3. The fields of a board with 25 rows and 125 columns are alternately colored black and white (like the fields on a chessboard), and knight figures are placed on it, which attack according to special rules. A knight on a white field attacks all white fields in the same row and all black fields in the same column, while a knight on a black field attacks all black fields in the same row and all white fields in the same column (even if there is another figure in between). Determine the maximum number of knights that can be placed on the board so that they do not attack each other.
## Solution:

First, note that in each row, there can be at most two knights, one on a white square and one on a black square - if there were three knights in one row, two of them would have to be on squares of the same color, thus attacking each other.

The board has 25 rows, so it can have at most $2 \cdot 25 = 50$ knights.

The problem will be solved if we show that it is indeed possible to place 50 knights on the board in the required manner. In a given column, all knights must be on squares of the same color. The image below shows one possible arrangement of 50 knights that do not attack each other:

![](https://cdn.mathpix.com/cropped/2024_05_30_4ba13bd558581b5f9252g-4.jpg?height=1071&width=1725&top_left_y=1663&top_left_x=180)