# Problem 8. (4 points)

Given a checkerboard $8 \times 8$, 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: 32
# Solution:

Notice that with a chessboard coloring, Pythagoras only attacks cells of the opposite color to the cell he is on.

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

Estimate: we mark the field as follows:

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$ | $H$ |
| $I$ | $J$ | $K$ | $L$ | $M$ | $O$ | $P$ | $Q$ |
| $R$ | $S$ | $T$ | $U$ | $V$ | $W$ | $X$ | $Z$ |
| 5 | 6 | 7 | 8 | 1 | 2 | 3 | 4 |
| $E$ | $F$ | $G$ | $H$ | $A$ | $B$ | $C$ | $D$ |
| $M$ | $O$ | $P$ | $Q$ | $I$ | $J$ | $K$ | $L$ |
| $V$ | $W$ | $X$ | $Z$ | $R$ | $S$ | $T$ | $U$ |

On each pair of cells marked with the same symbol, there can be at most one Pythagoras.

## 8th Grade

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