Editing Pocket Cube (section)

==Group Theory==
[[File:Pocket cube twisted.jpg|thumb|Pocket cube with one layer partially turned]]The [[group theory]] of the [[Rubik's Cube|3×3×3 cube]] can be transferred to the 2×2×2 cube.<ref name=":1">{{citation |author=Pina Kolling |date=2021 |language=de |location=Dortmund |url=https://www.researchgate.net/publication/368732507 |title=Gruppentheorie des 2×2×2 Zauberwürfels und dessen Lösungsalgorithmen}}<!-- auto-translated from German by Module:CS1 translator --></ref> 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 [[Tuple|2-tuple]], which is made up of the following parameters:

* Position of the corner pieces as a [[Bijection|bijective]] function ([[permutation]])
* Orientation of the corner pieces as [[Vector space|vector]] x

Two moves <math>M_1</math>and <math>M_2</math> from the set <math>A_M</math>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]] <math>\sim 
</math> is introduced with <math>M_1 \sim M_2 := M_1 </math> and <math>M_2</math> result in the same cube configuration (with optional rotation of the cube). This relation is [[Reflexive relation|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 relation|transitive]], as it is similar to the mathematical relation of [[Equality (mathematics)|equality]].

With this equivalence relation, [[equivalence class]]es can be formed that are defined with <math>[ M ] := \{ M' \in A_M | M' \sim M \} \subseteq A_M</math> on the set of all moves <math>A_M</math>. Accordingly, each equivalence class <math>[M]</math> contains all moves of the set <math>A_M</math> that are equivalent to the move with the equivalence relation. <math>[M]</math> is a subset of <math>A_M</math>. All equivalent elements of an equivalence class <math>[M]</math> are the representatives of its equivalence class.

The [[quotient set]] <math>A_M / \sim </math> 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 <math>A_M / \sim </math> are all equivalence classes with regard to the equivalence relation <math>\sim 
</math>. The following therefore applies: <math>A_M / \sim := \{ [M] | M \in  A_M \}</math>. 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[[factorial|!]] positions), and seven of them can be independently [[rotation|rotated]] with three possible orientations (3<sup>7</sup> 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 [[Permutations#Circular permutations|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

:<math>\frac{8! \times 3^7}{24}=7! \times 3^6=3,674,160.</math> This is the order of the group as well.

The largest [[order (group theory)|order]] of an element in this group is 45. For example, one such element of order 45 is 
:<math>(UR^2L')</math>.

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).<ref name="Jaapsch">[http://www.jaapsch.net/puzzles/cube2.htm Jaapsch.net: Pocket Cube]</ref>

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:

{| class="wikitable" style="margin: 1em auto; text-align:right;"
|- style="text-align:center;"
!''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.
<ref>
{{ cite web 
 | url=http://sporadic.stanford.edu/bump/match/morepolished.pdf
 | title=Unravelling the (miniature) Rubik's Cube through its Cayley Graph
 | date=13 October 2006
}}
</ref>

Code that generates these results can be found here.<ref>{{cite web | url=https://medium.com/@bradenripple/enumerating-all-possible-combinations-of-a-pocket-cube-using-golang-ad80d7af23b | title=Enumerating all permutations of a Pocket Cube using Golang | date=21 July 2022 }}</ref>