The number of ways of painting the faces of $a$ cube with six different colors is ________.

Step 1 (Break Rotational Symmetry): A cube has 6 identical faces. To paint them with 6 different colors, we place the first color on any arbitrary face (say, the bottom). Since all faces of a blank cube are identical and can be rotated into each other, there is only $1$ way to place the first color (breaking the 3D rotational symmetry).

Step 2 (Color the Opposite Face): Now, there is exactly one face directly opposite to the colored face (the top face). There are 5 remaining colors, so we have:

Step 3 (Color the Remaining 4 Side Faces): The remaining 4 faces form a ring around the vertical axis. Arranging 4 distinct colors in a circular ring in 2D has circular permutation symmetry $(n-1)!$:

Step 4 (Calculate Product): The total number of distinct color arrangements is:

Step 5 (Conclusion): The cube can be painted in exactly 30 distinct ways.