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To solve the problem of finding the number of distinct ways to paint a cube with (m) colors where adjacent faces have different colors (up to rotation), we use combinatorial arguments and known results from group theory (Burnside's lemma) for cube rotations.

Key Insight:

The cube's rotation group has 24 elements, but most rotations do not fix valid colorings (adjacent faces different). The valid colorings are those where opposite faces can share colors, but adjacent faces cannot. The formula for distinct colorings (up to rotation) is derived from counting valid configurations and accounting for rotational symmetry.

Final Formula:

The number of distinct ways to paint the cube is: [ \boxed{m(m-1)(m^2 -3m +3)} ]

Explanation:

• For (m=2): This gives (2(1)(4-6+3)=2) (correct: two distinct colorings where opposite faces are the same color).
• For (m=3): This gives (3(2)(9-9+3)=18) (valid distinct colorings with three colors).

This formula accounts for all valid colorings and removes duplicates due to cube rotations.

Answer: (\boxed{m(m-1)(m^2 - 3m + 3)})

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