Algorithms for Solving Rubik's Cubes

  title={Algorithms for Solving Rubik's Cubes},
  author={Erik D. Demaine and Martin L. Demaine and Sarah Eisenstat and Anna Lubiw and Andrew Winslow},
The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n×n×n Rubik's Cube, as well as the n×n× 1 variant, has a "God's Number" (diameter of the configuration space) of Θ(n2/ log n). The upper bound comes from effectively parallelizing standard Θ(n2) solution algorithms, while the… 
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
On Algorithms for Solving the Rubik's Cube
A novel algorithm and its three variations for solving the Rubik's cube more efficiently and its algorithmic complexity of $O(n^2)$ is presented.
Advanced Rubik's Cube Algorithmic Solver
  • Vasile Dan, G. Harja, I. Nascu
  • Computer Science
    2021 7th International Conference on Automation, Robotics and Applications (ICARA)
  • 2021
A technical solution for building an Advanced Rubik's Cube Algorithmic Solver (ARCAS), that uses a PC and an Arduino Due board as processing units and an implementation of Kociemba's algorithm and blindfolded method to solve the cube.
Zero Knowledge with Rubik's Cubes and Non-abelian Groups
A public key identification scheme based on the problem of the Rubik's cube when the number of moves is fixed to a given value, which includes an interactive protocol which is zero-knowledge argument of knowledge under the assumption of the existence of a commitment scheme.
Comparison of Rubik’s Cube Solving Methods Made for Humans
The conclusion is that the CFOP, Roux, and ZZ method are fairly equivalent when it comes to move span, but CFOP has the lowest average number of moves used to solve a Rubik’s Cube.
Manu Will Unfold The Rubik’s Manifold: 42 Easy Steps To Solving The Cube
We have provided a simple step by step guide to solving the Rubik’s cube. We have aimed for this document to be completely self contained and yet easy to follow. We have described all the required
Solving Rubik’s cube via quantum mechanics and deep reinforcement learning
Rubik’s cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 1019 possible configurations. Its mathematical description is expressed by the Rubik’s group, whose elements define
Rubik's for cryptographers
This study studies mathematical generalizations of the famous Rubik’s cube puzzle, namely the factorization, representation and balance problems in non-Abelian groups and demonstrates that Cayley hash functions deserve further interest by the cryptography community.
Hamiltonian cycle and related problems : vertex-breaking, grid graphs, and Rubik's Cubes
In this thesis, we analyze the computational complexity of several problems related to the Hamiltonian Cycle problem. We begin by introducing a new problem, which we call Tree-Residue Vertex-Breaking
Layers Method Implementation for Rubik’s Cube Solution
This article will be presented step by step how to solve Rubik’s cube 3x3 by using layers method and show each detail side of Rubik's cube to facilitate new users to use and learn an application made by applying layers method.


NxN Puzzle and Related Relocation Problem
The ( n 2-1 )-Puzzle and Related Relocation Problems
This paper extends the 8-puzzle and the 15puzzle to an n xn board and shows that finding a shortest solution for the extended puzzle is NP-hard and is thus believed to be computationally infeasible.
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge
The complexity of satisfiability problems
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
The Complexity of Finding Minimum-Length Generator Sequences
The Minimum-Length Generator Sequence Problem is NP-Hard
A Real-Time Algorithm for the (n²-1)-Puzzle
  • I. Parberry
  • Computer Science, Mathematics
    Inf. Process. Lett.
  • 1995
God's number is 20
There are many different algorithms for solving the Cube, varying in complexity and number of moves required, but those that can be memorized by a mortal typically require more than forty moves.
A Survey of NP-Complete Puzzles
This article surveys NP-Complete puzzles in the hope of motivating further research in this fascinating area, particularly for those puzzles which have received little scientific attention to date.
Polynomial-time algorithms for permutation groups
It is demonstrated that the normal closure of a subgroup can be computed in polynomial time, and that this proceaure can be used to test a group for solvability.