Zhendarov R.g.

What is the maximum number of knights that can be placed on a chessboard so that each one attacks no more than seven of the others

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Example of placing 60 knights: all cells are occupied except for the cells of the central $2 \times 2$ square.

We will show that with a placement satisfying the condition, no fewer than four cells will remain free (i.e., it is impossible to place more than 60 knights). Consider the central $4 \times 4$ square. If all four fields in it are free, the proof is complete.

Suppose there are no more than three free fields in it, then there are at least 13 knights there. Each of them attacks 8 cells on the board, at least one of which is free. It is easy to verify that a fixed cell is attacked by knights standing in the central square no more than four times. Therefore, the number of free cells is not less than $13: 4 > 3$, i.e., not less than 4.

## Answer

60 knights.