# Tagged Questions

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

2answers
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### combinations of Rubik cube

A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers? Can someone give me some hints? Thanks
1answer
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### Rubik's cube and counting

Inspired by this question, I formulate the following: Suppose I have a $3\times3\times3$ Rubik's cube, call each small square on the surface a piece, there are then $3*3*6 = 54$ pieces. Enumerate ...
1answer
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### Literature on group theory of Rubik's Cube

While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
3answers
811 views

### Why should Rubik's cube get attention from mathematicians?

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering ...
2answers
175 views

### Rubik Cube finite non-abelian group

I was reading the following paper from MIT: http://web.mit.edu/sp.268/www/rubik.pdf The paper is not difficult to understand, it is more or less a short introduction into group theory, taking the ...
1answer
87 views

### Rubik's cube as $2\times2\times2$ or $2\times2$? [closed]

I'm surprised why people call a Rubik's cube as $2 \times2\times2$ or $3\times3\times3$? It's a CUBE! Isn't saying $2\times2$ or $3\times3$ cube (or rather just $2$ or $3$) sufficient?
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### Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5). So I gave ...
4answers
510 views

### Minimum number of steps needed to solve a rubik cube

Long time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...
1answer
89 views

### The redundancy of Rubik's cube states [duplicate]

Possible Duplicate: Rubik’s Cube Not a Group? I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long ...
2answers
170 views

### Rubik's Cube Combination

Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of ...
2answers
229 views

### Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
3answers
679 views

### Brute force method of solving the cube: How many moves would it take?

Given that Rubik's cube has finitely many positions, one possible "brute force" method to solve it would be to determine once a sequence of moves which eventually reaches every possible position of ...
1answer
176 views

### Orders of moves in the Rubik's cube

Playing with my Rubik's cube, I was thinking of facts about it that are immediate to mathematicians but novel to others. Here's one: Given a Rubik's cube in the starting position, any sequence of ...
1answer
102 views

### Rubik's cube notation [closed]

Various Rubik cube guides use the notation F and R to describe the front and right faces of the cube. However, in the illustrations they show the cube in isometric projection. So which side is right, ...
3answers
573 views

### Is there a “most random” state in Rubik's cube?

Is there a state in Rubik's cube which can be considered to have the highest degree of randomness (maximum entropy?) asssuming that the solved Rubik's cube has the lowest?
1answer
190 views

### Are actions in the $3\times 3\times 3$ rubik cube a group?

Are actions in the $3\times 3\times 3$ rubik cube a group? You can see here Rubik's Cube Not a Group? that $4\times 4\times 4$ rubik cubes or higher arent groups. But what about \$3\times 3\times ...
1answer
636 views

### Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
2answers
346 views

### Rubik's cube puzzle

If we cut along the plane orthogonal to the largest diagonal of a Rubik's cube, what is the maximum number of small cubes can we cut? I thought this should be $9$, but apparently this is not the ...
2answers
2k views

### How to tell if a Rubik's cube is solvable

How can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experience ...
2answers
393 views

### All possible permutations on a Rubik cube ($3\times3\times 3$) can be reached from the initial state?

If I were to represent a state in the Rubik cube as a permutation of the colors on the 9 tiles per side on all sides of the cube, could I reach all possible states (i.e. colorings) by the permutations ...
1answer
229 views

### Does solving a Rubik's cube imply alignment?

Today, I got my hands on a Rubik's cube with text on it. It looks like this: Now, I would love to know whether solving the right cube will also always correctly align the text on the cube or ...
2answers
369 views

### Mathematics Behind the 4×4 and 5×5 Rubik's Cube

A lot is known about the math behind the 3×3 Rubik's cube (symmetries, generators, group structure etc...). Is the same true for the 4×4 and 5×5 cubes? I haven't had much success finding this ...
1answer
325 views

### Solving a scrambled $3 \times 3 \times 3$ Rubik's Cube with at most 20 moves!

I read somewhere that any scrambled form of $3 \times 3 \times 3$ Rubik's cube can be solved using at most $20$ moves, and I just said "wow"! I am wondering can we prove this by mathematical ways? Or ...
1answer
1k views

### What are the 2125922464947725402112000 symmetries of a Rubik's Cube?

In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry. What counts as a ...
1answer
529 views

### Rubik's cube interesting questions?

The upper bound for the number of moves required to solve a regular Rubik's cube has been shown to be 20. Two questions come to mind: Does this result have more general significance? What are the ...
3answers
490 views

### How many ways can I make six moves on a Rubik's cube?

I am writing a program to solve a Rubik's cube, and would like to know the answer to this question. There are 12 ways to make one move on a Rubik's cube. How many ways are there to make a sequence of ...
2answers
236 views

### How is done the calculation of the minimum number of movement to solve any configuration of Rubik's Cube?

I have read a few weeks ago that some mathematical researchers have discover that the minimum number of movements to solve any initial configuration of a Rubik's cube has been downsized to 20. How do ...
1answer
462 views

### How many disconnected graphs of the Rubik's cube exist?

Let us say that a Rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be ...
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