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Solving the Rubik's Cube Optimally is NP-complete
Abstract
In this paper, we prove that optimally solving an 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 Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square---an generalization of the Rubik's Cube---and then proceed with a similar but more complicated proof for the Rubik's Cube case.
... Finding the shortest paths on generic finite Cayley graphs is an NP-hard problem [30], as it is for many particular groups: the Rubik's Cube group [31] and some others [18,32]. Brute force breadth-first search, Dijkstra's, and related methods can find the shortest paths on graphs with billions of nodes, the bidirectional trick squares feasible sizes, but these methods require extremely large computational resources and are not practical for much larger sizes, which are of our interest. ...
The paper proposes a novel machine learning-based approach to the pathfinding problem on extremely large graphs. This method leverages diffusion distance estimation via a neural network and uses beam search for pathfinding. We demonstrate its efficiency by finding solutions for 4x4x4 and 5x5x5 Rubik's cubes with unprecedentedly short solution lengths, outperforming all available solvers and introducing the first machine learning solver beyond the 3x3x3 case. In particular, it surpasses every single case of the combined best results in the Kaggle Santa 2023 challenge, which involved over 1,000 teams. For the 3x3x3 Rubik's cube, our approach achieves an optimality rate exceeding 98%, matching the performance of task-specific solvers and significantly outperforming prior solutions such as DeepCubeA (60.3%) and EfficientCube (69.6%). Additionally, our solution is more than 26 times faster in solving 3x3x3 Rubik's cubes while requiring up to 18.5 times less model training time than the most efficient state-of-the-art competitor.
... In terms of related work, So Hirata has recently released two interesting papers concerning the diameter of Rubik's puzzles [12,13], which attack the problem from different angles; either considering the girth of the cube's graph or using estimations for its branching factor. On a more theoretical line of work, Demaine et al. proved that computing the diameter of anˆR ubik's cube is NP-hard [8]5, and that the diameter of theˆˆcube is Θ´2 log¯ [ 7]. ...
It is well-known by now that any state of the Rubik's Cube can be solved in at most 20 moves, a result often referred to as "God's Number". However, this result took Rokicki et al. around 35 CPU years to prove and is therefore very challenging to reproduce. We provide a novel approach to obtain a worse bound of 36 moves with high confidence, but that offers two main advantages: (i) it is easy to understand, reproduce, and verify, and (ii) our main idea generalizes to bounding the diameter of other vertex-transitive graphs by at most twice its true value, hence the name "demigod number". Our approach is based on the fact that, for vertex-transitive graphs, the average distance between vertices is at most half the diameter, and by sampling uniformly random states and using a modern solver to obtain upper bounds on their distance, a standard concentration bound allows us to confidently state that the average distance is around , from where the diameter is at most 36.
... A reconfiguration problem asks questions like: given two different states (configurations) of a system, is it possible to gradually transform (reconfigure) the first configuration to the second by modifying it in a slow, step-by-step manner? Two popular examples of reconfiguration problems are the 15-puzzle [2,3] and the Rubik's cube [4,5]. In both, we want to determine how to reach a "solved" final configuration using a sequence of "moves", starting from a given initial configuration. ...
Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming P≠NP), even for relaxed variants of the problem (assuming P≠PSPACE). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k≥2) contiguous vertices on a shortest path can be changed at a time.
... The Transitional Interface was perceived as slightly more usable during the Sorting Task, yet the scores for the Rubik's Cube Task are still very high. Even though solving a Rubik's Cube is way more complex than sorting shapes [10,27], the Sorting Task received higher mental load scores than the Rubik's Cube Task, with some outliers. The turning instructions made the Rubik's Cube Task pretty easy, while the heavy transitioning in the Sorting Task may have added to generally higher mental load scores. ...
The concept of Transitional Cross-Reality Interfaces where a user can seamlessly transition across the Reality-Virtuality-Continuum dates back to the introduction of the Magic Book in 2001 but recently gained new research momentum since head-mounted displays advanced by combining the former distinctive concepts of augmented reality and virtual reality into a single device. New technological capabilities require new ways of developing applications to satisfy user requirements. Especially in the context of operating in multiple realities, developing AR/VR applications is not a trivial task. In this context, the development of Transitional Cross-Reality Interfaces remains a complex engineering problem that requires specific methods, concepts, and tools. To address this problem and provide a systematic development approach, we propose a conceptual framework for both developing and evaluating Transitional Cross-Reality Interfaces. To evaluate our solution, we implemented a Transitional Cross-Reality Interface based on our framework and evaluated it with both the measurements provided by the framework and additional questionnaires. We found high usability and interesting transitional behavior of users indicating the usefulness of the proposed framework as an underlying software architecture for Transitional Cross-Reality Interfaces.
... Specifically, in our experiments, we use the Rubik's Cube, Sokoban, N-Puzzle, and Inequality Theorem Proving (INT) [60]. These are classical benchmarks, explored in various planning works [41,14], and are known to be NP-hard [15,13,46]. Descriptions of these environments are in Appendix A. ...
Efficiently tackling combinatorial reasoning problems, particularly the notorious NP-hard tasks, remains a significant challenge for AI research. Recent efforts have sought to enhance planning by incorporating hierarchical high-level search strategies, known as subgoal methods. While promising, their performance against traditional low-level planners is inconsistent, raising questions about their application contexts. In this study, we conduct an in-depth exploration of subgoal-planning methods for combinatorial reasoning. We identify the attributes pivotal for leveraging the advantages of high-level search: hard-to-learn value functions, complex action spaces, presence of dead ends in the environment, or using data collected from diverse experts. We propose a consistent evaluation methodology to achieve meaningful comparisons between methods and reevaluate the state-of-the-art algorithms.
... Shuffle strings will be generated using the World Cube Association's Java cube shuffler, TNoodle. This is the industry standard for performing random and fair cube shuffles for cubing competitions.The Rubik's Cube has been proven to be NP-Complete meaning that the validity of a solution can be tested quickly [14]. Solution validity will be tested using a verify script I have written using the cubeai Python library. ...
Since its introduction to the international market in 1980 finding methods to solve the Rubik's Cube TM has become a worldwide obsession. An increase in available processing power has allowed artificial intelligence to progressively deliver increasingly efficient solutions for the Cube. The speed with which the Cube can be solved has reduced exponentially over time for both humans and robots, with robots using a physical cube already capable of surpassing human speed performance by a factor of 7:1. Hobbyists and experts have toyed with programmatic solutions for the Rubik's Cube over the last 40 years. Many programs are no longer maintained, or hidden on obscure personal websites, untouched since their creation. This thesis will review the effectiveness of these long-since forgotten solutions and compare their performance with more modern solutions, exploring how machine learning to solve the Cube has evolved by benchmark testing the performance of older solutions on modern hardware. It will explore which data structure is most efficient for representing the Rubik's Cube programmatically to artificially intelligent alogrithms, and experiment with training a neural network on a cubeless solution, giving no domain specific information, but merely shuffles and solutions as the training input.
... The same property can be found in another one of the most famous puzzle: Rubic's cube. Recently, similar results have been shown for the generalized Rubic's cube problem for a cube of size n  n  n (see [20,22]): (1) Any arrangement has its parity, and two arrangements S and T can be transformed with each other if and only if they have the same parity, (2) if a pair of two arrangements S and T is a yes-instance, we can find a solution of polynomial length of n between S and T, however, (3) finding a shortest way from S to T is NP-complete. It may be worth for mentioning that when n ¼ 3, it is known that any pair of S and T in the same parity can be transformed in 20 moves and there exits a pair of S and T that requires 20 moves to transform. ...
Since the 1930s, mathematicians and computer scientists have been interested in computation. While mathematicians investigate recursion theory, computer scientists investigate computational complexity based on Turing machine model to understand what a computation is. Beside them, there is another approach of research on computation, which is the investigation of puzzles and games. Once we regard the rules used in puzzles and games as the set of basic operations of computation, we can perform some computation by solving puzzles and playing games. In fact, research on puzzles and games from the viewpoint of theoretical computer science has continued without any break in the history of theoretical computer science. Sometimes the research on computational complexity classes has proceeded by understanding the tons of puzzles. The wide collection of complete problems for a specific computational complexity class shares a common property, which gives us a deep understanding of the class. In this survey paper, we give a brief history of research on computational complexities of puzzles and games with related results and trends in theoretical computer science.
... Given two different configurations of a system, is it possible to gradually transform one to the other? The two most popular examples of reconfiguration problems are the 15-puzzle (Ratner and Warmuth 1986;Goldreich 2011) and the Rubik's cube (Demaine et al. 2011;Demaine, Eisenstat, and Rudoy 2018). In both, we want to determine how to reach a "solved" final configuration using a sequence of "moves", starting from a given initial configuration. ...
Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalise the problem to when at most k (for some k >= 2) contiguous vertices on a shortest path can be changed at a time.
... Many mathematicians have studied the cube, giving group theoretical analyses and solutions for certain cases [2,3,15]. Another well-studied facet of the Rubik's Cube is God's Number: the minimum number of face turns needed to solve the puzzle from any position [13,14,9,17,6]. Additionally, the Rubik's Cube can act as a physical model of chaotic behaviour, which has led to applications in cryptography [7,16] as well as physics [4,10,11]. ...
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 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 Rubik's Cube. Finally, we derive a lower bound for God's Number using the group theoretical results and a counting argument.
... A well studied extension [5,4,3,8] of the Rubik's cube is the n × n × n-cube where the 3-lanes of the cubes are extended to n. There has also been interest [10] in cryptographic applications of Rubik's Cube. ...
... A well studied extension [5,4,3,8] of the Rubik's cube is the n × n × n-cube where the 3-lanes of the cubes are extended to n. There has also been interest [10] in cryptographic applications of Rubik's Cube. ...
The Rubik's cube is a famous puzzle in which faces can be moved and the corresponding movement operations define a group. We consider here a generalization to any 3-valent map. We prove an upper bound on the size of the corresponding group which we conjecture to be tight.
... In this work, we present the design and evaluation of an AR based Rubiks cube solving application and compare it with paper-based instructions. The Rubik's Cube solving task was selected due to its complexity (i.e. standard 3x3 Rubik's Cube has 43 quintillion possible states -an NP-complete problem [8]) and its familiarity. For each instruction modality, we employed implicit and explicit data capture and analysis. ...
Augmented reality (AR) can support a user in performing an expert task by overlaying real world objects with the domain specific information required to complete the task. Understanding how users can process and use such information is very important for informing the design of AR technologies and applications. In this paper, the results of a quality of experience (QoE) evaluation of an AR application for the task of solving a Rubik’s Cube are presented. The Rubik’s Cube was selected based on its familiarity and the expertise needed to solve it unaided. An empirical approach was taken to identify the QoE features that affect the usability and utility of an AR head-mounted display (HMD) compared with paper-based instruction. The QoE evaluation methodology involved the capture and analysis of implicit and explicit QoE metrics. The utility (in terms of performance) of each mode of instruction was objectively measured using: (a) cube completion success rates; and (b) time-to-completion. The implicit metrics of electrodermal activity (EDA), skin temperature, heart rate and the novel use of facial action units (AUs) were recorded to infer emotional state during the task completion. Finally, with respect to explicit metrics, the test subjects completed a Likert scale questionnaire post the experience to subjectively report QoE as well as a self-assessment manikin (SAM) questionnaire to self-report emotional state upon task completion. The results show that AR yielded higher success rates and significantly lower time-to-completion rates. The AR group explicitly reported higher levels of positive valance (affective state) than the paper-based group. The physiological data showed that the AR group were less stressed (via EDA) than the paper-based group. Finally, analysis of the AU data reflected a greater than chance (total: 21.85%) accuracy when predicting affective state based on SAM questionnaires as ground-truth.
... Recently, there has been considerable interest in the complexity of finding shortest transformations between configurations. Examples include finding a shortest transformation between triangulations of planar point sets [21] and simple polygons [1], configurations of the Rubik's cube [6], and satisfying assignments of Boolean formulas [17]. For all of these problems, except the last one, we can decide efficiently if a transformation between two given configurations exists. ...
Imagine that unlabelled tokens are placed on the edges of a graph, such that no two tokens are placed on incident edges. A token can jump to another edge if the edges having tokens remain independent. We study the problem of determining the distance between two token configurations (resp., the corresponding matchings), which is given by the length of a shortest transformation. We give a polynomial-time algorithm for the case that at least one of the two configurations is not inclusion-wise maximal and show that otherwise, the problem admits no polynomial-time sublogarithmic-factor approximation unless P = NP. Furthermore, we show that the distance of two configurations in bipartite graphs is fixed-parameter tractable parameterized by the size d of the symmetric difference of the source and target configurations, and obtain a -factor approximation algorithm for every if additionally the configurations correspond to maximum matchings. Our two main technical tools are the Edmonds-Gallai decomposition and a close relation to the Directed Steiner Tree problem. Using the former, we also characterize those graphs whose corresponding configuration graphs are connected. Finally, we show that deciding if the distance between two configurations is equal to a given number is complete for the class , and deciding if the diameter of the graph of configurations is equal to is -hard.
... The Hamiltonian cycle problem (HCP) in grid graphs has been well studied and has led to application in numerous NP-hardness proofs for problems such as the milling problem [AFM93], Pac-Man [Vig14], finding optimal solutions to a Rubix Cube [DER17], and routing in wireless mesh networks [WGB12]. The problem has been of interest to computer scientists for many years and recently a number of variations on the problem have been investigated. ...
Finding Hamitonian Cycles in square grid graphs is a well studied and important questions. More recent work has extended these results to triangular and hexagonal grids, as well as further restricted versions. In this paper, we examine a class of more complex grids, as well as looking at the problem with restricted types of paths. We investigate the hardness of Hamiltonian cycle problem in grid graphs of semiregular tessellations. We give NP-hardness reductions for finding Hamiltonian paths in grid graphs based on all eight of the semiregular tessilations. Next, we investigate variations on the problem of finding Hamiltonian Paths in grid graphs when the path is forced to turn at every vertex. We give a polynomial time algorithm for deciding if a square grid graph admits a Hamiltonian cycle which turns at every vertex. We then show deciding if cubic grid graphs, even if the height is restricted to 2, admit a Hamiltonian cycle is NP-complete.
Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so that consecutive partitions differ by refinement? These are examples of Gray coding problems: Can a list of combinatorial objects (of a particular type and size) be ordered so that consecutive objects differ by a flip (of a particular type)? For example, 000, 001, 010, 100 is a no instance of the first question, while 1234, 1324, 1243 is a yes instance of the second question due to the order . We prove that a variety of Gray coding problems are NP-complete using a new tool we call a Gray code reduction.
Do individuals learn more effectively when given progressive or variable problem-solving experience, relative to consistent problem-solving experience? We investigated this question using a Rubik’s Cube paradigm. Participants were randomly assigned to a progression-order condition, where they practiced solving three progressively more difficult Rubik’s Cubes (i.e., 2 × 2 × 2 to 3 × 3 × 3 to 4 × 4 × 4), a variable-order condition, where they practiced solving three Rubik’s Cubes of varying difficulty (e.g., 3 × 3 × 3 to 2 × 2 × 2 to 4 × 4 × 4), or a consistent-order condition, where they consistently practiced on three 5 × 5 × 5 Rubik’s Cubes. All the participants then attempted a 5 × 5 × 5 Rubik’s Cube test. We tested whether variable training is as effective as progressive training for near transfer of spatial skills and whether progressive training is superior to consistent training. We found no significant differences in performance across conditions. Participants’ fluid reasoning predicted 5 × 5 × 5 Rubik’s Cube test performance regardless of training condition.
Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer ) contiguous vertices on a shortest path can be changed at a time.
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 how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the cube from its geometrical constraints. Cubies are described by single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the cube, as a geometrical object, shows symmetries which are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the cube is preserved. When all such symmetries are preserved, the configuration of the cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a deep reinforcement learning algorithm based on a Hamiltonian reward. The cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.
Rubik’s cube is one of the most challenging problems in the field of evolutionary algorithm and group theory. Several algorithms have been considered to solve this problem in recent years. The solution of Rubik’s cube is different from other search algorithms. Finding the goal state is the target of the normal search algorithms. However, in this problem, the start (origin) and goal (destination) states are known and the challenge is to provide a convenient and efficient method for finding the shortest path between origin and destination. Existing solutions can be divided into several categories. In the first category, by dividing problem into sub-problems, researchers have tried to simplify and solve the cube. Apart from the divide and conquer method, the second category by creating a similar concept to the lookup tables tries to reduce computational costs with dynamic programming. Last category focused on the evolutionary algorithms to solve the problem. Regarding the fact that Rubik’s cube is one of the models in the field of group theory, we have proposed a novel method for solving this problem by a combination of group theory and genetic algorithm. The main idea of the proposed algorithm is finding and removing repeated states by group theory, because there are several repeated individuals (states) in different generations of genetic algorithm. The experimental results show a huge sieve in individuals per generations. In these experiments, by a significant reduction in the populations size, the speed of finding solutions has been increased by more than .
Imagine that unlabelled tokens are placed on edges forming a matching of a graph. A token can be moved to another edge provided that the edges containing tokens remain a matching. The distance between two configurations of tokens is the minimum number of moves required to transform one into the other. We study the problem of computing the distance between two given configurations. We prove that if source and target configurations are maximal matchings, then the problem admits no polynomial-time sublogarithmic-factor approximation algorithm unless . On the positive side, we show that for matchings of bipartite graphs the problem is fixed-parameter tractable parameterized by the size d of the symmetric difference of the two given configurations. Furthermore, we obtain a -factor approximation algorithm for the distance of two maximum matchings of bipartite graphs for every . The proofs of our positive results are constructive and can hence be turned into algorithms that output shortest transformations. Both algorithmic results rely on a close connection to the Directed Steiner Tree problem. Finally, we show that determining the exact distance between two configurations is complete for the class , and determining the maximum distance between any two configurations of a given graph is -hard.
A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.
Single-player games (often called puzzles) have received considerable attention from the scientific community. Consequently, interesting insights into some puzzles, and into the approaches for solv- ing them, have emerged. However, many puzzles have been neglected, possibly because they are unknown to many people. In this article, we survey NP-Complete puzzles in the hope of motivat- ing further research in this fascinating area, particularly for those puzzles which have received little scientific attention to date.
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(n
2/logn). The upper bound comes from effectively parallelizing standard Θ(n
2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically optimal algorithm for solving a general Rubik’s Cube in the worst case. Given a specific starting state, we show how to find the shortest solution in an n ×O(1) ×O(1) Rubik’s Cube. Finally, we show that finding this optimal solution becomes NP-hard in an n ×n ×1 Rubik’s Cube when the positions and colors of some cubies are ignored (not used in determining whether the cube is solved).
The 8-puzzle and the 15-puzzle have been used for many years as a domain for testing heuristic search techniques. From experience it is known that these puzzles are “difficult” and therefore useful for testing search techniques. In this paper we give strong evidence that these puzzles are indeed good test problems. We extend the 8-puzzle and the 15-puzzle to an n×n board and show that finding a shortest solution for the extended puzzle is NP-hard and is thus believed to be computationally infeasible.We also sketch an approximation algorithm for transforming boards that is guaranteed to use no more than a constant times the minimum number of moves, where the constant is independent of the given boards and their side length n.The studied puzzles are instances of planar relocation problems where the reachability question is polynomial but efficient relocation is NP-hard. Such problems are natural robotics problems: A robot needs to efficiently relocate packages in the plane. Our research encourages the study of polynomial approximation algorithms for related robotics problems.
Can computers routinely discover mathematical proofs?
- A Stephen
- Cook
Stephen A. Cook. Can computers routinely discover mathematical proofs? Proceedings of
the American Philosophical Society, 128(1):40-43, 1984.
Is optimally solving the n×n×n Rubik's Cube NP-hard? Theoretical Computer Science Stack Exchange
- Jeff Erickson
Jeff Erickson. Is optimally solving the n×n×n Rubik's Cube NP-hard? Theoretical Computer Science Stack Exchange. URL: https://cstheory.stackexchange.com/q/783 (version:
2010-10-23).