Pocket Cube
The Pocket 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 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 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.
Permutations
Any permutation of the eight corners is possible, and seven of them can be independently rotated. 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 isThe maximum number of turns required to solve the cube is up to 11 half or quarter turns, or up to 14 quarter turns only.
The number a of positions that require n any 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 is of order 29,160.
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 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, also called the Varasano method, is an intermediate method. First a face is built, then the last layer is oriented and lastly both layers are permuted. The Ortega method requires a total of 12 algorithms.
The CLL method first builds a layer and then solves the second layer in one step by using one of 42 algorithms. 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 oriented. 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.
The most advanced method is the EG method. It also starts by building a layer, 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
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.The world record average of 5 solves 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, 1.09,, 1.47, and 1.07 seconds.
Top 5 solvers by single solve[World Cube Association] https://www.worldcubeassociation.org/results/events.php?eventId=222®ionId=&years=&show=100%2BPersons&single=Single Official 2x2x2 Ranking Single
Top 5 solvers by average of 5 solves[World Cube Association] https://www.worldcubeassociation.org/results/events.php?eventId=222®ionId=&years=&show=100%2BPersons&average=Average Official 2x2x2 Ranking Average
| Name | Average | Competition |
| Martin Vædele Egdal | 1.21s | Kjeller Open 2018 |
| Will Callan | 1.23s | CubingUSA Nationals 2019 |
| Jiazhou Li | 1.25s | Xi'an Cherry Blossom 2019 |
| Advay Sant | 1.31s | Oculus Cube Open 2019 |
| Zayn Khanani | 1.34s | ODU Big Blue Spring 2019 |