Example 4 A $9 \times 9$ chessboard is colored in black and white such that for each white square, the number of black squares adjacent to it is greater than the number of white squares adjacent to it, and for each black square, the number of white squares adjacent to it is greater than the number of black squares adjacent to it (two squares are considered adjacent if they share at least one side). Find the maximum difference in the number of black and white squares in all such coloring methods. (53rd Belarusian Mathematical Olympiad problem)
Analysis and Solution: To satisfy the coloring condition, each square can have at most one adjacent square of the same color, thus certain special cases cannot occur: (1) 3 squares in an L-shape are the same color. (2) 3 squares in a $1 \times 3$ rectangle are the same color.

If any two adjacent squares on the chessboard have different colors, then the difference in the number of black and white squares is no more than 1.

If there exist two adjacent squares $A$ and $B$ on the chessboard that are the same color, assume without loss of generality that $A$ and $B$ are in the same row (as shown in Figure 13-7). Consider the row adjacent to this one; since there are no 3-L shapes of the same color on the chessboard, the two squares adjacent to $A$ and $B$ in this row must be of a different color from $A$ and $B$. Continuing this reasoning, we find that in the two columns containing $A$ and $B$, the squares in the same row are the same color, and the squares in the same column alternate in color (adjacent squares are different colors). We now prove that there are no two adjacent squares of the same color in any column of the chessboard. Otherwise, suppose $P$ and $Q$ are two adjacent squares in a column, and $P$ and $Q$ are the same color. Similarly, we can prove that in the two rows containing $P$ and $Q$, the squares in the same column are the same color, and the squares in the same row alternate in color. In this case, consider the 4 squares where the columns of $A$ and $B$ intersect with the rows of $P$ and $Q$. Since the squares in the same row are the same color and the squares in the same column are the same color, the 4 squares are the same color, which is a contradiction. Therefore, the colors of the squares in each column of the entire chessboard alternate.

Removing the first row, the remaining chessboard has an equal number of black and white squares in each column, so the total number of black and white squares in the remaining chessboard is equal. In the first row, since there are no $1 \times 3$ rectangles of the same color, the difference in the number of black and white squares in every 3 squares is no more than 1, so the difference in the number of black and white squares in the first row is no more than 3.

Therefore, for any valid coloring, the difference in the number of black and white squares on the chessboard is no more than 3. As shown in Figure 13-8, the coloring meets the requirements, and at this time, the difference in the number of black and white squares is 3.

In summary, the maximum value sought is 3.