A warm-up on 2×2: a smaller cube, the same ideas

The Pocket Cube is a 2×2×2 cube. It has 8 corner pieces and nothing else: no edges and no centres. It shows 8 × 3 = 24 stickers.

There is no centre to tell you which colour belongs on which face. Instead you pick one corner and treat it as fixed. A neat trick is to never turn the faces that would move it: use only R, U and F and the bottom-back-left corner never moves. Everything else is solved around it.

How many positions does it have? The eight corners can be arranged in 8! = 40,320 ways and twisted in 3⁷ = 2,187 ways (the twist of the last corner is forced by the other seven). Turning the whole cube over does not change the puzzle, so divide by the 24 ways of holding a cube: 40,320 × 2,187 ÷ 24 = 3,674,160 positions, about 3.7 million. World 3 shows where such counts come from.

How would you solve it? First build one face by moving each corner into place around the fixed one. This part you can do by thinking, with no memorised moves. Then you finish the last layer. The corner moves you will learn in World 2 work on the 2×2 too, because they only ever need to move corners.