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

• The problem asks for the number of distinct colorings of a cube's faces using two colors (black and white), where rotations of the cube are considered identical.

• We classify the distinct cubes by the number of black faces (k), ranging from 0 to 6.

• For k=0 (0 black, 6 white) and k=6 (6 black, 0 white), there is exactly 1 distinct way each.

• For k=1 (1 black, 5 white) and k=5 (5 black, 1 white), the single colored face can be in any position, but all positions are rotationally equivalent, so there is 1 distinct way for each.

• For k=2 (2 black, 4 white), the two black faces can be either adjacent or opposite, resulting in 2 distinct ways.

• For k=4 (4 black, 2 white), the two white faces can be either adjacent or opposite, resulting in 2 distinct ways.

• For k=3 (3 black, 3 white), the three black faces can meet at a single corner (forming an L-shape) or form a 'U' shape (two opposite and one adjacent to both), resulting in 2 distinct ways.

• Summing these cases: 1 (k=0) + 1 (k=1) + 2 (k=2) + 2 (k=3) + 2 (k=4) + 1 (k=5) + 1 (k=6) = 10 distinct ways.