443 MATH -- Assignment #5
Below are short answers (hints) to assigned questions.
37A
Let C denote the group of rotations of the cube. Following the description of elements of C in Appendix 1 we find that the cycle index of the action of C on the six faces of the cube is ZC = 1 6 2 2 2 3 2 X1 + 6X1 X4 + 3X1 X2 + 6X2 + 8X3 . 24
After substituting Xi = n (i = 1, ..., 6) we get that the number of face colourings with n colours is 1/24 n6 + 1/8 n4 + 1/2 n3 + 1/3 n2 . In particular the answer is 10 for n = 2 and 57 for n = 3.
37B
After substituting Xi = r i + bi + g i (i = 1, ..., 6) into ZC and expanding the polynomial we find that the coefficient of rb2 g 3 is 3. (See Maple the worksheet.) Thus there are 3 ways of painting one face red, two faces blue and three faces green. In one painting the red face and two blue faces are pairwise (mutually) adjacent. In the second painting the blue faces are adjacent and exactly one blue face is adjacent to the red face. In the third painting two opposite faces are blue.
37E
If a rotation has a cycles on the faces and b cy...