Questions tagged [rubiks-cube]

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

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8
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1answer
92 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
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1answer
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 ...
1
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1answer
41 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... ...
4
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0answers
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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2answers
26 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?
0
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1answer
29 views

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 $\mathfrak C(A)=6$...
3
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1answer
72 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
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0answers
16 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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0answers
17 views

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 ...
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1answer
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 ...
4
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1answer
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 ...
2
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2answers
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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2answers
111 views

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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0answers
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 ...
0
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1answer
42 views

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
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1answer
423 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?
3
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2answers
84 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 ...
2
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1answer
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 ...
5
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1answer
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 ...
9
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1answer
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 ...
2
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0answers
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 ...
3
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1answer
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 ...
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2answers
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 ...
0
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1answer
74 views

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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0answers
33 views

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.
2
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2answers
94 views

Prove that the group of moves of the Rubik’s cube is not abelian.

I'm currently working in the following excercise: Remember that $G$ is the group of moves of the Rubik’s cube. Prove that this group is not abelian. I'm starting from picking two moves $M_1$ and $...
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0answers
92 views

Sylow Subgroups of the Rubik's Cube Group

The Rubik's Cube Group has order $2^{27} 3^{14} 5^3 7^2 11$. What is known about the Sylow subgroups of this group? Do they have an intuitive meaning with regards to the symmetry of the cube?
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257 views

Valid Megaminx Patterns

A Megaminx is a dodecahedral twisty puzzle similar to a Rubik's cube. Each of the twelve faces has a different color; the puzzle is cut into 50 pieces by slices parallel to the faces, which may be ...
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0answers
118 views

A single set of moves $S$ that, if repeated, solves the Rubik's cube from any state

I am looking for a proof verification. I often find these concepts simple, but struggle to communicate them clearly. Communication in mathematics is very important to me: Examples could be: Any ...
0
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1answer
47 views

Maximum cycle length of tiles in rubik cube

I tried to write a program to count the number of repeated sequence(s) to make NxNxN (for N=2 to 20) rubik cube back to it's initial state/placement. I solved it by counting the length of cycle of ...
8
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1answer
132 views

Apocorubik's Cube (10,000 blocks across): How long to solve if a quarter turn takes 2.8 seconds?

Suppose you have a Rubik's cube, $10,000$ blocks across (so, $600,000,000$ total tiles), scrambled. Assuming an ideal solving algorithm, approximately how long will the cube take to be solved, given ...
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2answers
697 views

High School Mathematical Research Project Ideas [closed]

i am currently looking out for some possible topics i could study for my research project in high school. Algebra, trigonometry, Pythagoras’ theorem, geometry, circles and their properties, etc. and ...
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0answers
100 views

Rubik's cube $5 \times 5 \times 5$ numbered arrangement

If you take a $5 \times 5 \times 5$ cube and enumerate each little square with a number, following an ascending order, then scramble it, how many possibilities you have to solve it? With all numbers ...
4
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1answer
96 views

Introductory book on Rubik's cube math

My 18-year-old brother really enjoys solving Rubik's cubes and other 3D puzzles, and enjoys math and physics at school. I'd like to get him a book on the math of the Rubik's cube, but one that makes ...
2
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0answers
417 views

Cayley graph of Rubik's cube group

(a) I would like to know whether there is a group theoretic approach for calculating the diameter of the Cayley graph of Rubik's Cube group. I know it's been proved that the above diameter is $20$ ...
2
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0answers
39 views

Confirmation on $2\times 2\times n$ Rubik's cube group

I believe that I have come up with formulae for the groups representing the $2\times 2\times n$ Rubik's "cubes," but I need someone to confirm that they are correct. Here are the groups that I came up ...
2
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1answer
60 views

What is generated by $(1265)$, $(2376)$, and $(3487)$? (Pocket cube group)

I am trying to find the group structure $G$ of the $2\times 2\times 2$ Rubik's cube, or the "pocket cube," and I have determined that it is isomorphic to the group of permutations on $8$ numbers ...
4
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1answer
159 views

Shortest Algorithm That Generates a Harlequin* Pattern

What is the shortest possible length of an algorithm that generates on each side of the cube a pattern that has: 6 colors at most 2 facelets of the same color no adjacent facelets of the same color ...
2
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2answers
280 views

Lower Bound on God's Number in the Rubik's Cube

I'm trying to write a project about the Rubik's Cube, and althought it's easy to find out the lower bound (using half turns) is 18, finding a precise proof hereof seems impossible. How does one proof ...
2
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0answers
42 views

Is it possible to calculate when we are close to solving the Rubik's cube?

While the Hamiltonian circuit can apparently represent the various combinations, I considered that if we can start with a solved Rubik's cube and start making twists and turns and record the 'path' ...
2
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2answers
95 views

Rubiks Cube function — how many configurations reachable in n moves.

I'm working on a relatively low-level math project, and for one part of it I need to find to a function that returns how many many configurations are reachable within n moves. from the solved state. ...
7
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0answers
227 views

Impossible permutations of the Gear Cube

If you're familiar with the group properties of the Rubik's Cube, you will probably know that, under the action of the standard moves, all possible permutations of the (unoriented) edge pieces are ...
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0answers
76 views

What use is group theory to understand and solve Rubik’s cubes? [duplicate]

This question has been asked before on other sites, but none of the answers were satisfying to me. I have heard people say that one can use group theory to solve rubik’s cubes. I understand why the ...
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0answers
122 views

How many solvable configurations of the Rubik's cube have no two squares of the same color touching?

The Superflip (image below) is an example of a configuration where no two squares of the same color are adjacent. I'm interested to know how many solvable configurations (that is, those that you can ...
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0answers
185 views

Cardinality of permutation group

Let's have $K$ permutations $P_1$, $P_2$, ... $P_K$ of the same set $M$. For example $M$ can be the set of stickers on Rubik's cube and $P_1$, $P_2$, ... $P_K$ are allowed cube moves. The set of all ...
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2answers
82 views

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'...
11
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3answers
1k views

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 ...
1
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1answer
79 views

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 ...
7
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0answers
148 views

Permutations of Rubik's cube such that no adjacent sticker is the same

I've always wondered, what is the number of possible permutations of the Rubik's cube such that any two adjacent stickers has a different color. By a permutation I mean a configuration of the cube ...
5
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2answers
205 views

Center of the Rubik's Cube Group

I can think of a Rubik's-Cube subgroup with four two elements which commute with all other Rubik's cube elements: Identity Super-flip (flip all the edges around) Super-swap (swap all the edges with ...