To solve the problem of finding the number of distinct ways to paint a cube with 6 different colors (each face a unique color), we consider rotational symmetry:

Key Observations:

A cube has rotational symmetries that make some colorings identical. We need to account for these symmetries:

1. Fix one face: Choose any color for the top face (6 choices), but since we can rotate the cube to fix the top face (regardless of its initial position), we divide by 6.
2. Fix adjacent face: For the fixed top face, there are 4 adjacent faces. We can rotate the cube around the axis through the top and bottom faces to fix one adjacent face, so divide by 4.

Calculation:

Total permutations of 6 colors: $6! = 720$.
Adjust for symmetries: $\frac{6!}{6 \times 4} = \frac{720}{24} = 30$.

Answer: $\boxed{30}$

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