3 [ Central symmetry helps solve the problem]

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 relative to the e4 square?

#
Evaluation. All fields of the board except for the a-file, the 8th rank, and the e4 square can be divided into pairs symmetrical relative to e4. Such pairs amount to 24. According to the condition, no more than one pawn can be placed on the fields of each pair. Additionally, no more than one pawn can be placed on the fields of the a-file and the 8th rank. There are 15 such fields. Pawns cannot be placed on the e4 square according to the condition. Therefore, no more than 39 pawns can be placed in total.

An example of placing 39 pawns is shown in the figure:

![](https://cdn.mathpix.com/cropped/2024_05_06_69924287c79077e28276g-20.jpg?height=578&width=506&top_left_y=2195&top_left_x=773)

Let $A, B$, and $C$ be three numbers greater than 0 and less than 1, and let $K$ be the largest of them. Prove that $1-(1-A)(1-B)(1-C) > K$.

## Solution

Write the inequality as $1-K < (1-A)(1-B)(1-C)$. Now it is obvious, as the positive number $1-K$ is multiplied by a positive number less than one.