Consider the labelled pentagon

For each of the following permutations, determine if the permutation is a symmetry of the pentagon in flatland, sphereland (3 dimenions), or if the transformation is never a symmetry.

1. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\1&2&3&4&5\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

2. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\2&3&4&5&1\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

3. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\2&1&5&4&3\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

4. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\3&2&4&1&5\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

5. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\4&5&1&2&3\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

6. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\5&4&3&2&1\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

7. Where is the permutation \begin{pmatrix} 1&2&3&4&5\\5&4&1&3&2\end{pmatrix} a symmetry of the above figure, if at all?
I do not know yet
Flatland
Sphereland, but not Flatland
This permutation is not a symmetry

8. Find the permutation representing the transformation
from to

9. Find the permutation representing the transformation
from to

10. Find the permutation representing the transformation
from to

11. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

12. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

13. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

14. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

15. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

16. Is the following statement true, false, or neither?

and are the same pentagon in sphereland

I don't know yet
True
False
Does not make sense

However, it's impossible. Show that given a combination $(x,y,z)$, $[x + z - y]_3$ does not change no matter what button is pushed. Use this to show that the 27 combinations can be partitioned into 3 equivalence classes of 9 combinations each, and that the initial combination $(1,1,1)$ is not in the same equivalence class as the solution combination $(3,1,3)$.