We have $4$ different colors. We have to color a cube such a way that no two adjacent sides have same color. In how many ways we can color the cube? (Do not consider rotations of the same coloration to be distinct. Consider mirrorings to be distinct)
2026-04-05 02:00:07.1775354407
In how many ways a cube can be colored with 4 different colors?
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You can't use a color more than twice, so the only possible strategies for getting all six faces colored are:
Use three colors two times each, and the fourth color not at all.
Use two colors two times each, and two colors one time each.
In each of these cases it turns out that as soon as you have selected which colors to use how many times, the entire coloring is determined up to rotations of the cube.
Can you now count how many possibilities there are for each case?