8. What is the maximum number of white and black pawns that can be placed on a 9x9 grid (a pawn, regardless of its color, can be placed on any cell of the board) so that no pawn attacks any other (including those of the same color)? A white pawn attacks two diagonally adjacent cells on the next higher horizontal row, while a black pawn attacks two diagonally adjacent cells on the next lower horizontal row (see the figure). (A. Antropov)
Answer: 56. Solution: An example with 56 pawns is shown in the figure. Evaluation: Note that in each rectangle of three rows and two columns, there are no more than 4 pawns. Indeed, if there are at least 5, then on one of the colors, all three cells are occupied, and the pawn in the middle row attacks one of the two remaining ones. Therefore, in any rectangle of 9 rows and 8 columns, there are no more than 48 pawns (since it can be divided into 12 rectangles $3 \times 2$). Suppose we managed to place 57 pawns. Then in the ninth column, there must be at least 9 pawns, i.e., exactly 9. But then in the eighth column, there are no more than 2 pawns (otherwise, there would be a pawn not in the first and not in the last row that would attack some pawn in the ninth column). Then in the eighth and ninth columns together, there are no more than 11 pawns, and in the 2nd to 7th columns - no more than 36 pawns (they can be divided into 9 rectangles $3 \times 2$). It follows that in the first column, there should be 10 pawns - a contradiction.

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