What is the maximum number of queens that can be placed on the black squares of an $8 \times 8$ chessboard so that each queen is attacked by at least one of the others?

#
Consider some arrangement of queens that satisfies the condition. A queen standing at the edge of the board cannot be attacked by any other queen, which means that all the queens are in the "inner" $6 \times 6$ square. Note that in none of the $3 \times 3$ squares can there be more than four queens. Indeed, if there are five queens in one of them, then the central cell of this square is black, and the queens are on all the black squares of this square. But then the queen standing in the center of the square is not attacked by any queen. Therefore, in the $6 \times 6$ square, there can be no more than $4 \cdot 4=$ 16 queens.

An example of the arrangement of sixteen queens is shown in the figure.

## Answer

16 queens.