What is the maximum number of a) rooks; b) queens that can be placed on an $8 \times 8$ chessboard so that each of these pieces is attacked by no more than one of the others?
a) Let $k$ rooks be placed while satisfying the condition. On each field where a rook stands, we write the number 0. In each of the $n$ columns, we perform the following operation: if there are two numbers in the column, we add 1 to both; if there is one number, we add 2 to it (we do nothing in an empty column). Then we perform the same operation for each row. Clearly, on the position of each of the $k$ rooks, the result will be either 3 or 4, so the sum $S$ of all written numbers is at least $3k$; on the other hand, since in each of the eight columns and then in each of the eight rows we added no more than 2, we have $S \leq 32$. Therefore, $3k \leq S \leq 4n$, from which $k \leq 10$.

An example of placing 10 rooks that satisfy the condition is shown in the left figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_de7bb834a1e75b584fa6g-23.jpg?height=732&width=1762&top_left_y=594&top_left_x=136)

b) Clearly, the number of queens that can be placed is no more than the number of rooks. The right figure shows how to place 10 queens.

## Answer

a) 10 rooks, b) 10 queens.