Math 173 Hand-In Assignment #5, Fall 2008
The assignment is due at the beginning of class on Monday, December 8, 2008. It is okay to work on the assignment with
classmate(s), provided that you each hand in your own work, write explanations in your own words, and identify who
you worked with. It is not okay to copy. Show your work!
Write up your work neatly in Mathematica; hand in a print-out. Include a title with your name, section / subsection
labels for each question, text cells explaining your work, and input and output for all calculations.
1. Find the number of ways to 3-color the 64 squares of an 8 by 8 game board that rotates but does not flip.
(See Figure 1.)
2. Refer to Figure 2. The numbers refer to the vertices; the letters refer to edges. The figure rotates and flips.
(a) Find the orbit of edge g, and the stabilizer of edge g.
(b) What is the size of the group of symmetries for the figure?
(c) Find the cycle index if the symmetry group acts on the edges of the figure, and use it to determine the number of
4-colorings of the edges.
(d) Find the cycle index if the symmetry group acts on the vertices, and use it to determine the number of 4-colorings
of the vertices.
(e) Find the cycle index if the symmetry group acts simultaneously on the vertices and edges. Use it to determine the
number of 4-colorings of edges and vertices.
(f) Explain why the answer to (e) does not equal the answer to (c) times the answer to (d). Give a non-technical
explanation of what is happening with the colorings. Illustrate with a specific example of colorings that shows why
the multiplication principle fails.
3. A necklace has 8 equally spaced beads, which can be colored red, yellow, blue, green , or purple.
(See Figure 3.)
(a) Find the cycle index, and use it to determine the number of colorings for the necklace.
(b) Find the complete pattern inventory of colorings, and multiply it out. (Include it in your printout. Really.)
(c) Use the pattern inventory to determine the number of distinct necklaces with:
(i) 2 red beads, 3 blue, 1 yellow, 1 green , and 1 purple
(ii) 4 red beads, 2 blue, and 2 green
(iii) exactly 3 purple beads
See next page for figures. I will also post a Mathematica file with figures you can copy/paste into your own Mathematica
notebook. You can use the Drawing Tools to draw on them in Mathematica.
