# Problem 8. (4 points)

Given a $7 \times 7$ checkerboard, where the side length of each cell is one centimeter. A chess piece called Pythagoras, standing on cell $A$, attacks cell $B$ if the distance between the centers of cells $A$ and $B$ is five centimeters. What is the maximum number of non-attacking Pythagorases that can be placed on the board?

Answer: 25.
# Solution:

Notice that with a chessboard coloring of the board, Pythagoras only attacks squares of the opposite color to the square he is on.

Example: we color the board in a checkerboard pattern and place Pythagorases on the squares of the color that is more prevalent.

Estimate: we mark the field as follows:

| $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $H$ | $I$ | $J$ | $K$ | $L$ | $M$ | $N$ |
| $O$ | $P$ | $Q$ | $R$ | $S$ | $O$ | $P$ |
| $T$ | $U$ | $G$ |  | $V$ | $T$ | $U$ |
| $W$ | $X$ | $N$ | $A$ | $H$ | $W$ | $X$ |
| $S$ | $B$ | $C$ | $D$ | $E$ | $F$ | $Q$ |
| $V$ | $I$ | $J$ | $K$ | $L$ | $M$ | $R$ |

On each pair of squares marked with the same letter, a maximum of one Pythagoras can stand. Plus one more can stand in the central square.

## 2nd Variant

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