Permutations of the Pocket Cube

Problem

The 2x2x2 Rubik's cube is also called the Pocket Cube. Calculate the number of arrangements of the Pocket Cube if the cube is solved only by twisting the faces (a.k.a. making legal moves).

Solution

Keep one corner piece of the Pocket Cube stationary and twist the other 7 corner pieces around the stationary corner. The non-stationary corner pieces can be permuted in $7!$ ways. These non-stationary corner pieces can be in multiple orientations, so if we fix other 6 corners can be oriented in 3 directions. Therefore, the number of possible arrangements is $7! \times 3^6$.

$$\begin{align} 7! \times 3^6 &= 5040 \times 729 \\[ 5 pt] &= 3674160 \end{align}$$

Thus, the number of arrangements on a Pocket Cube is 3,674,160.