3. What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square), if: 1) a pawn cannot be placed on the $e4$ square; 2) no two pawns can stand on squares that are symmetric with respect to the $e4$ square?

Answer: 39 pawns.
Solution. All fields of the board except for the vertical $a$, the horizontal 8, and the field e4 can be divided into pairs that are symmetrical relative to e4. Such pairs form 24. According to the condition, no more than one pawn can be placed on the fields of each pair. In addition, no more than one pawn can be placed on the fields of the vertical $a$ and the horizontal 8. There are 15 such fields. On the field $e 4$, according to the condition, no pawns can be placed. Therefore, in total, no more than

![](https://cdn.mathpix.com/cropped/2024_05_06_704e715e936cf2f990c2g-2.jpg?height=457&width=465&top_left_y=520&top_left_x=1407)
39 pawns can be placed. An example of placing 39 pawns is shown in the figure (the fields for pawns are shaded).

Criteria. If the solution is incorrect - 0 points.

If only the correct example - 3 points.

If only the correct estimate - 3 points.

If the correct solution (any correct example and justification of the estimate) - 7 points.