8.3. What is the maximum number of rooks that can be placed on an 8x8 chessboard so that each rook attacks no more than one other? A rook attacks all squares on the same row and column it occupies.
Answer: 10.

Solution: It is clear that in each column and row there are no more than two rooks. Let $k$ rooks be placed while satisfying the condition. On each square where a rook is placed, write the number 0. In each of the 8 columns, perform the following operation: if there are two numbers in the column, add 1 to both; if there is one number, add 2 to it (do nothing in an empty column). Then perform the same operation for each row. Clearly, on the square of each of the $k$ rooks, the result will be either 3 or 4 (if 2 is written, then this rook is under attack by two others), so the sum $S$ of all the written numbers is at least $3k$. On the other hand, since we added no more than 2 in each of the 8 columns and then in each of the 8 rows, $S \leq 32$. Therefore, $3k \leq S \leq 32$, from which $k \leq 10$. An example is easily constructed.

![](https://cdn.mathpix.com/cropped/2024_05_06_9af6344cc278069a4da7g-03.jpg?height=491&width=485&top_left_y=796&top_left_x=1439)
Criterion: Only the answer - 0 points. An example is worth 2 points, the estimation of the number of rooks is worth 5 points (points are cumulative).