Determine the number of ways the faces of a cube can be colo

Determine the number of ways the faces of a cube can be colored with three colors. Show all of your work and adequately explain your reasoning for your solution.

We first find the cycle index of the group of FACE permutations induced by the rotational symmetries of the cube.

Looking down on the cube, label the top face 1, the bottom face 2 and the side faces 3, 4, 5, 6 (clockwise)

You should hold a cube and follow the way the cycle index is calculated as described below. The notation

(1)(23)(456) = (x1)(x2)(x3)

means that we have a permutation of 3 disjoint cycles in which face 1

remains fixed, face 2 moves to face 3 and 3 moves to face 2, face 4 moves to 5, 5 moves to 6 and 6 moves to 4.

(This is not a possible permutation for our cube; it is just to illustrate the notation.) We now calculate the cycle index.

(1)e = (1)(2)(3)(4)(5)(6); index = (x1)^6

(2)3 permutations like (1)(2)(35)(46); index 3(x1)^2.(x2)^2

(3)3 permutations like (1)(2)(3456); index 3(x1)^2.(x4)

(4)3 further as above but counterclockwise; index 3(x1)^2.(x4)

(5)6 permutations like (15)(23)(46); index 6(x2)^3

(6)4 permutations like (154)(236); net index 4(x3)^2

(7)4 further as above but counterclockwise; net index 4(x3)^2

Then the cycle index is

P[x1,x2,...x6] =(1/24)[x1^6 + 3x1^2.x2^2 + 6x2^3 + 6x1^2.x4 + 8x3^2]

and the pattern inventory for these configurations is given by the generating function:

(I shall use r=red and b=blue, y=yellow as the three colors) f(r,b,y) = (1/24)[(r+b+y)^6 + 3(r+b+y)^2.(r^2+b^2+y^2)^2 + 6(r^2+b^2+y^2)^3 + 6(r+b+y)^2.(r^4+b^4+y^4) + 8(r^3+b^3+y^3)^2] and putting r=1, b=1, y=1 this gives =(1/24)[3^6 + 3(3^2)(3^2) + 6(3^3) + 6(3^2)(3) + 8(3^2)] = (1/24)[729 + 243 + 162 + 162 + 72] = 1368/24 = 57

However please answer these question below: Description of method: In 1–2 paragraphs, describe the method you used. Explain why you chose that method.

Description of method: In 1–2 paragraphs, describe the method you used. Explain why you chose that method.

Solution

This is a problem requiring the use of Burnside\'s lemma. You consider
the 24 symmetries of a cube and sum all those symmetries that keep
colours fixed. The number of non-equivalent configurations is then the
total sum divided by the order of the group (24 in this case).

We first find the cycle index of the group of FACE permutations
induced by the rotational symmetries of the cube.

Looking down on the cube, label the top face 1, the bottom face 2 and
the side faces 3, 4, 5, 6 (clockwise)

You should hold a cube and follow the way the cycle index is
calculated as described below. The notation

(1)(23)(456) = (x1)(x2)(x3)

means that we have a permutation of 3 disjoint cycles in which face 1
remains fixed, face 2 moves to face 3 and 3 moves to face 2, face 4
moves to 5, 5 moves to 6 and 6 moves to 4. (This is is not a possible
permutation for our cube, it is just to illustrate the notation.) We
now calculate the cycle index.

(1)e = (1)(2)(3)(4)(5)(6); index = (x1)^6
(2)3 permutations like (1)(2)(35)(46); index 3(x1)^2.(x2)^2
(3)3 permutations like (1)(2)(3456); index 3(x1)^2.(x4)
(4)3 further as above but counterclockwise; index 3(x1)^2.(x4)
(5)6 permutations like (15)(23)(46); index 6(x2)^3
(6)4 permutations like (154)(236); net index 4(x3)^2
(7)4 further as above but counterclockwise; net index 4(x3)^2

Then the cycle index is

P[x1,x2,...x6] =(1/24)[x1^6 + 3x1^2.x2^2 + 6x2^3 + 6x1^2.x4 + 8x3^2]

and the pattern inventory for these configurations is given by the
generating function:

(I shall use r=red and b=blue, y=yellow as the three colours)

f(r,b,y) = (1/24)[(r+b+y)^6 + 3(r+b+y)^2.(r^2+b^2+y^2)^2
+ 6(r^2+b^2+y^2)^3 + 6(r+b+y)^2.(r^4+b^4+y^4)
+ 8(r^3+b^3+y^3)^2]

and putting r=1, b=1, y=1 this gives

=(1/24)[3^6 + 3(3^2)(3^2) + 6(3^3) + 6(3^2)(3) + 8(3^2)]

= (1/24)[729 + 243 + 162 + 162 + 72]

= 1368/24

= 57