Related papers: Twenty-Five Moves Suffice for Rubik's Cube
Scrambling the standard 3x3x3 Rubik's Cube corresponds to a random walk on a group containing approximately 43 quintillion elements. Viewing the random walk as a Markov chain, its mixing time determines the number of random moves required…
The Rubix Cube is a 3-dimensional single-player combination puzzle attracting attention in the reinforcement learning community. A Rubix Cube has six faces and twelve possible actions, leading to a small and unconstrained action space and a…
In this paper we give a mathematical model for a game that we call picture cube puzzle and investigate its properties. The central question is the number of moves required to solve the puzzle. A mathematical discussion is followed by the…
It is well-known by now that any state of the $3\times 3 \times 3$ 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…
In this paper, we present a novel algorithm and its three variations for solving the Rubik's cube more efficiently. This algorithm can be used to solve the complete $n \times n \times n$ cube in $O(\frac{n^2}{\log n})$ moves. This algorithm…
Rubik's Revenge, a 4x4x4 variant of the Rubik's puzzles, remains to date as an unsolved puzzle. That is to say, we do not have a method or successful categorization to optimally solve every one of its approximately $7.401 \times 10^{45}$…
Rubik's Cube is one of the most famous combinatorial puzzles involving nearly $4.3 \times 10^{19}$ possible configurations. Its mathematical description is expressed by the Rubik's group, whose elements define how its layers rotate. We…
Rubik's Cube is an easily-understood puzzle, which is originally called the "magic cube". It is a well-known planning problem, which has been studied for a long time. Yet many simple properties remain unknown. This paper studies whether…
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…
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…
A generally intelligent agent must be able to teach itself how to solve problems in complex domains with minimal human supervision. Recently, deep reinforcement learning algorithms combined with self-play have achieved superhuman…
Permutation puzzles, such as the Rubik's Cube and the 15 puzzle, are enjoyed by the general public and mathematicians alike. Here we introduce quantum versions of permutation puzzles where the pieces of the puzzles are replaced with…
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…
This work describes in detail how to learn and solve the Rubik's cube game (or puzzle) in the General Board Game (GBG) learning and playing framework. We cover the cube sizes 2x2x2 and 3x3x3. We describe in detail the cube's state…
This paper presents a systematic method to solve difficult 9 x 9 Sudoku puzzles by hand. While computer algorithms exist to solve these puzzles, these algorithms are not good for human's to use because they involve too many steps and…
In 1979, David Fabian found a complete game of two-person Chinese Checkers in 30 moves (15 by each player) [Martin Gardner, Penrose Tiles to Trapdoor Ciphers, MAA, 1997]. This solution requires that the two players cooperate to generate a…
Learning to solve a Rubik's Cube requires the learners to repeatedly practice a skill component, e.g., identifying a misplaced square and putting it back. However, for 3D physical tasks such as this, generating sufficient repeated practice…
The sudoku minimum number of clues problem is the following question: what is the smallest number of clues that a sudoku puzzle can have? For several years it had been conjectured that the answer is 17. We have performed an exhaustive…
Ultimate Tic-Tac-Toe is a variant of the well known tic-tac-toe (noughts and crosses) board game. Two players compete to win three aligned "fields", each of them being a tic-tac-toe game. Each move determines which field the next player…
The diameter of the Cayley graph of the Rubik's Cube group is the fewest number of turns needed to solve the Cube from the hardest initial configuration. For the 2$\times$2$\times$2 Cube, the diameter is 11 in the half-turn metric, 14 in…