page_title: Rubik's Revenge
section_title: Permutations
hierarchical_section_title: Rubik's Revenge / Mechanics / Permutations
caption_attribution_description: English: This is my en:Rubik's Revenge with a tilted side. Released under the en:GFDL.
context_page_description: The Rubik's Revenge is a 4×4×4 version of Rubik's Cube. It was released in 1981. Invented by Péter Sebestény, the Rubik's Revenge was nearly called the Sebestény Cube until a somewhat last-minute decision changed the puzzle's name to attract fans of the original Rubik's Cube. Unlike the original puzzle, it has no fixed facets: the centre facets are free to move to different positions.
  Methods for solving the 3×3×3 cube work for the edges and corners of the 4×4×4 cube, as long as one has correctly identified the relative positions of the colours — since the centre facets can no longer be used for identification.
context_section_description: There are 8 corners, 24 edges and 24 centres.
  Any permutation of the corners is possible, including odd permutations. Seven of the corners can be independently rotated, and the orientation of the eighth depends on the other seven, giving 8!×3⁷ combinations. 
  There are 24 centres, which can be arranged in 24! different ways. Assuming that the four centres of each colour are indistinguishable, the number of permutations is reduced to 24!/(24⁶) arrangements. The reducing factor comes about because there are 24 (4!) ways to arrange the four pieces of a given colour. This is raised to the sixth power because there are six colours. An odd permutation of the corners implies an odd permutation of the centres and vice versa; however, even and odd permutations of the centres are indistinguishable due to the identical appearance of the pieces. There are several ways to make the centre pieces distinguishable, which would make an odd centre permutation visible.
  The 24 edges cannot be flipped, because the internal shape of the pieces is asymmetrical. Corresponding edges are distinguishable, since they are mirror images of each other. Any permutation of the edges is possible, including odd permutations, giving 24! arrangements, independently of the corners or centres.
  Assuming the cube does not have a fixed orientation in space, and that the permutations resulting from rotating the cube without twisting it are considered identical, the number of permutations is reduced by a factor of 24. This is because all 24 possible positions and orientations of the first corner are equivalent because of the lack of fixed centres. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centres which identify the cube's spatial orientation.
  This gives a total number of permutations of 
  The full number is 7401196841564901869874093974498574336000000000 possible permutations (about 7401 septillion, 7.4 septilliard on the long scale or 7.4 quattuordecillion on the short scale).
  Some versions of Rubik's Revenge have one of the centre pieces marked with a logo, distinguishing it from the other three of the same colour. This increases the number of distinguishable permutations by a factor of four to 2.96×10⁴⁶, although any of the four possible positions for this piece could be regarded as correct.
caption: Rubik's Revenge with a tilted side
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