# Questions tagged [rubiks-cube]

Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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### Generalised Rubik's cube algorithm to go from any valid cube state A to any valid cube state B.

For a $3 \times 3 \times 3$ cube, there are several known algorithms to go from any valid state of the cube to the solved state, $S$, of the cube (same color on each side). How about a more general ...
94 views

### Rubik's cube repetitions, weird behaviour (Best if you know cube rotation notation)

In this question, I'll refer to standard cube rotation notation, seen here: J perm website - cube notation. The move U, can be performed 4 times and return a solved cube (or any cube) to its original ...
1 vote
97 views

### Rubik's Cube Group: Number of Elements With Each Order [closed]

In the Rubik's Cube group, for each order, how many elements are there that have that order?
80 views

### God's number for one face of a 3x3x3 Rubik's cube

I'm working on a program to solve a single face on a Rubik's cube. In trying to find a way to reach the optimal number of moves I found God's number (the minimum number of moves needed to solve the ...
1 vote
184 views

### Seeking Proof: Can Rubik's Cube States be Mapped to a Euclidean Space of a Certain Dimension?

I am currently researching a problem that involves mathematics and computer science, and I am hoping to get advice or guidance from experts in the field. My question pertains to finding a mapping that ...
224 views

### Is the Rubik's Cube group normal in the assembly/disassembly group?

I found myself wondering whether the Rubik's Cube group (of order $2^{27} \cdot 3^{14} \cdot 5^3 \cdot 7^2 \cdot 11$) is normal as a subgroup of the slightly larger group where assembly/disassembly of ...
215 views

### Practically implementable map from $\mathbb{Z}_{\left(\!{n\choose k}\!\right)}$ into a bag of scrambled Rubik's Cubes?

What would it take to create a feasibly computable injective function sending elements of $\mathbb{Z}_{k}$ (for the largest possible $k$) into, say, the set of distinct states of a bag of $n$ freely ...
120 views

### Rubik's Cube algorithm with Optimal Algorithm [closed]

I'm confused about Thistlewaite's, Kociemba's, and Korf's algorithms for optimal Rubik's cube solving. What are the differences and how is each of the groups defined for each algorithm? I understood ...
1 vote
38 views

### Is there a useable error metric for rubics cube non-linear solver?

There exist step-by-step algorithms for solving rubics cubes. However, in the midst of an ongoing solution, often the state of the rubics cube can become more convoluted before the increased ...
178 views

### Rearranging the tiles in a Rubik's cube net

Consider the following image: Given that there are 54 tiles in this net--9 of each of the 6 colors--we can say that there are $\frac{54!}{9!^6}$ unique ways to arrange the tiles in this net. Now, ...
52 views

### How many steps before reaching an already-seen permutation of Rubik's cube?

Given a 3x3 Rubik's cube with an initial state, how many maximum moves will it take before a state is repeated (The number of moves after which, we can no longer make a move which generates a new ...
184 views

### Pocket Cube (2x2x2 Rubik's cube) as Quotient Group

The 2x2x2 Rubik's cube (also called Pocket cube) consists of $6 \cdot 4 = 24$ small squares. A rotation of one of the faces of the cube can therefore be described as an element of $S_{24}$ (giving ...
117 views

### Is there a loop consisting of an odd number of moves for the Rubik's cube?

I'm considering a loop to be a sequence of moves that brings a Rubik's cube configuration back to its original state. One example is if you turn any face a quarter turn four times, or if you turn one ...
351 views

### Can you compute the state of a 3x3x3 Rubik's cube if you only see three adjacent faces?

I've learned from How many colors of a Rubik's Cube must be known to locate all? that you need to know a minimum of 17 stickers (spread across all its faces) for a 3x3x3 cube in order to compute ...
604 views

### Proving the repeatedness in the sequence of moves for a Rubik's cube

I was trying to reason out in Rubik's cube why any specific set of moves works, like: Showing that applying the sequence of moves $\{R,D,R',D'\}$ $6$ times will restore the cube back to originally it ...
54 views

### False Rubik Cube and math

imagine you are given an scrambled Rubik cube :D , how would you know that this cube has a solution ? imagine i am a bad person and put the colors around the cube :d in a certain form that it has no ...
1 vote
151 views

### How many unique permutations exist on this type of rubik's cube?

NOW WAIT,I am not asking about a normal rubik's cube, I know there are 43 quintillion unique scrambles for the regular cube, but what I am asking goes beyond that. I am asking, how many unique ...
94 views

### number of paths between opposite boundaries of a cube

There is a calculation of the number of surface paths (with no backtracking allowed) between opposite corners of a Rubik's cube. I am interested in paths on an $L\times L\times L$ cube, where $L$ is ...
1 vote
51 views

### Prove that set of Rubik's cube rotations equipped with composition forms a group.

Tl;dr: I am trying to show intuitively that the set of Rubik's cube rotations forms a group. (I am not talking about literal rotation of the cube, but, of the edges.) The group axioms it must follow ...
256 views

### Probability of a Rubik's Cube being solvable in two moves.

So, I have recently gotten into speed-cubing, and I ran into a very interesting problem. According to the World Cube Association, a cube is legal if it takes at least two moves to solve. So, I want to ...
1 vote
509 views

### 4x4 Rubik's cube centers permutations count

About a 4x4 Rubik's cube. We know the count of permutations. But how about the permutation of the centers only ? X X X X X O O X X O O X X X X X There are 6 ...
66 views

### Solvability of a randomly put together Rubiks’s cube [duplicate]

After taking my Rubik’s cube apart, (leaving me with 12 edge pieces, 8 corner pieces and the center piece) and putting it back together randomly, I noticed that the cube is only solvable sometimes, ...
191 views

### How can I obtain or sample a random Rubiks Cube shuffle?

I was thinking of how to obtain a random shuffle of the Rubik's cube with uniform probability. Simply trying a randomly generated sequence of turns will not necessarily produce a uniform distribution ...
1 vote
83 views

### Can you uniquely define a cube knowing the color of the faces but not its orientation?

I'm doing an university project for one of my subjects. I have the 6 faces of the Rubik's cube but i don't know if they have been rotated or not, i just know their colors. Is that enough information ...
444 views

### Can a shuffled Rubik’s cube have 2 faces with the same number of pieces of the same color?

I couldn’t find any resources regarding this, so I’ve 2 questions: Can a shuffled Rubik’s cube have 2 faces - that for each color on a face, there is a matching number of pieces of the same color on ...
54 views

### Is there an algorithm for the Rubik’s cube that gets every configuration? [duplicate]

I was watching a video on Youtube of a youtuber named “Mathologer” (he’s a great guy, gi check it out!) about pigeonhole principle. It gave an example of how the pigeonhole principle could be applied ...
1 vote
40 views

### What is the average number of moves it takes to solve every 3x3 rubik’s cube position if only random moves are made? [duplicate]

Hope my question is clear enough. I’m just curious.
1 vote
622 views

### Solvability of a 9-card puzzle game with a similar concept as a rubik's cube

I challenged myself to create a card game that simulates the experience of solving a 3x3 Rubik's cube. I have a first prototype but I'm now stuck at knowing if every random initial state will be ...
405 views

### Rubik's cube, elements of order $7$ and $11$

The order of the Rubik's cube group is $$43 252 003 274 489 856 000 = 2^{27} \times 3^{14} \times 5^3 \times 7^2 \times 11$$ Cauchy's theorem guarantees an element of order 7, as well as one of order ...
562 views

### The possible orientations of a $2 \times 2 \times 2$ Rubik’s cube

So a $2 \times 2 \times 2$ cube has $8$ distinct pieces. With each of them having 3 colours(one on each of their exposed edges). Thus, the as echo piece has $3$ different orientations and there are $8$...
38 views

### How do I find the lexicographical index for 24 permutations?

I'm developing a Rubik's Cube Solver and require a pruning table for Kociemba's G1 to G2. I already have a table but the search is still very slow... I have 4 edge permutation values and I need to ...
1 vote
258 views

### How can I generate a pruning table for tetrads?

I'm currently creating a Rubik's Cube Solver and am having some difficulty generating pruning tables. Pruning tables contain information that is used to prune search tree branches, exponentially ...
140 views

### Non-oriented Rubiks Cube Group

Consider the (non-oriented, so we do not care about the center of each side) rubiks cube group, $G$, which is a subgroup of $S_{48}$ and is generated by the letters: $\{F,B,U,D,L,R\}$, i.e. I mean ...
122 views

### Is there a name for a graph in which every vertex is both a central vertex and a peripheral vertex

Is there a name for a graph for which every vertex is simultaneously in the center and in the periphery of the graph? If I'm not mistaken the graph representing the states of the Rubik's cube whose ...
1 vote
182 views

### Inverse Rubik's Cube

If I am given five faces of a rubik's cube, is it possible to a) Determine if these are five sides of an actually solvable cube b) Extend this to the sixth face in a unique way Assuming one eliminated ...
321 views

### GAP: Rubik's Cube Group - Thistlethwaite's algorithm

In GAP let $G_0$ be the Rubik's Cube Group defined by the six moves ...
379 views

### Rubik's Cube: Number of Permutations of the Corner Position Orientations

The Rubik's Cube has 8 corners, and each corner has 3 stickers. A corner can be in 1 of 3 orientations, i.e. any of the three stickers can point up, giving $3^8$ possible permutations of the corner ...
638 views

### Generating lookup tables for Thistlethwaite's algorithm

I'm trying to make a Rubik's cube solver using Thistlethwaite's algorithm. I've already made a working program that solves the cube in 3~ minutes without the lookup tables. I'll summarize the ...
1 vote
314 views

### What is the Shortest Set of Moves Needed to Solve a Rubik's Cube?

The question is asking what is the shortest sequence of moves that will have the cube solved at some point during the sequence. The most obvious such sequence would be one that has every solution to ...
1 vote
306 views

### In a Rubik's cube what is the probability of randomly getting a face blue?

When we solve a Rubik's cube we make planned moves and easily get the desired results but here I am talking about getting a colour by randomly moving a Rubik's cube without making an effort to solve ...
5k views

### Is there (or can there be) a general algorithm to solve Rubik's cubes of any dimension?

I love solving Rubik's cube (the usual 3D one). But, a lecture by Matt Parker at the Royal Institute (YouTube Link) led me to an app that can simulate a four dimensional rubik's cube. But ...
1 vote
54 views

### What is a cycle index notation for this group? Does a homomorphism onto a slice of the Rubiks cube exist?

I'm considering a group acting on a colored 2x2 block. Where the group element $t$ interchanges the top two blocks. Similarly $b$, $r$, and $l$ interchanges the bottom two blocks, right two blocks, ...
254 views

### No Rubik's cube can be solved in 13 moves

I've seen a very interesting paper/article few hours ago where it says: there is NO permutation of the Rubiks Cube from where there is a solution in 13 moves, but no solution in less moves. i.e., ...
1 vote
198 views

### Number of essentially different Rubik's cube permutations, or rather arrangements

I can find everywhere (e.g. wikipedia, ruwix.com and MIT) the information that the standard $3\times3$ Rubik's cube can be scrambled in $4.3 \times 10^{19}$ different configurations. These are ...
1 vote
734 views

### What is the number of possible configurations of a 2*2*2 Rubik's cube?

This is a follow-up question of Is there an equation for permutations with different numbers of element available? For a regular permutation, the number of possible configurations for a 2 * 2 * 2 cube(...
1k views

### Proving impossible Rubik's Cube permutations

It turns out that the Rubik's Cube has 12 orbits, because there are a few impossible cases just by turning the edges: cannot have just one corner twisted (/3) cannot have just one edge flipped (/2) ...
265 views

### Rate of Growth of Permutations of Rubik's Cubes

I'd like to know how fast the number of permutations grows on an $n\times n\times n$ Rubik's Cube as $n$ increases. I'm well aware of the $\frac{3^88!2^{12}12!}{12}$ calculation for the permutations ...
1 vote
690 views

### How to calculate the minimum number of moves for resolve the Rubik's cube (3x3) from any position or the level of difficulty?

I want to shuffle the 3x3 cube randomly in quarter turn metric and determine the minimum number of moves of each position with some mathematical solution. I know that the minimum moves from the most ...
1 vote
Consider the generalization of the Rubik cube problem, where we are given an initial state $A$ and a final state $B$, and we search for a path between the two. We can easily show that given $A$ and $B$...