Consider the labelled pentagon

_anonymous_0 1 1 2 2 1--2 3 3 2--3 4 4 3--4 5 5 4--5 5--1

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 2 1 3 4 5 to 1 4 5 3 2

  9. Find the permutation representing the transformation
    from 1 3 5 2 4 to 2 5 1 4 3

  10. Find the permutation representing the transformation
    from 2 5 4 3 1 to 1 2 5 3 4

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

    2 3 5 1 4 and 3 5 1 4 2 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?

    4 2 3 1 5 and 3 4 2 1 5 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?

    2 3 5 1 4 and 4 1 5 3 2 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?

    1 4 3 5 2 and 1 4 5 2 3 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?

    5 1 4 2 3 and 1 5 3 2 4 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?

    1 2 4 3 5 and 3 5 4 2 1 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.