Homework for Week 9

  1. How many rotational symmetries does a dodecahedron have?
  2. (Artin, Chapter 5, Problem 5.5) Let G be the dihedral group of symmetries of the square (meaning the eight rotational and reflectional symmetries). For the purposes of this problem, an edge is considered stabilized if it's sent to itself regardless of what direction it's drawn.
    1. What is the stabilizer of a vertex of the square? Of an edge of the square?
    2. G acts on the set of diagonal lines. How many elements are in the set of diagonal lines? What is the stabilizer of a diagonal?
  3. (Tucker, Applied Combinatorics, 9.2.1) How many different 4-bead necklaces are there using beads of red, white, blue, and green (assume necklaces can rotate but cannot flip over)?
  4. (Tucker, Applied Combinatorics, 9.2.5) A merry-go-round can be built with three different styles of horses. How many five-horse merry-go-rounds are there?
  5. (Modified from Tucker, Applied Combinatorics, 9.3) How many ways are there to 3-color the corners of a pentagon that are:
    1. Distinct with respect to rotations only?
    2. Distinct with respect to rotations and reflections?
    3. Find two 3-colorings of the pentagon that are different in the first part but equivalent in the second.