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

There are many cases possible. Basic assumption is that all faces are identical and not identified by different numbers. Case I: All the faces are painted black. ONE case. Case II: Five faces are painted black and one face is painted white. ONE case (Since cube is symmetrical figure on all its faces, that is why six cases will reduce to ONE) Case III:Four faces are painted black and two faces is painted white. Two cases. First when the two white faces are on opposite faces and second when the two white faces are on adjacent faces. Case IV: Three faces are painted black and three faces is painted white. Two cases. This can be done in two ways. Case V: Two faces are painted black and four faces is painted white. Similar to case III. Two cases. First when the two white faces are on opposite faces and second when the two white faces are on adjacent faces. Case VI: One face is painted black and five faces are painted white. Similar to case II. ONE case (Since cube is symmetrical figure on all its faces, that is why six cases will reduce to ONE) Case VII: All the faces are painted white. ONE case. So, total 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10 cases are possible. A simpler way to look at it will be – 0 sides White : 1 way (all sides Black) 1 side White : 1 way (all other sides Black) 2 sides White : 2 ways (1 way with adjacent sides White, and 1 way with opposite sides White) 3 sides White : 2 ways (1 way where three White sides have same corner, and 1 way where opposite sides are White and one center side is White) 4 sides White : 2 ways (same as 2 sides Black) 5 sides White : 1 way (same as 1 side Black) 6 sides White : 1 way (same as 0 sides Black) So total=1+1+2+2+2+1+1 = 10 ways.