Pocket Cube

From left to right: original Pocket Cube, Eastsheen cube, V-Cube 2, V-Cube 2b.

The Pocket Cube (also known as the Mini Cube or the Ice Cube) is the 2×2×2 equivalent of a Rubik's Cube. The cube consists of 8 pieces, all corners.

Permutations

Pocket Cube in different forms. From top (to bottom):
i. Solved pocket cube.
ii. Scrambled pocket cube.
iii.Pocket cube with one side tilted.

Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). There is nothing identifying the orientation of the cube in space, reducing the positions by a factor of 24. This is because all 24 possible positions and orientations of the first corner are equivalent due to the lack of fixed centers. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. The number of possible positions of the cube is

$\frac{8! \times 3^7}{24}=7! \times 3^6=3,674,160.$

The maximum number of turns required to solve the cube is up to 11 full turns, or up to 14 quarter turns.[1]

The number f of positions that require n full twists and number q of positions that require n quarter turn twists are:

n f q
0 1 1
1 9 6
2 54 27
3 321 120
4 1847 534
5 9992 2256
6 50136 8969
7 227536 33058
8 870072 114149
9 1887748 360508
10 623800 930588
11 2644 1350852
12 0 782536
13 0 90280
14 0 276

For the miniature (2 × 2 × 2) Rubik’s cube, the two-generator subgroup (the number of positions generated just by rotations of two adjacent faces) is of order 29,160. [2]

Records

Vicente Albíter of Mexico solving it in 1.55 seconds at the Mexican Open 2008

Christian Kaserer holds the current world record of solving the Pocket Cube in competition, with a time of 0.69 seconds set at the Trentin Open 2011, using only three turns to solve the cube.[3][4] For the best average time of 5 solves, Rami Sbahi and Lucas Etter hold the world record with a time of 1.69 seconds including the individual times of 1.58, 1.82, (1.91), (1.50), and 1.66 for Rami, while Lucas's times are: 1.27 1.50 2.16 1.40 3.65. [4]