Problem 3. (6 points) What is the maximum number of knights that can be placed on an $8 \times 8$ chessboard such that each of them threatens at most seven other knights?
Solution: We will prove that the maximum number of knights is 60.

An example of such an arrangement is obtained by placing knights in all fields except the four fields of the central $2 \times 2$ square. It is not difficult to verify that each knight threatens no more than 7 other knights.

We will prove that the number of knights cannot be increased. Consider the central $4 \times 4$ square. If 12 or fewer knights are placed in the central $4 \times 4$ square, then the total number of knights will be no more than 60.

Suppose that no fewer than 13 knights are placed in the central $4 \times 4$ square. Then each of them threatens 8 fields on the board, one of which is empty. Then the empty fields are no fewer than $13: 4$ (since each empty field is threatened by no more than 4 knights from the central square), i.e., no fewer than 3.25.

Therefore, the empty fields are at least 4, and the knights are no more than 60.