# A Derivation of an Upper Bound for the Number of Configurations of an N×N×N×N Rubik's Cube

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A Derivation of an Upper Bound for the Number of Configurations of an n×n×n×n Rubik 's Cube
By David Smith

1. Introduction C4(n) is a formula for an upper bound of the number of distinguishable configurations of an n×n×n×n Rubik 's Cube, which will be derived in this paper. It will be assumed that the reader is familiar with a 4-dimensional Rubik 's Cube. Online, one can find the free computer program Magic Cube 4D, developed by Melinda Green, Don Hatch, and Jay Berkenbilt, which is a completely interactive representation of a 4-dimensional Rubik 's Cube, and which was the inspiration for this paper and much of my other work.1 An FAQ page has been provided to help familiarize new users with the necessary concepts of higher dimensions and how Rubik 's Cubes would function in these spaces. Additionally, a solution guide has been provided by Roice Nelson, who is another pioneer in the research of higherdimensional puzzles. His creations include the free programs MagicCube5D, which was written along with Charlie Nevill, and Magic120Cell, which are representations of a 5-dimensional Rubik 's Cube and a puzzle based on the 120-cell, respectively.2,3 I would like to thank Roice in particular for his continual support and encouragement, which includes both hosting this paper and my other work on his website, and proofreading this paper while it was being developed. Roice found many oversights and errors, all of which have been corrected, and provided simplifications and new ideas. His creations MagicCube5D and Magic120Cell have also inspired me, and my work is focused on these programs as well. It should also be mentioned that my discoveries would not have been possible without the previous investigations of H. J. Kamack and T. R. Keane in their paper, "The Rubik Tesseract"; it was used extensively in developing sections 3 and 4 of this paper.4 Eric Balandraud 's article, "Calculating the Permutations of 4D Magic Cubes", was also helpful, and greatly assisted me in

References: [1] Nelson, R., Magic120Cell, http://www.gravitation3d.com/magic120cell [2] Wikipedia, http://en.wikipedia.org/wiki/120-cell [3] Yacas, http://yacas.sourceforge.net [4] Kamack, H. J. and Keane, T. R. (1982), "The Rubik Tesseract," Newark

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