Pocket Cube

The Pocket Cube (also known as the Mini Cube) is a 2×2×2 combination puzzle invented in 1970 by American puzzle designer Larry D. Nichols. The cube consists of 8 pieces, all corners.

History
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 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.

Group Theory
The group theory of the 3×3×3 cube can be transferred to the 2×2×2 cube. 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 $$M_1$$and $$M_2$$ from the set $$A_M$$of 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.

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:

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

Code that generates these results can be found here.

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 most 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, which involves inspecting the entire cube and planning as many solutions as possible and choosing the best one before starting the solve by predicting where the pieces will go after finishing a side.

Notation
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

World records
The world record for the fastest single solve time is 0.43s, achieved by Teodor Zajder at Warsaw Cube Masters 2023.

The world record average of 5 solves (excluding fastest and slowest) is 0.78 seconds, set by Yiheng Wang (王艺衡) of The People's Republic of China on Jun 22, 2024 at Johor Cube Open 2024, with the times 0.74, (0.70), (0.97), 0.78, and 0.81 seconds.

=== Top 5 solvers by single solve ===

=== Top 5 solvers by Olympic average of 5 solves ===