Answer:
Let's assume that the colors on the cube are numbered as follows:
Red
Blue
Green
Yellow
Purple
Orange
Without loss of generality, we can assume that the face numbered 1 is painted red. We can then use the following table to determine the colors of the other faces:
Face Number Color Opposite Face Number Opposite Color
1 Red 6 Orange
2 Blue 5 Purple
3 Green 4 Yellow
4 Yellow 3 Green
5 Purple 2 Blue
6 Orange 1 Red
We can start by choosing the color for face number 2, which must be one of the colors opposite to red (i.e., purple, green, or yellow). Let's say we choose purple. Then, the color for face number 5 must be one of the colors opposite to blue (i.e., green or yellow). Let's say we choose green. Then, the color for face number 3 must be one of the colors opposite to yellow (i.e., purple or orange). Let's say we choose orange. Now, the remaining three colors must be the ones that have not been used yet, which are blue, green, and yellow. We can then assign them to faces 4, 6, and 1, respectively.
Therefore, the numbering of the faces is as follows:
Face Number Color
1 Yellow
2 Purple
3 Orange
4 Green
5 Green
6 Blue
Note that we could have started by choosing green for face number 2 instead of purple, which would have led to a different numbering. However, we would have ended up with the same set of six colors.
In general, there are 3 choices for the color of face number 2, 2 choices for the color of face number 5, and 2 choices for the color of face number 3. After those choices are made, there is only one way to assign the remaining three colors to the remaining three faces. Therefore, there are:
3 x 2 x 2 x 1 = 12
different ways to number the faces of the cube as described in the problem.
Explanation:
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