# 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.

## History

In March 1970, Larry Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted U.S. Patent 3,655,201 on April 11, 1972, two years before Rubik invented his Cube.

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal in 1982. In 1984, Ideal lost the patent infringement suit and appealed. In 1986, the appeals court affirmed the judgment that Rubik's 2×2×2 Pocket Cube infringed Nichols's patent, but overturned the judgment on Rubik's 3×3×3 Cube.[1]

## 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

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

The maximum number of turns required to solve the cube is up to 11 half or quarter turns, or up to 14 quarter turns only.[2]

The number a of positions that require n any (half or quarter) turns and number q of positions that require n quarter turns only are:

n a 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. [3]

The two algorithms that can be used to solve the last layer:

1.To fix the positions of the corners:(U R U' L' U R' U' L)

2.To adjust the corners:(D R' D' R)

## Records

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

Maciej Czapiewski (Poland) holds the current world record for solving the Pocket Cube in competition, with a time of 0.49 seconds at the Grudziądz Open 2016. [4]

The best average of five consecutive solves in competition is 1.51 seconds, set by Lucas Etter (USA) at the Music City Speedsolving 2015 competition. The times in his average, of which the best and worst are dropped, were (1.24), 1.69, (2.21), 1.45, and 1.39. [5]