\section*{Task 1 - 231021}

On a chessboard, a queen can move such that from her position, she can reach all fields in the horizontal and vertical rows and the fields of the two diagonals intersecting at her position.

In the diagram, the queen's position is marked by a black field, and the reachable fields are marked with dots. Letters and numbers at the edges are to help name the fields (here, the queen is on d2).

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1471.jpg?height=440&width=425&top_left_y=380&top_left_x=1295)

On a \(5 \times 5\) square of fields, 5 queens are to be placed such that no queen stands on a field that can be reached by another.

Determine whether this is possible, and if so, find all such placements that cannot be transformed into each other by rotation or reflection!
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I. If an arrangement of 5 queens satisfies the condition that no queen stands on a field that can be reached by another, it follows initially that in each row and each column, exactly one queen must stand.

Furthermore, in the arrangement, only one of the two possibilities (1), (2) can apply:

(1) A queen stands on the field c3.

In row 1, a queen must then stand either on b1 or d1. By possible reflection, it can be achieved that a queen stands on b1. From the image, it is evident that three more queens can only stand on a4, d5, and e2.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1471.jpg?height=294&width=303&top_left_y=1412&top_left_x=317)

(2) No queen stands on the field c3.

If no queen stands on any of the fields b1, d1, a2, a4, b5, d5, e2, e4 (*), then one of the two rows 1 and 5 must be occupied in a corner field (column a or e), and the other in its middle field (column c). The same would apply to the two columns a and e.

This would result in a contradiction, as the queen on the middle field of row 1 or 5 could reach the queen on the middle field of column a or e.

Therefore, a queen stands on one of the fields (*).

By possible rotation or reflection, it can be achieved that a queen stands on b1. Then no queen can stand on e3, as the queens on b1 and e3 would cover all fields in row 2.

Thus, a queen must stand on a3 in row 3. From the image, it is then evident:

A queen cannot stand on d4 (no possibility for row 5), so one must stand on c4, and then two more queens can only stand on d2 and e5.

Thus, it is proven:

There are at most the two arrangements shown in the images, up to reflection and rotation.
![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1471.jpg?height=304&width=664&top_left_y=2122&top_left_x=702)

II. These two arrangements also satisfy the condition that no queen stands on a field that can be reached by another.

They cannot be transformed into each other by rotation or reflection, as in any rotation or reflection of the square, the field c3 remains in place, which is occupied in the left image but not in the right image.

Thus, there are exactly the two arrangements of the required type, as indicated in the last two images, up to rotation and reflection.