Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?

A cube has 6 faces, and each face can be painted in one of two colors (black or white). We need to find the number of distinct ways to paint the cube, considering that rotations of the cube result in the same painting.

We can enumerate the distinct patterns by considering the number of black faces:

1. 0 Black faces (all white): There is only 1 way (all faces are white).
2. 1 Black face: All positions for a single black face are equivalent by rotation. So, 1 way.
3. 2 Black faces:
   • The two black faces can be adjacent (sharing an edge). All such pairs are rotationally equivalent. (1 way)
   • The two black faces can be opposite (sharing no edge). All such pairs are rotationally equivalent. (1 way) Total = 2 distinct ways.
4. 3 Black faces:
   • The three black faces can meet at a corner (like the three faces visible from one vertex). No two of these faces are opposite. (1 way)
   • Two black faces can be opposite, and the third face is adjacent to both of them. (1 way) Total = 2 distinct ways.

Now, we can use symmetry. Painting 'k' faces black is equivalent to painting '6-k' faces white. So:

5. 4 Black faces: This is symmetric to having 2 white faces. So, there are 2 ways (same as for 2 black faces).
6. 5 Black faces: This is symmetric to having 1 white face. So, there is 1 way (same as for 1 black face).
7. 6 Black faces (all black): This is symmetric to having 0 white faces. So, there is 1 way (all faces are black).

Adding up all the distinct ways: 1 (for 0 black) + 1 (for 1 black) + 2 (for 2 black) + 2 (for 3 black) + 2 (for 4 black) + 1 (for 5 black) + 1 (for 6 black) = 10 ways.

The final answer is 10.

The final answer is B