## Task A-1.4.

For a square tile whose fields are colored black or white, we say it is beautiful if its appearance does not change when rotated by $90^{\circ}$.

How many different beautiful tiles of dimensions $5 \times 5$ are there?
## First Solution.

The central field of the board does not change upon rotation. We can color it arbitrarily.

Notice that by rotating the board four times in the same direction, any field transitions to at most three different fields and eventually returns to the starting field. From the problem's condition, it follows that these four fields must be the same color (for example, all the fields in the corners of the board must be the same color).

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-03.jpg?height=517&width=514&top_left_y=844&top_left_x=700)

The remaining 24 fields can be divided into four rectangles of dimensions $2 \times 3$ as shown in the figure: with each rotation, these four rectangles cyclically map to one another. The fields in one of these rectangles can be colored arbitrarily, and the other rectangles are then uniquely determined.

Thus, on the entire board, we can color 7 fields arbitrarily, and for each of these fields, we have two possibilities. Therefore, there are $2^{7}$ different colorings, or different beautiful boards.