## Task 3 - 250523

Twelve queen stones are to be distributed on the perimeter of a square such that the following conditions are met:

(1) Each corner of the square has the same number of queen stones. It is permissible for the corners to be free of queen stones; however, multiple queen stones can also be stacked on the corners.

(2) Each side of the square (including its two corner points) has the same number of queen stones. All queen stones that lie on a side of the square but between its corner points should be stacked on top of each other.

a) Provide four different distributions of the twelve queen stones such that each of these distributions meets conditions (1) and (2)!

b) Prove that there cannot be more than four different distributions of this kind!
a) The following distributions satisfy conditions (1) and (2):

| 30 | 121 | 212 | 30 |
| :---: | :---: | :---: | :---: |
| 3 | 2 | 1 | 0 |
| 030 | 121 | 212 | 30 |

b) For each distribution of the required type, the following holds:

If exactly \( x \) stones lie on one corner, then according to (1), \( x \) stones lie on each corner. If further, on one side, apart from the \( 2x \) stones that lie on both endpoints, there are exactly \( y \) stones, then according to (2), this applies to each side with the same number \( y \).

Therefore, a total of \( 4x + 4y \) stones are distributed, so \( 4 \cdot x + 4 \cdot y = 12 \), i.e., \( x + y = 3 \).

This can only be satisfied with the numbers \( x \) and \( y \) by \( 0 + 3 = 3, 1 + 2 = 3, 2 + 1 = 3 \) or \( 3 + 0 = 3 \). Therefore, there can only be the four distributions mentioned in a).