A variation on the Rubik's cube

On the nxnxn Rubik's Cube

We describe in details the nxnxn Rubik's Cube, namely a Rubik's Cube with n rotating slices in each face. Then we state and prove the "first law of Cubology", i.e. the solvability criterion, for it

The Completeness of 2D Rubik's Shapes

The Rubik's cube was invented in 1974 by Erno Rubik, who had no idea of the incredible popularity and mathematical fascinations his toy would bring. Through the years of study on the mathematical properties of the cube, the Rubik's Cube group was introduced to represent all possible moves one could perform on the cube. In this paper, we define a planar analogue to the Rubik's cube, which we dub the Rubik's Square, and prove that the Rubik's square is complete in the sense that given any two configurations there is a sequence of moves which changes one to the other. The Rubik's cube does not ha…

$n \times n \times n$ Rubik's Cubes and God's Number

The Rubik's Cube is the most popular puzzle in the world. Two of its studied aspects are God's Number, the minimum number of turns necessary to solve any state, and the first law of cubology, a solvability criterion. We modify previous statements of the first law of cubology for $n \times n \times n$ Rubik's Cubes, and prove necessary and sufficient solvability conditions. We compute the order of the Rubik's Cube group and the number of distinct configurations of the $n \times n \times n$ Rubik's Cube. Finally, we derive a lower bound for God's Number using the group theoretical results and a …

A New Upper Bound on Rubik's Cube Group

In this paper we give a upper bound of 40 on Rubik's cube group. The previously known upper bound has been 42 since 1995. In order to prove our claim we use computational methods. The program used is GAP computer algebra. Further more we give the number of elements of given length in two important subgroups of Rubik's cube group.

Solving the Rubik's Cube Optimally is NP-complete

In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an $n \times n \times n$ Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square---an $n \times n \times 1$ generalization of the Rubik's Cube---and then proceed with a similar but more complicated proof for the Rubik's Cube case.

魔方的抽象多面体

Rubik's Abstract Polytopes

本文将魔方及其构型群推广至任意抽象正多面体，完整刻画任意维魔方单纯形的相关群，并概述魔方超立方体的类似结论。