5. What is the maximum number of chips that can be placed in the cells of a chessboard (no more than one chip per cell) so that no more than three chips are placed on each diagonal?
Answer: 38

Solution. Consider the eight-cell diagonal consisting of black cells, and the parallel diagonals consisting of black cells. These diagonals have $2, 4, 6, 8, 6, 4,$ and 2 cells. On each of the outermost diagonals, no more than two chips can be placed, and on each of the middle diagonals, no more than three. Therefore, a total of no more than $2 \cdot 2 + 5 \cdot 3 = 19$ chips can be placed on the black cells.

![](https://cdn.mathpix.com/cropped/2024_05_06_52ba2bd370bb2f80da76g-11.jpg?height=294&width=297&top_left_y=413&top_left_x=1348)
chips. Similarly, no more than 19 chips can be placed on the white cells. Therefore, a total of no more than 38 chips can be placed on the board.

An example of placing 38 chips is shown in the figure.