21. In a $9 \times 9$ grid, each small square is colored either black or white, such that the number of black squares adjacent to each white square is greater than the number of white squares, and the number of white squares adjacent to each black square is greater than the number of black squares (squares sharing a common edge are considered adjacent), find the maximum difference in the number of black and white squares in such a coloring method.
21. A coloring scheme of a grid that satisfies the problem's conditions is called a good scheme. For a good coloring scheme, it is easy to prove that the following conclusions hold: (1) If two adjacent cells have the same color and are not in the same row (or column), then any two adjacent cells in the same column (row) of these two rows (columns) have the same color, and these colors alternate (Figure a); (2) Three consecutive cells cannot have the same color; (3) If a corner cell is black (white), then the two adjacent cells are white (black); (4) If there are two adjacent cells of the same color in a certain column (row), then it is impossible to have two adjacent cells of the same color in any row (column). From these conclusions, we can see that if there are no two adjacent cells of the same color in a certain row (column), then the colors of the other cells in the grid depend only on the coloring of this row (column). Therefore, we only need to check the color distribution of the first row (column). It is evident that if the grid is colored like a chessboard, the maximum difference between the number of black and white cells is 1. If there are two adjacent cells of the same color in the first row, then by (1)-(4), the number of black and white cells in any two adjacent rows is the same. Therefore, the difference between the number of black and white cells is equal to the difference in the first row, and it is easy to see that the difference in the number of black and white cells in the first row is at most 3 (Figure b). When there are two adjacent cells of the same color in the first column, the same conclusion can be drawn. By (4), it is impossible for any good coloring scheme to have two adjacent cells of the same color in both the first row and the first column. In summary, the maximum difference between the number of black and white cells is 3.