# Pocket Cube

A scrambled Pocket Cube

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

## History

Solved versions of, from left to right: original Pocket Cube, Eastsheen cube, V-Cube 2, V-Cube 2b.

In March 1970, Larry D. 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 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 a(%) q(%)
0 1 1 0.000027% 0.000027%
1 9 6 0.00024% 0.00016%
2 54 27 0.0015% 0.00073%
3 321 120 0.0087% 0,0033%
4 1847 534 0.050% 0.015%
5 9992 2256 0.27% 0.061%
6 50136 8969 1.36% 0.24%
7 227536 33058 6.19% 0.90%
8 870072 114149 23.68% 3.11%
9 1887748 360508 51.38% 9.81%
10 623800 930588 16.98% 25.33%
11 2644 1350852 0.072% 36.77%
12 0 782536 0% 21.3%
13 0 90280 0% 2.46%
14 0 276 0% 0.0075%

The two-generator subgroup (the number of positions generated just by rotations of two adjacent faces) is of order 29,160.[3]

## Methods

A pocket cube can be solved with the same methods as a 3x3x3 Rubik's cube, simply by treating it as a 3x3x3 with solved (invisible) centers and edges. More advanced methods combine multiple steps and require more algorithms. These algorithms designed for solving a 2x2x2 cube are often significantly shorter and faster than the algorithms one would use for solving a 3x3x3 cube.

The Ortega method,[4] also called the Varasano method,[5] is an intermediate method. First a face is built (but the pieces may be permuted incorrectly), then the last layer is oriented (OLL) and lastly both layers are permuted (PBL). The Ortega method requires a total of 12 algorithms.

The CLL method[6] first builds a layer (with correct permutation) and then solves the second layer in one step by using one of 42 algorithms.[7] A more advanced version of CLL is the TCLL Method also known as Twisty CLL. One layer is built with correct permutation similarly to normal CLL, however one corner piece can be incorrectly orientated. The rest of the cube is solved, and the incorrect corner orientated in one step. There are 83 cases for TCLL, however algorithms have not been generated for solving all of them.[8]

The most advanced method is the EG method,[9] named after Eric Gunner. It also starts by building a layer (in any permutation), but then solves the rest of the puzzle in one step. It requires knowing 128 algorithms, 42 of which are the CLL algorithms.

## World records

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

The world record fastest solve is 0.49 seconds, set by Maciej Czapiewski of Poland on 20 March 2016 at Grudziądz Open 2016 in Grudziądz, Poland.[10]

The world record average of 5 solves (excluding fastest and slowest) is 1.21 seconds, set by Martin Vædele Egdal of Denmark on 21 October 2018 at Kjeller Open 2018, in Kjeller, Norway, with the times (1.06), 1.09, (1.64), 1.47, and 1.07 seconds.[10]

### Top 5 solvers by single solve[11]

Name Fastest solve Competition
Maciej Czapiewski 0.49s Grudziądz Open 2016
Sameer Aggarwal 0.51s Puget Sound Spring 2019
Michał Rzewuski 0.52s Grudziądz Open 2016
Jody Jones 0.53s Koalafication Melbourne 2019
Abraham Torres Ortíz Aguirre 0.54s ArCubingFest 2018

### Top 5 solvers by average of 5 solves[12]

Name Average Compeition
Martin Vædele Egdal 1.21s Kjeller Open 2018
Jiazhou Li (李佳洲) 1.25s Xi'an Cherry Blossom 2019
Advay Sant 1.31s Oculus Cube Open 2019
Zayn Khanani 1.34s ODU Big Blue Spring 2019
Maciej Czapiewski 1.35s Warsaw Cube Masters 2018

## References

1. ^ "Moleculon Research Corporation v. CBS, Inc". Digital-law-online.info. Retrieved 2012-06-20.
2. ^ Jaapsch.net: Pocket Cube