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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
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

What is your answer?
17. In the 1993 video game, "Myst", one of the puzzles involved setting the lock combination below to 3-1-3.
1 - 1 - 1
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 $(3,1,3)$.

18. Can you turn the combination lock below to combination 1-2-1-1?
1 - 1 - 1 - 1
Explain.