Editing Rubik's Cube (section)

===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}}
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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>