Editing Rubik's Cube (section)

==Mathematics==
The puzzle was originally advertised as having "over 3,000,000,000 (three [[1000000000 (number)|billion]]) combinations but only one solution".<ref>{{Cite AV media |url=https://www.youtube.com/watch?v=jTkhA3RO5fU |title=Rubik's Cube Commercial 1981 |date=23 October 2008 |access-date=10 October 2017 |archive-url=https://ghostarchive.org/varchive/youtube/20211211/jTkhA3RO5fU |archive-date=11 December 2021 |url-status=live |via=YouTube}}{{cbignore}}</ref>  Depending on how combinations are counted, the actual number is significantly higher.

===Permutations===
{{main|Rubik's Cube group}}

[[File:Rubik's cube colors.svg|thumb|The current colour scheme of a Rubik's Cube – yellow opposes white, blue opposes green, orange opposes red, and white, green, and red are positioned in anti-clockwise order around a corner.]]
The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are [[factorial|8!]] (40,320) ways to arrange the corner cubes. Each corner has three possible orientations, although only seven (of eight) can be oriented independently; the orientation of the eighth (final) corner depends on the preceding seven, giving 3<sup>7</sup> (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, restricted from 12! because edges must be in an [[even permutation]] exactly when the corners are. (When arrangements of centres are also permitted, as described below, the rule is that the combined arrangement of corners, edges, and centres must be an even permutation.) Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 2<sup>11</sup> (2,048) possibilities.<ref>{{Cite web |last=Schönert |first=Martin |title=Analyzing Rubik's Cube with GAP |url=https://www.gap-system.org/Doc/Examples/rubik.html |access-date=30 December 2022 |website=gap-system.org |archive-date=28 September 2017 |archive-url=https://web.archive.org/web/20170928194734/http://www.gap-system.org/Doc/Examples/rubik.html |url-status=dead }}</ref>

:<math> {8! \times 3^7 \times \frac{12!}{2} \times 2^{11}} = 43{,}252{,}003{,}274{,}489{,}856{,}000</math>

which is approximately 43 [[quintillion]].<ref>{{Cite web |date=March 17, 2009 |title=The Mathematics of the Rubik's Cube |url=https://web.mit.edu/sp.268/www/rubik.pdf |website=Massachusetts Institute of Technology}}</ref><!-- 12!8!*2^11*3^7/2 expanded: 12! = 479,001,600, 8! = 40,320, 2^11 = 2,048, 3^7 = 2,187, 479,001,600 * 40,320 * 2,048 * 2,187 / 2 = 43,252,003,274,489,856,000 (approximately 4.33 x 10^19) -->  To put this into perspective, if one had one standard-sized Rubik's Cube for each [[permutation]], one could cover the Earth's surface 275 times, or stack them in a tower 261 [[light-year]]s high.

The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times larger:

:<math> {8! \times 3^8 \times 12! \times 2^{12}} = 519{,}024{,}039{,}293{,}878{,}272{,}000</math>

which is approximately 519&nbsp;quintillion<ref name="Vaughen">{{Cite web |author=Scott Vaughen |title=Counting the Permutations of the Rubik's Cube |url=http://faculty.mc3.edu/cvaughen/rubikscube/cube_counting.ppt |url-status=dead |archive-url=https://web.archive.org/web/20110719235049/http://faculty.mc3.edu/cvaughen/rubikscube/cube_counting.ppt |archive-date=19 July 2011 |access-date=19 January 2011 |website=Montgomery County Community College |language=en}}</ref> possible arrangements of the pieces that make up the cube, but only one-twelfth of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus, there are 12 possible sets of reachable configurations, sometimes called "universes" or "[[orbit (group theory)|orbits]]", into which the cube can be placed by dismantling and reassembling it.

The preceding numbers assume the centre faces are in a fixed position.  If one considers turning the whole cube to be a different permutation, then each of the preceding numbers should be multiplied by 24.  A chosen colour can be on one of six sides, and then one of the adjacent colours can be in one of four positions; this determines the positions of all remaining colours.

===Centre faces===
The original Rubik's Cube had no orientation markings on the centre faces (although some carried the "Rubik's Cube" mark on the centre square of the white face), and therefore solving it does not require any attention to orienting those faces correctly. However, with marker pens, one could, for example, mark the central squares of an unscrambled Cube with four coloured marks on each edge, each corresponding to the colour of the adjacent face; a cube marked in this way is referred to as a "supercube". Some Cubes have also been produced commercially with markings on all of the squares, such as the [[Lo Shu Square|Lo Shu]] [[magic square]] or [[playing card]] [[suit (cards)|suits]]. Cubes have also been produced where the nine stickers on a face are used to make a single larger picture, and centre orientation matters on these as well. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centres as well.

Marking Rubik's Cube's centres increases its difficulty, because this expands the set of distinguishable possible configurations. There are 4<sup>6</sup>/2 (2,048) ways to orient the centres since an even permutation of the corners implies an even number of quarter turns of centres as well. In particular, when the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of centre squares requiring a quarter turn. Thus orientations of centres increases the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×10<sup>19</sup>) to 88,580,102,706,155,225,088,000 (8.9×10<sup>22</sup>).<ref>{{cite journal
 | last = Hofstadter | first = Douglas R. | author-link = Douglas Hofstadter
 | date = March 1981
 | issue = 3
 | journal = [[Scientific American]]
 | jstor = 24964321
 | pages = 20–39
 | title = Metamagical themas: The Magic Cube's cubies are twiddled by cubists and solved by cubemeisters
 | volume = 244| doi = 10.1038/scientificamerican0381-20 }}; see p. 22</ref>

When turning a cube over is considered to be a change in permutation then we must also count arrangements of the centre faces. Nominally there are 6! ways to arrange the six centre faces of the cube, but only 24 of these are achievable without disassembly of the cube. When the orientations of centres are also counted, as above, this increases the total number of possible Cube permutations from 88,580,102,706,155,225,088,000 (8.9×10<sup>22</sup>) to 2,125,922,464,947,725,402,112,000 (2.1×10<sup>24</sup>).

===Algorithms===
In Rubik's cubers' parlance, a memorised sequence of moves that have a desired effect on the cube is called an "algorithm". This terminology is derived from the mathematical use of ''[[algorithm]]'', meaning a list of well-defined instructions for performing a task from a given initial state, through well-defined successive states, to a desired end-state. Each method of solving the Cube employs its own set of algorithms, together with descriptions of what effect the algorithm has, and when it can be used to bring the cube closer to being solved.

Many algorithms are designed to transform only a small part of the cube without interfering with other parts that have already been solved so that they can be applied repeatedly to different parts of the cube until the whole is solved. For example, there are well-known algorithms for cycling three corners without changing the rest of the puzzle or flipping the orientation of a pair of edges while leaving the others intact.

Some algorithms do have a certain desired effect on the cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side effects and are employed early on in the solution when most of the puzzle has not yet been solved and the side effects are not important. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead.

===Relevance and application of mathematical group theory===
Rubik's Cube lends itself to the application of [[Rubik's Cube group|mathematical group theory]], which has been helpful for deducing certain algorithms – in particular, those which have a ''[[commutator]]'' structure, namely ''XYX''<sup>−1</sup>''Y''<sup>−1</sup> (where ''X'' and ''Y'' are specific moves or move-sequences and ''X''<sup>−1</sup> and ''Y''<sup>−1</sup> are their respective inverses), or a ''[[Conjugation (group theory)|conjugate]]'' structure, namely ''XYX''<sup>−1</sup>, often referred to by speedcubers colloquially as a "setup move".<ref name="Singmaster" /> In addition, the fact that there are well-defined [[subgroup]]s within the Rubik's Cube group enables the puzzle to be learned and mastered by moving up through various self-contained "levels of difficulty". For example, one such "level" could involve solving cubes that have been scrambled using only 180-degree turns. These subgroups are the principle underlying the computer cubing methods by [[Morwen Thistlethwaite#Thistlethwaite's algorithm|Thistlethwaite]] and [[Optimal solutions for Rubik's Cube#Kociemba's algorithm|Kociemba]], which solve the cube by further reducing it to another subgroup.

===Unitary representation===
The Rubik's group can be endowed with a [[unitary representation]]: such a description allows the Rubik's Cube to be mapped into a quantum system of few particles, where the rotations of its faces are implemented by unitary operators. The rotations of the faces act as generators of the [[Lie group]].<ref>{{cite journal |author1=S. Corli |author2=L. Moro |author3=D.E. Galli |author4=E. Prati | title=Solving Rubik's Cube via Quantum Mechanics and Deep Reinforcement Learning| journal=[[Journal of Physics A: Mathematical and Theoretical]]| year=2021| volume=54 | issue=5 | pages=425302 |arxiv=2109.07199|doi=10.1088/1751-8121/ac2596|bibcode=2021JPhA...54P5302C |s2cid=237513509  |issn=1751-8113 }}</ref>