rubik's revenge

Rubik's Revenge is the 4x4x4 version of Rubik's Cube. Unlike the original puzzle, it has no fixed facets: the centre facets (four per face) are free to move to different positions. The internal mechanics are rather different: the centre cubelets slide in grooves on an internal ball, which cannot be seen unless the puzzle is taken to pieces. The edge and corner cubelets glide on tracks formed by the edges of the centre cubelets in much the same way as in the 3x3x3 version. Methods for solving the 3x3x3 cube work for the edges and corners of the 4x4x4 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.

Number of permutations

There are 8 corner cubelets, 24 edge cubelets and 24 centre cubelets. Any permutation of the corner cubelets is possible, including odd permutations, giving 8! possible arrangements. Seven of the corner cubelets can be independently rotated, and the eighth cubelet's orientation depends on the other seven, giving 3⁷ combinations. Assuming the 4 centre cubelets of each colour are indistinguishable, there are 24!/(4!⁶) arrangements, all of which are possible, independently of the corner cubelets. (An odd permutation of the corner cubelets implies an odd permutation of the centre cubelets, and vice versa; however, even and odd permutations are indistinguishable because of identically coloured centre cubelets.) The 24 edge cubelets cannot be flipped. The two edge cubelets in each matching pair are distinguishable, since the colours on a cubelet are reversed relative to the other. Any permutation of the edge cubelets is possible, including odd permutations, giving 24! arrangements, independently of the corner or centre cubelets. Assuming the cube does not have a fixed orientation in space, and that the permutations resulting from rotating the cube without twisting the cube are considered identical, the number of permutations is reduced by a factor of 24. This gives a total number of permutations of

\frac{8! \cdot 3^7 \cdot 24! \cdot 24!}{4!^6 \cdot 24}=\frac{7! \cdot 9 \cdot 23!^2}{2^{12}} \approx 7.4 \cdot 10^{45}

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