5. How many ways are there to color the cells of a $10 \times 10$ board in blue and green so that in each $2 \times 2$ square there are two blue and two green cells?
Answer: $2^{11}-2=2046$.

Solution. Note that if the colors of three cells in a $2 \times 2$ square are known, the color of the remaining cell is uniquely determined. Also, the colors of the two remaining cells are uniquely determined if there are two adjacent cells of the same color in a $2 \times 2$ square.

Consider the first row of the table. Suppose there are no adjacent cells of the same color in this row. Then the color of the first cell in the second row uniquely determines the colors of the other cells in the second row: they either all match the colors of the cells in the first row or are opposite to them. Therefore, the second row can be painted in two ways. In any of these ways, there are also no adjacent cells of the same color. Similarly, the color of the first cell in the third row uniquely determines its coloring. Thus, the third row can also be painted in two ways, and so on. Therefore, the number of ways to paint the table with such a fixed first row is exactly $2^{9}$. The number of ways to paint the table so that there are no adjacent cells of the same color in the first row is $2^{10}$.

Suppose there are two adjacent cells of the same color in the first row. Let's assume for definiteness that they are green. Then there must be two blue cells under them. Then the colors of the cells in the second row are uniquely determined. Therefore, there must be two green cells under the two adjacent blue cells. Thus, the colors of the cells in the third row are also uniquely determined, and so on. Therefore, the first row with adjacent cells of the same color uniquely determines the coloring of the entire table. The number of ways to paint the first row in two colors is $2^{10}$, but in two of them there are no adjacent cells of the same color. Therefore, the number of colorings of the table with such first rows is $2^{10}-2$, and the total number of colorings is $2^{10}+\left(2^{10}-2\right)=2^{11}-2=2046$.