page_title: V-Cube 7
section_title: Permutations
hierarchical_section_title: V-Cube 7 / Mechanics / Permutations
caption_attribution_description: English: A size comparison between an "ordinary" 3x3x3 cube, and a 7x7x7 v-cube.
context_page_description: The V-Cube 7 is a combination puzzle in the form of a 7×7×7 cube. The first mass-produced 7×7×7 was invented by Panagiotis Verdes and is produced by the Greek company Verdes Innovations SA. Other such puzzles have since been introduced by a number of Chinese companies, some of which have mechanisms which improve on the original. Like the 5×5×5, the V-Cube 7 has both fixed and movable center facets.
context_section_description: There are 8 corner "cubies", 60 edges and 150 centers (6 fixed, 144 movable).
  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 144 movable centers, consisting of six sets of 24 pieces each. Within each set there are four centers of each color. Centers from one set cannot be exchanged with those from another set. Each set can be arranged in 24! different ways. Assuming that the four centers of each color in each set are indistinguishable, the number of permutations of each set is reduced to 24!/(24⁶) arrangements, all of which are possible. The reducing factor comes about because there are 24 (4!) ways to arrange the four pieces of a given color. This is raised to the sixth power because there are six colors. The total number of permutations of all movable centers is the permutations of a single set raised to the sixth power, 24!⁶/(24³⁶). 
  There are 60 edge pieces, consisting of 12 central, 24 intermediate, and 24 outer edges. The central edges can be flipped but the rest cannot (because the internal shape of the pieces is asymmetrical), nor can an edge from one set exchange places with one from another set. The five edges in each matching quintet are distinguishable, since corresponding non-central edges are mirror images of each other. There are 12!/2 ways to arrange the central edges, since an odd permutation of the corners implies an odd permutation of these pieces as well. There are 2¹¹ ways that they can be flipped, since the orientation of the twelfth edge depends on the preceding eleven. Any permutation of the intermediate and outer edges is possible, including odd permutations, giving 24! arrangements for each set or 24!² total, regardless of the position or orientation of any other pieces.
  This gives a total number of permutations of
  The entire number is 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 (roughly 19,501 sexvigintillion or 19.5 sexvigintilliard on the long scale or 19.5 duoquinquagintillion on the short scale).
  One of the fixed center pieces is usually marked with the logo of the manufacturer, such as V in a cube by V-Cube. This center piece can be oriented in four different ways, which increases the number of patterns by a factor of four to 7.80×10¹⁶⁰. Any orientation of the fixed center piece is usually regarded as solved.
caption: A size comparison between an original size 3×3×3 cube, and a 7×7×7 V-Cube 7
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