# Questions tagged [rubiks-cube]

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

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### Does a Fixed Sequence of Moves Solve Any Rubik's Cube Configuration When Repeated Enough Times?

Question: Can a fixed sequence of moves restore any Rubik's Cube configuration to its solved state when repeated enough times? I’m wondering if it's possible to use a fixed sequence of moves (a "...
• 317
1 vote
82 views

### Unique reconstruction of rubik's cube state from images [closed]

Suppose I have 6 images of a rubik's cube. Each representing one face. The orientation of each image is arbitrary. Given that the images are from a valid cube state can I always reconstruct the rubik'...
• 33
51 views

### Is there some higher-dimensional polytope that represents the Rubik's Cube group?

I recently found a Pocket Cube and while trying to find instructions on how to solve it, I got sucked into the whole Rubik maths rabbit hole. I was wondering if the Rubik's group can be represented by ...
• 101
1 vote
54 views

### Is it possible to realize a mechanical puzzle in between the $2\times 2\times 2$ pocket cube and the $3\times 3\times 3$ rubiks cube?

I am curious if there then exists a finite subgroup that is a proper subgroup of the $3\times 3 \times 3$ Rubik's cube group AND itself contains a copy of the $2 \times 2 \times 2$ pocket cube as a ...
• 17.5k
1 vote
71 views

### I don't understand the last part of the paper "Group Theory and the Rubik’s Cube" by Janet Chen (final theorem leads to the conclusion of valid moves) [closed]

(link to paper) I was trying to read through the paper a long time ago, I think I managed to understand the core math of the paper. However, I am still struggling to see wheat the missing knowledge ...
113 views

### $n \times n \times n$ Rubik's cube [closed]

Is there any paper or blog which explains the maths behind $n\times n\times n$ Rubik's cube in detail along with proofs ? I want to understand all possible configuration of $n \times n \times n$ Rubik'...
64 views

### Impossible Rubik's cube position (2 corners swapped!) [closed]

Left side, white up, Red, green, white corner swapped with Red, blue, white corner Right side, white up, Red, blue, white corner swapped with Red, green, white corner Right, Down, Back view of cube. ...
1 vote
112 views

### Are there any unanswered questions regarding math behind Rubik's cubes? [closed]

I'm a student who has to do either a capstone(as in release a product of some sort) or research project as part of graduating. Right now I'm planning on doing something related to computer science and ...
• 35
99 views

### Why do we divide by 12 when calculating the number of different instances of the Rubik's cube? [duplicate]

$\frac{8!\cdot3^8\cdot12!\cdot2^{12}}{3\cdot2\cdot2}$ This is the solution, do you know a paper in which this formula is explained, why we calculate like this? Then I heard that the numerator of the ...
• 21
332 views

### What is diameter and growth of group generated by two cyclic permutations having only two common elements (rotation group of two-circle puzzle)?

Consider subgroup of S_n generated by two cyclic permutations of the form described below. (Roughly speaking cycles having only two common elements). Question: what is known about its growth and ...
65 views

### MInimal number of faces to generate full rubik's cube state [closed]

I am wondering what's the minimal number of faces we need to know in advance to be assured of the rest of the state of the cube. For a solved cube I believe just the 3 solved faces is enough to infer ...
• 33
138 views

### 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 ...
171 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
169 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?
• 2,079
95 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
232 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 ...
• 11
264 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 ...
• 21.2k
223 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 ...
145 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
41 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 ...
• 834
286 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, ...
• 357
54 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 ...
• 105
302 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 ...
• 695
158 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 ...
• 41
399 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 ...
• 153
941 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 ...
61 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 ...
• 8,536
1 vote
188 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 ...
112 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 ...
• 409
1 vote
53 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 ...
272 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
651 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 ...
• 111
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, ...
• 31
216 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 ...
• 1,290
1 vote
85 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 ...
• 41
506 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 ...
• 133
55 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 ...
• 3,366
1 vote
42 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.
• 61
1 vote
806 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 ...
491 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 ...
769 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 $8$...
39 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
283 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 ...
152 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 ...
• 1,042
142 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 ...
• 509
1 vote
204 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 ...
• 573
386 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 ...
475 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 ...
• 143