Six colored cube puzzle
Suppose we have the colors 1 thru 6.
Each color is on exactly one face.
To determine the number of possibilities we rotate the cube so that color 1 is on the bottom face.
The top face can be any of the remaining 5 colors.
We know rotate the cube so that color 2 is on the left face (and color 1 still at the bottom).
If color 2 is at the top face we rotate so that color 3 is the left face.
Rotating this way doesn’t change the number of possibilities.
For the front, right and back face we have 3 colors left. And there are 6 ways they can be arranged (3x2x1).
The total number of unique ways to paint the cube is 30 (5×6).
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