Editing Rubik's Cube (section)

==Solutions==
===Singmaster notation===
Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by [[David Singmaster]] to denote a sequence of moves, referred to as "Singmaster notation" or simple "Cube notation".<ref name="advgroup">{{Cite book |last=Joyner |first=David |url=https://archive.org/details/adventuresingrou0000joyn/page/7 |title=Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys |publisher=Johns Hopkins University Press |year=2002 |isbn=0-8018-6947-1 |location=Baltimore |page=[https://archive.org/details/adventuresingrou0000joyn/page/7 7]}}</ref> Its relative nature allows algorithms to be written in such a way that they can be applied regardless of which side is designated the top or how the colours are organised on a particular cube.
* ''F'' (Front): the side currently facing the solver
* ''B'' (Back): the side opposite the front
* ''U'' (Up): the side above or on top of the front side
* ''D'' (Down): the side opposite the top, underneath the Cube
* ''L'' (Left): the side directly to the left of the front
* ''R'' (Right): the side directly to the right of the front
* ''f'' (Front two layers): the side facing the solver and the corresponding middle layer
* ''b'' (Back two layers): the side opposite the front and the corresponding middle layer
* ''u'' (Up two layers): the top side and the corresponding middle layer
* ''d'' (Down two layers): the bottom layer and the corresponding middle layer
* ''l'' (Left two layers): the side to the left of the front and the corresponding middle layer
* ''r'' (Right two layers): the side to the right of the front and the corresponding middle layer
* ''x'' (rotate): rotate the entire Cube on ''R''
* ''y'' (rotate): rotate the entire Cube on ''U''
* ''z'' (rotate): rotate the entire Cube on ''F''

When a [[Prime (symbol)|prime symbol]] (&nbsp;′&nbsp;) follows a letter, it indicates an anticlockwise face turn; while a letter without a prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face.  A letter followed by a&nbsp;2 (occasionally a superscript&nbsp;<sup>2</sup>) denotes two turns, or a 180-degree turn. For example, ''R'' means to turn the right side clockwise, but ''R′'' means to turn the right side anticlockwise. The letters ''x'', ''y'', and ''z'' are used to indicate that the entire Cube should be turned about one of its axes, corresponding to R, U, and F turns respectively. When ''x'', ''y'', or ''z'' is primed, it is an indication that the cube must be rotated in the opposite direction. When ''x'', ''y'', or ''z'' is squared, the cube must be rotated 180 degrees. <!-- Which axis (x, y, z) is which? user Nikevich -->

One of the most common deviations from Singmaster notation, and in fact the current official standard, is to use "w", for "wide", instead of lowercase letters to represent moves of two layers; thus, a move of ''Rw'' is equivalent to one of ''r''.<ref name="wcarules">{{Cite web |title=World Cube Association Competition Regulations |url=http://worldcubeassociation.org/regulations/ |access-date=5 May 2012 |publisher=World Cube Association}}</ref>

For methods using middle-layer turns (particularly corners-first methods), there is a generally accepted "MES" extension to the notation where letters ''M'', ''E'', and ''S'' denote middle layer turns. It was used e.g. in Marc Waterman's Algorithm.<ref>{{Cite book |last1=Treep |first1=Anneke |title=Marc Waterman's Algorithm, Part 2 |last2=Waterman |first2=Marc |publisher=Nederlandse Kubus Club |year=1987 |series=Cubism For Fun 15 |page=10}}</ref>
* ''M'' (Middle): the layer between L and R, turn direction as L (top-down)
* ''E'' (Equator): the layer between U and D, turn direction as D (left-right)
* ''S'' (Standing): the layer between F and B, turn direction as F

The 4×4×4 and larger cubes use an extended notation to refer to the additional middle layers. Generally speaking, uppercase letters (''F B U D L R'') refer to the outermost portions of the cube (called faces). Lowercase letters (''f b u d l r'') refer to the inner portions of the cube (called slices). An asterisk (L*), a number in front of it (2L), or two layers in parentheses (Ll), means to turn the two layers at the same time (both the inner and the outer left faces) For example: (''Rr'')<nowiki>'</nowiki>&nbsp;''l''2&nbsp;''f''<nowiki>'</nowiki> means to turn the two rightmost layers anticlockwise, then the left inner layer twice, and then the inner front layer anticlockwise. By extension, for cubes of 6×6×6 and larger, moves of three layers are notated by the number 3, for example, 3L.

An alternative notation, Wolstenholme notation,<ref>{{Cite web |title=Rubik's Cube - Solution and Wolstenholme Notation |url=http://www.topaccolades.com/notation/rubikscube.htm |access-date=19 February 2022 |website=topaccolades.com}}</ref> is designed to make memorising sequences of moves easier for novices.  This notation uses the same letters for faces except it replaces U with T (top), so that all are consonants.  The key difference is the use of the vowels O, A, and I for cl'''o'''ckwise, '''a'''nticlockwise, and tw'''i'''ce (180-degree) turns, which results in word-like sequences such as LOTA RATO LATA ROTI (equivalent to LU′ R′ U L′ U′ R U2 in Singmaster notation). The addition of a C implies rotation of the entire cube, so ROC is the clockwise rotation of the cube around its right face. Middle layer moves are denoted by adding an M to the corresponding face move, so RIM means a 180-degree turn of the middle layer adjacent to the R face.

Another notation appeared in the 1981 book ''[[The Simple Solution to Rubik's Cube]]''. Singmaster notation was not widely known at the time of publication. The faces were named Top (T), Bottom (B), Left (L), Right (R), Front (F), and Posterior (P), with + for clockwise, – for anticlockwise, and 2 for 180-degree turns.

Another notation appeared in the 1982 "The Ideal Solution" book for Rubik's Revenge. Horizontal planes were noted as tables, with table 1 or T1 starting at the top. Vertical front to back planes were noted as books, with book 1 or B1 starting from the left. Vertical left to right planes were noted as windows, with window 1 or W1 starting at the front. Using the front face as a reference view, table moves were left or right, book moves were up or down, and window moves were clockwise or anticlockwise.

===Period of move sequences===
The repetition of any given move sequence on a cube which is initially in solved state will eventually return the cube back to its solved state: the smallest number of iterations required is the period of the sequence. For example, the 180-degree turn of any side has period 2 (e.g. {{nowrap|{U<sup>2</sup>}<sup>2</sup>}}); the 90-degree turn of any side has period 4 (e.g. {{nowrap|{R}<sup>4</sup>}}). The maximum period for a move sequence is 1260:<ref name="advgroup" /> for example, allowing for full rotations, {{nowrap|{F x}<sup>1260</sup>}} or {{nowrap|{R y}<sup>1260</sup>}} or {{nowrap|{U z}<sup>1260</sup>}}; not allowing for rotations, {{nowrap|{D R' U<sup>2</sup> M}<sup>1260</sup>}}, or {{nowrap|{B E L' F<sup>2</sup>}<sup>1260</sup>}}, or {{nowrap|{S' U' B D<sup>2</sup>}<sup>1260</sup>}}; only allowing for clockwise quarter turns, {{nowrap|{U R S U L}<sup>1260</sup>}}, or {{nowrap|{F L E B L}<sup>1260</sup>}}, or {{nowrap|{R U R D S}<sup>1260</sup>}}; only allowing for lateral clockwise quarter turns, {{nowrap|{F B L F B R F U}<sup>1260</sup>}}, or {{nowrap|{U D R U D L U F}<sup>1260</sup>}}, or {{nowrap|{R L D R L U R F}<sup>1260</sup>}}.

===Optimal solutions===
{{Main|Optimal solutions for Rubik's Cube}}

Although there are a significant number of possible permutations for Rubik's Cube, a number of solutions have been developed which allow solving the cube in well under 100 moves.

Many general solutions for the Cube have been discovered independently. [[David Singmaster]] first published his solution in the book ''Notes on Rubik's "Magic Cube"'' in 1981.<ref name="Singmaster">{{Cite book |last=Singmaster |first=David |title=Notes on Rubik's Magic Cube |publisher=Penguin Books |year=1981 |isbn=0-907395-00-7 |location=Harmondsworth, Eng}}</ref> This solution involves solving the Cube layer by layer, in which one layer (designated the top) is solved first, followed by the middle layer, and then the final and bottom layer. After sufficient practice, solving the Cube layer by layer can be done in under one minute. Other general solutions include "corners first" methods or combinations of several other methods. In 1982, David Singmaster and Alexander Frey hypothesised that the number of moves needed to solve the Cube, given an ideal algorithm, might be in "the low twenties".<ref>{{Cite book |last1=Frey |first1=Alexander H. Jr. |url=https://archive.org/details/handbookofcubikm0000frey |title=Handbook of Cubik Math |last2=Singmaster, David |publisher=Enslow Publishers |year=1982 |isbn=0-89490-058-7 |location=Hillside, N.J. |url-access=registration}}</ref> In 2007, Daniel Kunkle and Gene Cooperman used computer search methods to demonstrate that any 3×3×3 Rubik's Cube configuration can be solved in 26 moves or fewer.<ref name="D.Kunkle">{{Cite conference |last1=Kunkle |first1=D. |last2=Cooperman, C. |year=2007 |title=Twenty-Six Moves Suffice for Rubik's Cube |url=http://www.ccs.neu.edu/home/gene/papers/rubik.pdf |publisher=ACM Press |book-title=Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '07)}}</ref><ref>{{Cite conference |year=2008 |title=Rubik's cube proof cut to 25 moves |url=http://arxivblog.com/?p=332}}</ref><ref>{{Cite web |last=Julie J. Rehmeyer |title=Cracking the Cube |url=http://blog.sciencenews.org/mathtrek/2007/08/cracking_the_cube.html |archive-url=https://web.archive.org/web/20071011211147/http://blog.sciencenews.org/mathtrek/2007/08/cracking_the_cube.html <!-- Bot retrieved archive --> |archive-date=11 October 2007 |access-date=9 August 2007 |publisher=MathTrek}}
</ref>
In 2008, Tomas Rokicki lowered that number to 22 moves,<ref>{{Cite arXiv |eprint=0803.3435 |class=cs.SC |author=Tom Rokicki |title=Twenty-Five Moves Suffice for Rubik's Cube |year=2008}}</ref><ref>{{Cite web |title=Rubik's Cube Algorithm Cut Again, Down to 23 Moves | date=5 June 2008 |url=http://science.slashdot.org/article.pl?sid=08/06/05/2054249 |access-date=5 June 2008 |publisher=[Slashdot]}}</ref><ref>{{Cite web |last=Rokicki |first=Tom |title=Twenty-Two Moves Suffice |url=http://cubezzz.duckdns.org/drupal?q=node/view/121 |access-date=20 August 2008}}</ref> and in July 2010, a team of researchers including Rokicki, working with computers provided by [[Google]], proved that the so-called "[[God's algorithm|God's number]]" for Rubik's Cube is 20.<ref name="BBC">{{Cite news |last=Fildes |first=Jonathan |date=11 August 2010 |title=Rubik's Cube quest for speedy solution comes to an end |work=[[BBC News]] |url=https://www.bbc.co.uk/news/technology-10929159}}</ref><ref>{{Cite web |last=Flatley |first=Joseph F. |date=9 August 2010 |title=Rubik's Cube solved in twenty moves, 35 years of CPU time |url=https://www.engadget.com/2010/08/09/rubiks-cube-solved-in-twenty-moves-35-years-of-cpu-time/ |access-date=10 August 2010 |website=[[Engadget]]}}</ref><ref>{{Cite web |last1=Davidson |first1=Morley |last2=Dethridge |first2=John |last3=Kociemba |first3=Herbert |last4=Rokicki |first4=Tomas |title=God's Number is 20 |url=http://www.cube20.org/ |access-date=10 August 2010 |publisher=cube20.org}}</ref> This means that all initial configurations can be solved in 20 moves or less, and some (in fact millions) require 20.<ref name=BBC/> More generally, it has been shown that an ''n''×''n''×''n'' Rubik's Cube can be solved optimally in [[Big O notation|Θ(''n''<sup>2</sup>&nbsp;/&nbsp;log(''n''))]] moves.<ref>{{Cite arXiv |eprint=1106.5736v1 |class=cs.DS |first1=Erik D. |last1=Demaine |first2=Martin L. |last2=Demaine |title=Algorithms for Solving Rubik's Cubes |last3=Eisenstat |first3=Sarah |last4=Lubiw |first4=Anna |author4-link=Anna Lubiw |last5=Winslow |first5=Andrew |year=2011}}</ref>

===Speedcubing methods===
A solution commonly used by speedcubers was developed by [[Jessica Fridrich]]. This method is called [[CFOP method|CFOP]] standing for "Cross, F2L, OLL, PLL".  It is similar to the [[Layer by Layer|layer-by-layer]] method but employs the use of a large number of algorithms, especially for orienting and permuting the last layer. The cross is solved first, followed by first layer corners and second layer edges simultaneously, with each corner paired up with a second-layer edge piece, thus completing the first two layers (F2L). This is then followed by [[Orientation (geometry)|orienting]] the last layer, then [[permutation|permuting]] the last layer (OLL and PLL respectively). There are a total of 120 algorithms for Fridrich's method, however they are not all required to use the [[CFOP method|CFOP]] method. Most dedicated cubers will learn as many of these algorithms as possible, and most advanced cubers know all of them. If a cuber knows every algorithm for OLL they may be described as knowing full OLL. It is the same for PLL and F2L.

A now well-known method was developed by [[Lars Petrus]]. In this method, a 2×2×2 section is solved first, followed by a 2×2×3, and then the incorrect edges are solved using a three-move algorithm, which eliminates the need for a possible 32-move algorithm later. The principle behind this is that in layer-by-layer, one must constantly break and fix the completed layer(s); the 2×2×2 and 2×2×3 sections allow three or two layers (respectively) to be turned without ruining progress. One of the advantages of this method is that it tends to give solutions in fewer moves. For this reason, the method is also popular for fewest move competitions.<ref name="larspatrus">{{Cite web |title=Rubik's Cube Solution - Petrus Method |url=https://lar5.com/cube/ |access-date=8 November 2018 |publisher=lar5.com}}</ref>

The Roux Method, developed by [[Gilles Roux]], is similar to the Petrus method in that it relies on block building rather than layers, but derives from corners-first methods. In Roux, a 3×2×1 block is solved, followed by another 3×2×1 on the opposite side. Next, the corners of the top layer are solved. The cube can then be solved using only moves of the U layer and M slice.<ref name="rouxmeth">{{Cite web |title=Introduction |url=http://grrroux.free.fr/method/Intro.html |access-date=20 June 2012 |publisher=Grrroux.free.fr}}</ref>

===Beginners' methods===
Most beginner solution methods involve solving the cube [[Layer by Layer|one layer at a time]] ("layer-by-layer" method or "beginner's method"), using algorithms that preserve what has already been solved. The easiest layer by layer methods require only 3–8 algorithms.<ref>{{Cite web |title=How to solve a Rubik's Cube |url=https://how-to-solve-a-rubix-cube.com/ |access-date=28 June 2016 |publisher=how-to-solve-a-rubix-cube.com}}</ref><ref>{{Cite web |title=Beginner Solution to the Rubik's Cube (available in multiple languages) |url=http://ozcubegirl.com/rubikscubesolution.html |access-date=17 July 2017 |publisher=Jasmine Lee}}</ref>

{{anchor|patrick_bossert}}In 1981, thirteen-year-old Patrick Bossert developed a solution for solving the cube, along with a graphical notation, designed to be easily understood by novices.<ref>{{Cite journal |last=Evans |first=Rob |date=24 September 1981 |title=Restore your cube |url=https://books.google.com/books?id=_0beD5LVNugC&pg=PA802 |archive-url=https://web.archive.org/web/20200818162232/https://books.google.com/books?id=_0beD5LVNugC&pg=PA802 |url-status=dead |archive-date=18 August 2020 |journal=New Scientist |page=802 }}</ref> It was subsequently published as ''You Can Do The Cube'' and became a best-seller.<ref>{{Cite journal |year=1982 |title=Rubik's Cube |url=https://books.google.com/books?id=yvomAQAAIAAJ&q=%22you+can+do+the+cube%22 |journal=Newsweek |volume=99 |page=16}}</ref>

In 1997, Denny Dedmore published a solution described using diagrammatic icons representing the moves to be made, instead of the usual notation.<ref>{{Cite web |title=Website with solutions created by Denny Dedmore |url=http://www.helm.lu/cube/solutions/rubikscube/ |access-date=20 June 2012 |publisher=Helm.lu}}</ref>

Philip Marshall's ''The Ultimate Solution to Rubik's Cube'' takes a different approach, averaging only 65 twists yet requiring the memorisation of only two algorithms. The cross is solved first, followed by the remaining edges (using the Edge Piece Series FR'F'R), then five corners (using the Corner Piece Series URU'L'UR'U'L, which is the same as the typical last layer corner permutation algorithm), and finally the last three corners.<ref>{{Cite web |last=Marshall |first=Philip |date=2005 |title=The Ultimate Solution to Rubik's Cube |url=http://helm.lu/cube/MarshallPhilipp/ |access-date=30 December 2022 |website=helm.lu}}</ref>
<!-- *A solution developed by Ryan Heise teaches a set of underlying principles that can be used to solve the Cube in fewer than 40 moves.{{Citation needed|date=December 2008}} -->

===Rubik's Cube solver programs===
The most move optimal online Rubik's Cube solver programs use [[Optimal solutions for Rubik's Cube#Kociemba's algorithm|Herbert Kociemba's two-phase algorithm]] which can typically determine a solution of 20 moves or fewer. The user has to set the colour configuration of the scrambled cube, and the program returns the steps required to solve it.<ref>{{Cite web |title=Rubik's Cube solver |url=https://rubiks-cube-solver.com/ |access-date=28 June 2016 |publisher=rubiks-cube-solver.com}}</ref>