Pocket Cube

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A scrambled Pocket Cube

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

History[edit]

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[edit]

Pocket cube with one layer 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 (similar to what happens in circular permutations). 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

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]

Code that generates these results can be found here.[4]

Methods[edit]

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 2×2×2 cube are often significantly shorter and faster than the algorithms one would use for solving a 3×3×3 cube.

The Ortega method,[5] also called the Varasano method,[6] 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[7] first builds a layer (with correct permutation) and then solves the second layer in one step by using one of 42 algorithms.[8] 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. Algorithms have been generated for solving all of them.[9]

The most advanced method is the EG method.[10] 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.

Top-level speedcubers may also 1-look the puzzle, [11] which involves inspecting the entire cube and planning as much of the solution as possible before starting the solve by predicting where the pieces will go after finishing a side.

Notation[edit]

Notation is based on 3×3×3 notation but some moves are redundant (All moves are 90°, moves ending with ‘2’ are 180° turns):

  • R represents a clockwise turn of the right face of the cube
  • U represents a clockwise turn of the top face of the cube
  • F represents a clockwise turn of the front face of the cube
  • R' represents an anti-clockwise turn of the right face of the cube
  • U' represents an anti-clockwise turn of the top face of the cube
  • F' represents an anti-clockwise turn of the front face of the cube

[12]

World records[edit]

Guanbo Wang China solving it in 0.47 seconds at the Northside Spring Saturday

The world record fastest solve is Guanbo Wang China solving it in 0.47 seconds at the on 30 November 2022 at Northside Spring Saturday 2022 in Brisbane, Australia.[13]

The world record average of 5 solves (excluding fastest and slowest) is 1.02 seconds, set by Zayn Khanani of The United States on 12 February 2022 at Cape Fear 2022, with the times 1.11, (0.71), 1.04, 0.90 and (1.21) seconds.[14]

Top 5 solvers by single solve[edit]

Name Fastest solve Competition
China Guanbo Wang 0.47s Australia Northside Spring Saturday 2022
Poland Maciej Czapiewski 0.49s Poland Grudziądz Open 2016
United States Zayn Khanani 0.50s United States Babylon Summer 2022
United States Luke Garrett 0.51s United States Cubing in the Ville 2022
United States Sameer Aggarwal 0.51s United States Puget Sound Spring 2019

Top 5 solvers by average of 5 solves[15][edit]

Name Average Competition Times
United States Zayn Khanani 1.01s United States Pioneer Valley Cubing B 2023 0.91, 0.94, (0.71), 1.16, (2.91)
United States Brian Johnson 1.18s United States Mt. Prospect Open 2022 (0.94), 1.19, 1.09, 1.26, (2.44)
Netherlands Antonie Paterakis 1.18s Spain Legazpi Open 2022 1.03, (0.95), (1.64), 1.15, 1.37
Denmark Martin Vædele Egdal 1.21s Norway Kjeller Open 2018 (1.06), 1.09, (1.64), 1.47, 1.07
United States Will Callan 1.23s United States CubingUSA Nationals 2019 (0.79), (1.78), 1.17, 1.00, 1.52

See also[edit]

References[edit]

  1. ^ "Moleculon Research Corporation v. CBS, Inc". Digital-law-online.info. Retrieved 2012-06-20.
  2. ^ Jaapsch.net: Pocket Cube
  3. ^ "Unravelling the (miniature) Rubik's Cube through its Cayley Graph" (PDF). 13 October 2006.
  4. ^ "Enumerating all permutations of a Pocket Cube using Golang". 21 July 2022.
  5. ^ Ortega method tutorial by Bob Burton
  6. ^ What is Varasano?
  7. ^ What is CLL?
  8. ^ CLL tutorial by Christopher Olson
  9. ^ What is Twisty CLL?
  10. ^ Description of the EG method
  11. ^ "2x2: How To Get Faster".
  12. ^ "How to solve the 2×2×2 pocket cube speedcube puzzle".
  13. ^ World Cube Association Official Results - 2×2×2 Cube.
  14. ^ World Cube Association Official Results – 2×2×2 Cube.
  15. ^ World Cube Association Official 2×2×2 Ranking Average

External links[edit]