Cube coloring with 3 colors.

Q. The six faces of a cube are coloured using all three colours red, blue and green. Two colourings are considered the same if one can be obtained from the other by rotating the cube. How many distinct colourings are there?
S21.
Since all 3 colors need to be used, there are 3 cases:
4,1,1 3,2,1 2,2,2

Case 1: 4,1,1
3 ways to choose the color with 4 faces.
Now the remaining 2 faces with 2 different colors have 2 options, opposite or adjacent.
So 3*2 = 6 ways.

Red
Blue
Yellow
Yellow
Yellow
Yellow

Drag the cube. The red and blue faces are adjacent. The other four faces have the same colour.


Case 2: 3,2,1
Choose first color in 3 ways, second in 2 and third in 1 way: 3*2*1 = 6 total.
For each of them 3 sub cases as mentioned below: so 6*3 = 18 total.
Total so far: 6(Case 1) + 18(Case 2) = 24
Look at the color with 2 faces. There are 3 sub cases.
A. Those 2 faces are opposite.

Case 2: 3,2,1 — Two Blue Faces Opposite

Blue
Blue
Red
Red
Red
Green

Drag the cube around. The two blue faces are on opposite sides of the cube. The remaining four faces contain three red faces and one green face.


B. adjacent, with single color face opposite one of them.

Case 2: 3,2,1 — Two Blue Faces Adjacent, Green Opposite One of Them

Blue
Blue
Green
Red
Red
Red

Drag the cube. The two blue faces are adjacent. The green face is opposite one of the blue faces. The other three faces are red.

C. adjacent, with single color face touching both of them.

     Case 2: 3,2,1 — Single Face Touches Both Blue Faces

Blue
Blue
Green
Red
Red
Red

Drag the cube. The two blue faces are adjacent. The single green face touches both blue faces. The remaining three faces are red.



Case 3: 2,2,2
 A. Each same color pair is on opposite faces - quite simple - one case. (Total so far: 25)
B. Exactly one color pair is opposite, choose that color in 3 ways. (Total: 28)

2,2,2 — Exactly One Colour Pair is Opposite

Blue
Blue
Red
Red
Green
Green

Here, the two blue faces are opposite. The other two colour-pairs are adjacent, not opposite.

Since the opposite pair can be blue, red, or green, there are 3 choices.


C. No colour-pair is opposite: there are 2 different arrangements (the two possible “twists” of the colours around the cube).
In this case, the 2 arrangements are mirror images of each other. They are fixed except for the Left/Right => mirror images.

2,2,2 — No Colour Pair is Opposite

There are two different arrangements. Use the buttons to compare the two twists. (Total: 30)

Red
Green
Red
Blue
Blue
Green

Currently showing: Twist A

Check the three opposite pairs:
Top ↔ Bottom: Red ↔ Blue
Front ↔ Back: Red ↔ Green
Left ↔ Right: Blue ↔ Green

Notice that in either twist, every pair of opposite faces has two different colours. Therefore no colour-pair is opposite.

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