24.32 * Place chess pieces on the squares of an $8 \times 8$ chessboard, with at most one piece per square, such that each row, each column, and each diagonal contains an even number of pieces. How many pieces can be placed at most?
Parse: The maximum number of chess pieces that can be placed is 48.
On the one hand, as shown in the figure, there are exactly 16 diagonals (indicated by dashed lines) on an $8 \times 8$ chessboard, each containing an odd number of squares, and no square is shared between any two diagonals. To ensure that each diagonal has an even number of chess pieces, at least 16 squares must remain empty, meaning that a maximum of 48 chess pieces can be placed.
On the other hand, it is possible to place 48 chess pieces such that each row, each column,

and each diagonal has an even number of chess pieces. The construction is as follows:
Simply do not place any chess pieces on the 16 squares along the two main diagonals, and place 1 chess piece on each of the other squares.