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math problem involving cubes and two solid colors

Mrs. Jones had some white paint and some green paint, and a bunch of wooden cubes. Her class decided to paint the cubes by making each face either solid white or green. Juan painted his cube with all 6 faces white- Julie painted her cube solid green. Hector painted 4 faces white and 2 faces green. How many cube could be painted in the fashion, so that each cube is different from the others? Two cubes are alike if one can be turned so that it exactly matches, color for color on each side, the other cube.

Answer:_________________ cubes can be painted so they are different.

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1 Answer

there are 6 general categories
 
0 sides white and 6 sides green: this can be done only 1 way
1 side white and 5 sides green: this can be done only 1 way
2 sides white and 4 sides green: this can be done 2 ways. Either white sides touch or face each other
3 sides white and 3 sides green: this can be done 2 ways. Either 2 white sides face each other and the third is between them or all 3 white sides share a common side
4 sides white and 2 sides green: this can be done 2 ways. Either green sides touch or face each other
5 sides white and 1 side green: this can be done only 1 way
6 sides white and 0 sides green: this can be done only 1 way
 
Total is 10