# [TMJ] Rubik’s 4-Cube

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New THE MATHEMATICA JOURNAL article:

## Rubik’s 4-Cube

by TAKASHI YOSHINO

ABSTRACT: Rubik’s cube has a natural extension to four-dimensional space. This article constructs the basic concepts of the puzzle and implements it in a program. The well-known three-dimensional Rubik’s cube consists of 27 unit subcubes. Each face of determines a set of nine subcubes that have a face in the same plane as . The set can be rotated around the normal through the center of . Rubik’s 4-cube (or 4D hypercube) consists of 81 unit 4-subcubes, each containing eight 3D subcubes. Each 3-face of determines a set of 27 4-subcubes that have a cube in the same hyperplane as . The set can be rotated around the normal (a plane) through the center of . Projecting the whole 4D configuration to 3D exhibits Rubik’s 4-cube as a four-dimensional extension of Rubik’s cube. Starting from a random coloring of the 4-cube, the goal of the puzzle is to return to the initial coloring of the 3-faces.