5. A $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is selected, and all zeros in it are replaced with crosses, and all crosses with zeros. We will call any arrangement of crosses and zeros in the table a "pattern." How many different patterns can be obtained as a result of such moves? Patterns that can be obtained from each other by rotating $90^{\circ}$ or $180^{\circ}$ are considered different.
Solution. First method. Different drawings differ by at least one cell in the table, either plus or minus. The symbol in a given specific cell is determined by the parity (even or odd) of the number of swaps performed in each $2 \times 2$ square containing that cell. Each $2 \times 2$ square contains exactly one corner cell of the table, and each such square has its own unique corner cell. The parity of the number of swaps in each square corresponds to the symbol in its unique corner cell (if there is a cross in the corner cell, the number of swaps was odd; if there is a circle, the number of swaps was even). Therefore, each combination of symbols in the table uniquely corresponds to a combination of circles and crosses in the corner cells of the table. Since the number of different combinations in four cells is $2^{4}=16$, there are 16 drawings in total.

Second method. One can create a transition table and show that when swaps are performed multiple times, cycles arise, and some drawings transition into others. The images below show all possible ways to place crosses and circles, and the number of corresponding drawings in one group. Drawings in the same group differ from each other by rotations of $90^{\circ}$ or $180^{\circ}$. The presence of cycles proves that the number of drawings is limited, and fewer than $2^{9}$.
![](https://cdn.mathpix.com/cropped/2024_05_06_82d58631c382ea5da31eg-3.jpg?height=426&width=186&top_left_y=1418&top_left_x=684)

![](https://cdn.mathpix.com/cropped/2024_05_06_82d58631c382ea5da31eg-3.jpg?height=174&width=186&top_left_y=1421&top_left_x=952)

1 drawing

![](https://cdn.mathpix.com/cropped/2024_05_06_82d58631c382ea5da31eg-3.jpg?height=217&width=174&top_left_y=1628&top_left_x=952)

![](https://cdn.mathpix.com/cropped/2024_05_06_82d58631c382ea5da31eg-3.jpg?height=194&width=171&top_left_y=1416&top_left_x=1202)

1 drawing

![](https://cdn.mathpix.com/cropped/2024_05_06_82d58631c382ea5da31eg-3.jpg?height=182&width=177&top_left_y=1628&top_left_x=1199)

4 drawings

Answer. 16 drawings.