# Questions tagged [rubiks-cube]

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

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### How do i generate cubic coordinate-space traversal functions?

Goal- generation of a function f(x) such that for each x = 0 to infinite, f(x) outputs a point in a coordinate space ...
1 vote
1 answer
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### 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 ...
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### 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 ...
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### 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 ...
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3 votes
1 answer
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### 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
1 answer
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### 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 ...
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2 votes
0 answers
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### 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, ...
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4 votes
2 answers
85 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 ...
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1 vote
1 answer
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### 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 ...
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3 votes
1 answer
109 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 ...
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### 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 ...
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### 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.
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1 vote
1 answer
190 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 ...
6 votes
2 answers
203 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 ...
4 votes
3 answers
134 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 $8$...
0 votes
1 answer
28 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
1 answer
130 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 ...
4 votes
1 answer
99 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 ...
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2 votes
1 answer
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### 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 ...
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1 vote
1 answer
73 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 ...
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1 vote
1 answer
139 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 ...
4 votes
1 answer
142 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 ...
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1 vote
1 answer
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### 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
1 answer
91 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
2 answers
176 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 ...
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36 votes
3 answers
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 ...
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1 vote
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### 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, ...
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1 answer
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### 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., ...
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1 vote
1 answer
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### 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 ...
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1 vote
1 answer
176 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(...
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1 vote
2 answers
420 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) ...
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9 votes
1 answer
212 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 ...
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1 vote
1 answer
339 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 ...
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1 vote
1 answer
61 views

### Number of times a pattern needs to be repeated on a Rubik's cube until it returns to solved

Click on this link if you are not familiar with Rubik's cube notation Suppose I want to repeat some arbitrary pattern of moves on a solved Rubik's cube, until it is solved again, i.e. F R F R F R... ...
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4 votes
0 answers
224 views

### Solving the Rubik cube with given initial and target states (generalization of standard Rubik cube)

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 $B$...
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2 answers
95 views

### What is the order of the subgroup of G (rubiks cube group) generated by <FF,RR>?

I got: =1,FF,RR,FFRR,RRFF But in my text book the answer is 12? Does any one else know the other elements? I assumed since FFFFRRRR=1 there were no more?
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### Longest Rubik's cube algorithm - maximization problem

For a given Rubik's cube algorithm $A$ let $\mathfrak C(A)$ be the number of times, we have to repeat algorithm $A$ to get back to where we've started. For example if $A=RUR'U'$ then $\mathfrak C(A)=6$...
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4 votes
1 answer
115 views

### How does $H$ act on $G^t$ in the wreath product $G^t \wr H$?

I'm reading this expository paper about group theory in the Rubik's cube. I'm a little confused by the definition of the wreath product in this paper. Example 3.12 on page 12 states that the elements ...
2 votes
0 answers
42 views

### What can be said about the stationary distribution of the Latch Cube?

Katsuhiko Okamoto's Latch Cube is similar to the standard $3\times 3$ Rubik's cube with the added features that on one of the faces of each of the edge cubies, there is an arrow identifying a ...
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### Oriented Rubiks Cube 4x4 re-arrange center pieces

There are a lot solutions for oriented 3x3. They rotate all center pieces. I'm trying to solve oriented 4x4 with GabbasSoft Cube Demo. Luckily blue and green ...
0 votes
1 answer
65 views

### Rubik's cube solving algorithm [closed]

I am aware of God's number and how there is a lower bound of 26 quarter turns on such algorithms. However, I wished to know if there is a certain algorithm that can look at the faces of the cube and ...
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4 votes
1 answer
57 views

### Add Relation To Permutation Group in GAP.

I suspect this is basic but I don't know how to do it. Using the GAP system, how would I go about adding a relation to a given permutation group? For example, maybe I start with modeling a Rubik's ...
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2 votes
2 answers
299 views

### Prove cycling in a Rubik's cube

How can I prove that if you apply some algorithm over and over again on a solved Rubik's cube, the cube will be solved? I mean mathematically not conceptually.
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1 vote
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### About orders of any given element of the Rubik's cube group

I realize this is an old question and very similar to mine, but I'll ask anyway. I have found a lot of literature about the element of largest order 1260 and how to derive this number. But I've yet ...
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### God's number for the 3x3x3 Picture Cube

I've a small question. I've read that God's number for the regular 3x3x3 Rubik's cube is 20 and in quarter-turn metric it's 26. Here are some links: https://en.wikipedia.org/wiki/God%27s_algorithm ...
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1 answer
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### Applications of Finding the Order of a Rubik's Cube Algorithm

I recently answered the question on How to find the order of a Rubik's cube algorithm. As you can see in my answer to Creating a Rubik's Cube Algorithm, I mentioned that using repetition is one of ...
10 votes
1 answer
906 views

### How to find the order of a Rubik's cube algorithm?

For example, the algorithm $R U R' U'$ has an order of $6$, that is, repeat the algorithms $6$ times to return to the original position. How would I go about finding the order of any other algorithm?
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3 votes
2 answers
137 views

### Let $H$ be the subgroup of Rubik's cube moves generated by a half-turn of the downward face and a half-turn of the rightmost face. What is $|H|$?

I was going through Janet Chen's "Group Theory and the Rubik's Cube" (PDF), and I came across this question: Let $D$ denote a clockwise quarter turn on the downward face and $R$ denote a quarter ...
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2 votes
1 answer
156 views

### "Permutation of a set" vs "permutation of a Rubik's cube": are these uses of "permutation" equivalent?

So my book defines a permutation as follows : "By a permutation of a set A we mean a bijective function from A to A, that is, a one-to-one correspondence between A and itself." This is ...
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5 votes
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### How many colors of a Rubik's Cube must be known to locate all?

I wonder if I can apply some kind of formula given a few colors with the location on a Rubik's Cube to calculate the locations of all other colors on it. Best would be if this formula would be so ...
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