\section*{Task 3 - 341033}

Antje and Beate decide, after finding ordinary dice too boring, to replace them with regular octahedra numbered from 1 to 8.

Before they start making these octahedral dice, they agree that (analogous to the dice) the sum of the numbers on any two opposite faces should be 9.

When they compare their self-made octahedra the next day, they notice:

Their octahedra are different from each other, even when the agreement about opposite faces is adhered to, in the sense that no rotation can make the arrangement of the numbers match.

a) Determine how many different arrangements of the numbers there are in the aforementioned sense, under the condition that the agreement about opposite faces is adhered to!

b) Determine how many different arrangements there are in the aforementioned sense, if the agreement about opposite faces does not need to be adhered to!
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a) We place the octahedron on the table so that the face with the 1 is facing down. Then the face with the 8 is on top. Now we rotate the octahedron around its vertical axis of symmetry until the face with the 2 is facing forward.

Then the face opposite to it shows the 7. Furthermore, the position of the octahedron is now uniquely determined, while the remaining four faces still need to be labeled with the numbers from 3 to 6:

If we first look for a place for the 3, we have four different possibilities, while the corresponding opposite face is labeled with the 6. For the remaining two faces, there are still two possibilities for which one is labeled with the 4, and the other with the 5. In total, there are exactly \(4 \cdot 2 = 8\) different octahedron labelings where the sum of opposite faces is 9, which cannot be converted into each other by rotation.

b) Again, we place the octahedron with the face showing the 1 facing down on the table. Then there are 7

possibilities for which number the face pointing upwards shows. Now we rotate the octahedron so that of the remaining six faces, the one with the smallest number is facing forward.

This uniquely determines the position of the octahedron, and the remaining five numbers can be distributed in any order on the remaining five faces of the octahedron. There are \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\) possibilities for this, so there are a total of \(7 \cdot 120 = 840\) different octahedron labelings that cannot be converted into each other by rotation.