Pocket Cube
2x2x2 combination puzzle
title: "Pocket Cube" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["rubik's-cube"] description: "2x2x2 combination puzzle" topic_path: "general/rubik-s-cube" source: "https://en.wikipedia.org/wiki/Pocket_Cube" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary 2x2x2 combination puzzle ::
::callout[type=note] the 2×2×2 rotatable puzzle ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/2f/Pocket_cube_scrambled.jpg" caption="A scrambled Pocket Cube (having the Japanese color scheme)"] ::
The Pocket Cube (also known as the 2×2×2 Rubik's Cube and Mini Cube) is a 2×2×2 combination puzzle invented in 1970 by American puzzle designer Larry D. Nichols. The cube consists of 8 external pieces, which are all corners.
History
::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/23/Pocket_Cube_size_comparison.jpg" caption="Solved versions of, from left to right: original Pocket Cube, Eastsheen cube, V-Cube 2, V-Cube 2b"] ::
In February 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 the 3×3×3 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.
Notation
Notation is based on Singmaster notation. Since turning a layer is functionally equivalent to turning the opposite layer in the opposite direction followed by a cube rotation, only three letters are necessary to represent every possible turn:
- 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
- R2 represents a 180-degree turn of the right face of the cube
- U2 represents a 180-degree turn of the top face of the cube
- F2 represents a 180-degree turn of the front face of the cube
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 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, also called the Varasano method, 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 first builds a layer (with correct permutation) 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.
One of the more advanced methods is the EG method. It starts by building a face like in the Ortega method, 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, |url=https://jperm.net/2x2/faster |title=2x2: How To Get Faster which involves inspecting the entire cube and planning out the entire solution in the 15 seconds of inspection allotted to the solver before the solve, with the best solvers being able to plan more than one solution, considering movecount and ergonomics of each.
Group Theory
::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a9/Pocket_cube_twisted.jpg" caption="title=Gruppentheorie des 2×2×2 Zauberwürfels und dessen Lösungsalgorithmen}} The elements of the group are typically the moves of that can be executed on the cube (both individual rotations of layers and composite moves from several rotations) and the group operator is a concatenation of the moves."] ::
To analyse the group of the 2×2×2 cube, the cube configuration has to be determined. This can be represented as a 2-tuple, which is made up of the following parameters:
- Position of the corner pieces as a bijective function (permutation)
- Orientation of the corner pieces as vector x
Two moves M_1and M_2 from the set A_Mof all moves are considered equal if they produce the same configuration with the same initial configuration of the cube. With the 2×2×2 cube, it must also be considered that there is no fixed orientation or top side of the cube, because the 2×2×2 cube has no fixed center pieces. Therefore, the equivalence relation \sim is introduced with M_1 \sim M_2 := M_1 and M_2 result in the same cube configuration (with optional rotation of the cube). This relation is reflexive, as two identical moves transform the cube into the same final configuration with the same initial configuration. In addition, the relation is symmetrical and transitive, as it is similar to the mathematical relation of equality.
With this equivalence relation, equivalence classes can be formed that are defined with [ M ] := { M' \in A_M | M' \sim M } \subseteq A_M on the set of all moves A_M. Accordingly, each equivalence class [M] contains all moves of the set A_M that are equivalent to the move with the equivalence relation. [M] is a subset of A_M. All equivalent elements of an equivalence class [M] are the representatives of its equivalence class.
The quotient set A_M / \sim can be formed using these equivalence classes. It contains the equivalence classes of all cube moves without containing the same moves twice. The elements of A_M / \sim are all equivalence classes with regard to the equivalence relation \sim . The following therefore applies: A_M / \sim := { [M] | M \in A_M }. This quotient set is the set of the group of the cube.
The 2×2×2 Rubik's Cube, has eight permutation objects (corner pieces), three possible orientations of the eight corner pieces and 24 possible rotations of the cube, as there is no unique top side.
Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated with three possible orientations (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
:\frac{8! \times 3^7}{24}=7! \times 3^6=3,674,160. This is the order of the group as well.
The largest order of an element in this group is 45. For example, one such element of order 45 is :(UR^2L').
Any cube configuration can be solved in up to 14 turns (when making only quarter turns) or in up to 11 turns (when making half turns in addition to quarter turns).
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:
::data[format=table]
| 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. | url=http://sporadic.stanford.edu/bump/match/morepolished.pdf | title=Unravelling the (miniature) Rubik's Cube through its Cayley Graph | date=13 October 2006
Code that generates these results can be found here.
Number of Unique States
Here is a table of the number of unique states at each depth under different degrees of symmetry reduction, with one corner fixed. In 6-fold symmetry, the location of the fixed corner is preserved and allows mirrors. In 24-fold symmetry, all reorientations of the cube are allowed, but not mirrors. In 48-fold symmetry, all reorientations of the cube are allowed, including mirrors.
::data[format=table]
| depth | no symmetry | 6-fold symmetry | 24-fold symmetry | 48-fold symmetry |
|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 |
| 1 | 9 | 2 | 3 | 2 |
| 2 | 54 | 9 | 9 | 5 |
| 3 | 321 | 54 | 36 | 19 |
| 4 | 1847 | 309 | 132 | 68 |
| 5 | 9992 | 1670 | 529 | 271 |
| 6 | 50136 | 8361 | 2276 | 1148 |
| 7 | 227536 | 37943 | 9768 | 4915 |
| 8 | 870072 | 145046 | 36582 | 18364 |
| 9 | 1887748 | 314710 | 79006 | 39707 |
| 10 | 623800 | 104076 | 26137 | 13225 |
| 11 | 2644 | 449 | 129 | 77 |
| Total | 3674160 | 612630 | 154608 | 77802 |
| :: |
World records
The world record for single solve is 0.39 seconds, set by Ziyu Ye (叶梓渝) of China at Hefei Open 2025 on October 25, 2025.
The world record for average of 5 solves (excluding fastest and slowest) is 0.86 seconds, set by Sujan Feist of USA at Kids America Christmas Clash OH 2025 with times of 0.86, 1.02, (0.56), (1.42), and 0.70 seconds.
Top 10 solvers by single solve
::data[format=table]
| Rank | Name | Result | Competition |
|---|---|---|---|
| 1 | CHN Ziyu Ye (叶梓渝) | 0.39s | CHN Hefei Open 2025 |
| 2 | HKG Sky Guo (郭建欣) | 0.41s | |
| CHN Jiazhou Li (李佳洲) | CHN Beijing Winter 2026 | ||
| 4 | POL Teodor Zajder | 0.43s | POL Warsaw Cube Masters 2023 |
| 5 | GEO Vako Marchilashvili (ვაკო მარჩილაშვილი) | 0.44s | GEO Tbilisi April Open 2024 |
| 6 | CHN Tian Xia (夏天) | 0.45s | CHN Hefei Open 2025 |
| CHN Yiheng Wang (王艺衡) | CHN Beijing Winter 2026 | ||
| 8 | NZL Connor Johnson | 0.47s | NZL Queenspark O'Clock 2025 |
| CHN Guanbo Wang (王冠博) | AUS Northside Spring Saturday 2022 | ||
| 10 | SPA Aitor Ibañez Larrea | 0.49s | SPA León Open 2025 |
| POL Maciej Czapiewski | POL Grudziądz Open 2016 | ||
| AUS Sebastian Lee | AUS NSW State Championship 2025 | ||
| :: |
Top 10 solvers by [[Olympic average]] of 5 solves
::data[format=table]
| Rank | Name | Result | Competition | Times |
|---|---|---|---|---|
| 1 | USA Sujan Feist | 0.86s | USA Kids America Christmas Clash OH 2025 | 0.86, 1.02, (0.56), (1.42), 0.70 |
| 2 | CHN Yiheng Wang (王艺衡) | 0.87s | CHN Beijing Winter 2026 | (0.55), 0.78, 0.97, (1.28), 0.85 |
| 3 | SGP Nigel Phang | 0.90s | SGP Singapore Skewby March 2025 | 0.80, 1.05, (1.17), 0.85, (0.72) |
| 4 | USA Zayn Khanani | 0.92s | USA New-Cumberland County 2024 | 0.84, (2.69), (0.71), 1.04, 0.88 |
| 5 | NLD Antonie Paterakis | 0.97s | ESP Warm Up Portugalete 2024 | 0.93, 1.05, (0.66), (1.43), 0.92 |
| POL Teodor Zajder | POL Energy Cube Białołęka 2024 | 0.96, 1.16, 0.78, (2.30), (0.77) | ||
| POL Cube4fun in Bełchatów 2025 | 1.02, 0.82, (1.06), 1.06, (0.71) | |||
| 7 | GBR Max Tully | 1.00s | GBR Stevenage July 2025 | (1.35), (0.91), 1.10, 0.99, 0.91 |
| 8 | AUS Roman Rudakov | 1.02s | AUS Melbourne Cube Days 2024 | 1.16, 0.96, 0.94, (1.24), (0.91) |
| POL Olaf Kuźmiński | POL Cube4fun Lublin Winter 2026 | 1.20, 0.91, (1.39), (0.90), 0.94 | ||
| 10 | CHN Ziyu Ye (叶梓渝) | 1.06s | CHN Nanchang Winter 2025 | 0.99, (0.77), 0.82, 1.38, (DNF) |
| :: |
References
References
- "All About The Rubik's Cube - Cubelo".
- "Moleculon Research Corporation v. CBS, Inc". Digital-law-online.info.
- [http://www.cubewhiz.com/ortega.php Ortega method tutorial] by Bob Burton
- [http://www.cyotheking.com/ortega/ What is Varasano?]
- [http://www.cyotheking.com/cll2-2/ What is CLL?]
- [http://www.cyotheking.com/cll2-2/ CLL tutorial] by Christopher Olson
- [http://www.cyotheking.com/tcll What is Twisty CLL?]
- [https://www.speedsolving.com/wiki/index.php/EG_Method Description of the EG method]
- Pina Kolling. (2021). "Gruppentheorie des 2×2×2 Zauberwürfels und dessen Lösungsalgorithmen".
- [http://www.jaapsch.net/puzzles/cube2.htm Jaapsch.net: Pocket Cube]
- (21 July 2022). "Enumerating all permutations of a Pocket Cube using Golang".
- "Rankings {{!}} World Cube Association".
- [[World Cube Association]] [https://www.worldcubeassociation.org/results/rankings/222/average Official Results – 2×2×2 Cube].
- "Rankings {{!}} World Cube Association".
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::