## Task 1 - 250611

On a $(3 \times 3)$-board, three game pieces are to be placed such that they do not threaten each other. A game piece is to threaten exactly those fields that lie in the same horizontal or the same vertical row as it does.

a) Draw all possible positions of the required kind for three such game pieces!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=251&width=243&top_left_y=477&top_left_x=1495)

b) How many different positions are there if two positions are considered different if and only if one cannot be obtained from the other by rotating around the center field?
a) There are exactly the following six positions of the required type:
a)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=177&width=171&top_left_y=979&top_left_x=380)
b)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=180&width=189&top_left_y=978&top_left_x=611)
c)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=180&width=192&top_left_y=975&top_left_x=835)
d)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=180&width=175&top_left_y=978&top_left_x=1089)
e)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=174&width=185&top_left_y=978&top_left_x=1324)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0249.jpg?height=186&width=209&top_left_y=975&top_left_x=1529)

b) Since in position (a) the center field is occupied, while in position (b) it is not, there can be no rotation around the center field that transforms one of these two positions into the other.

Position (c), (d), or (e) can be derived from position (b) by rotating $180^{\circ}$, $90^{\circ}$, or $270^{\circ}$ around the center field, respectively. Position (f) can be derived from position (a) by rotating $180^{\circ}$ around the center field.

Therefore, there are exactly two distinct positions (namely positions (a) and (b)).