 points. The estimate should not rely on an example. Reasoning such as: "in my example, it is impossible to reduce the number of colored cells" is not an estimate.

![](https://cdn.mathpix.com/cropped/2024_05_06_14de7b75f2b36c26555ag-2.jpg?height=409&width=434&top_left_y=732&top_left_x=1531)