# Questions tagged [rubiks-cube]

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

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### 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 ...
64 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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### 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... ...
198 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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### 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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### 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 ...
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### 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 ...
33 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 ...
40 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 ...
88 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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### 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 ...
110 views

### 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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### 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 ...
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### 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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### 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 ...
123 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 ...
126 views

### 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 ...
169 views

### Solving $3\times 3\times 3$ Rubik's cube positions using a $4\times 4\times 4$ cube?

Someone hands you a $4\times 4\times 4$ cube for solving. It is scrambled as if it were a $3\times 3\times 3$, i.e. without ever separating the $4$ central pieces of each face (which implies that edge ...
61 views

### Random shuffle of a rubik’ cube properties

While shuffling a Rubik’s cube, I noticed that if my shuffling had a pattern I subconsciously assumed it isn't shuffled enough so I do couple more rotations. But, knowing that pattern can and do ...
72 views

### How many legal positions are there on a Rubik's cube with two pieces glued together?

Legal here means reachable from the solved state without separating the two glued pieces. Answer depends on the type of pieces glued together (egde+corner or edge+center). Is it just all normally ...
297 views

### Understanding Rubik's cube [duplicate]

I learnt how to solve a 3x3 Rubik's cube 10 years ago. Every now and then, I picked up a cube, scrambled it, and solved it for fun. I used to work on speed-solving, and memorised lots of formulae for ...
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### Identity element or neutral element in a Rubik's Cube

I read that the Rubik's Cube is a permutation group. Here, it says that "Neutral Element - *there is a permutation which doesn't rearrange the set: ex. $RR'$ *" (For Rubik's Cube notations see this.) ...
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### Can someone show a configuration of rubik's cube which is exactly 20 optimal moves away from the solved state.

It takes atmost 20 optimal moves to solve a Rubik's cube, was just curious to see the cube on other end of the diameter.
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### Articles and Papers about the math behind Rubik's Cube [closed]

I'm very interested in the math behind the Rubik's Cube. I already found this: https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf Are there some others texts that deal with the Rubik'...
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### Corner Swappings on Rubiks Cube

My brother likes to solve Rubik's cubes. Occasionally I flip a corner or two as an added challenge, and he eventually is able to see where the illegal corner is. The other day, I turned all eight ...
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### Observation on Rubik's Cube's tiles

Might be a naive question. While playing with a Rubik's cube, my cousin noticed that at least 2 tiles of same colour (any one of the six colours) always remain adjacent to each other. Is this always ...