Editing Rubik's Cube (section)

===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.