Determine the number of ways the faces of a cube can be colored with three color

Question

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:

1- Type of problem: Identify the type of problem. Does this problem involve computations, problem solving, proving, or a mix? Explain.

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

3- Approach to problem: In 1–2 paragraphs, discuss the motivation behind your solution, including what research you conducted and how you approached the problem.

Explanation / Answer

1.

The problem here is to find the number of ways the faces of a cube can be colored with three colors where the solution was also clearly explained. Computation means a kind of calculation that follows a well-defined model which is understood and can be expressed (ex: algorithm). This problem doesn't involve this kind of well-defined model so this is not computation. The problem involves problem solving and proving as the calculations and configurations need to be explained and proved, so mix is involved in this type of problem.

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