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Questions tagged [rubiks-cube]

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

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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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Solving Rubik cube with imperfect information

What about playing Rubik cube when we can see only the same three facets which share the same vertice ? View of configuration on three facets don't fully define the another three facets, so until we ...
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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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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 ...
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1answer
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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 ...
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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
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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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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 ...
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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 ...
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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 $20$ ...
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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 ...
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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 ...
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1answer
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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 ...
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2answers
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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 ...
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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' ...
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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. ...
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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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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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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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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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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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1answer
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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 ...
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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 ...
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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 ...
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Is it possible to go through all Rubik's Cube combinations without repetition? [duplicate]

Starting at a certain combination (starting point), making one movement at a time, is it possible to go through all Rubik's Cube combinations without repeating a combination did before? If so, is the ...
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How can I calculate the number of permutations of an irregular rubik's cube?

Recently I made a 1x2x3 Rubik's cuboid (example here) for my little brother, thinking that the number of possible permutations of the cuboid would be small enough that he could solve it simply by ...
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1answer
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Number of unique permutations of a 3x3x3 cube, including transforms

I asked the question Number of unique permutations of a 3x3x3 cube but now I want to know a similar but related question. If I have the numbers 1-27 and I assign each small-cube in a 3x3x3 matrix one ...
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3answers
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Rubik's cube function

I'm thinking in a function and if it's possible to solve that. I have been playing with the cube using the following move: $R U L' U'.$ I notice that the cube solves itself with a certain number of ...
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Rubik's Cube and the symmetric group

I have a slightly unique question: Using the fact that no permutation of $S_n$ can be both odd or even, why can we conclude that there is no way to rearrange a Rubik's Cube so that it swaps two ...
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1answer
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What is the probability of this Rubik's cube configuration?

What percentage of the total number of combinations on a Rubik's Cube have the following property: On every face, no two adjacent squares have the same color. By adjacent I mean: adjacent: ...
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1answer
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Is it possible to achieve a Rubiks Cube with Complementary colors as center piece? [closed]

Info Achieving a Rubik's Cube with different center pieces (like a dot) is fairly simple when you have a correctly solved Cube. From the top of my head, you can get something like orange edges with a ...
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1answer
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Can my algorithm solve this problem?

Ok... this whole problem started with a 7x7x7 Rubik’s cube. But I’m only telling you that to give some context. Not to ask how to solve the cube. When solving the cube’s edges you have to math up the ...
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How many permutations of a Rubik's cube, with three fixed (adjacente) faces?

So suppose I have a Rubik's, but I can only rotate the faces Top, Left and Front. How many configurations are there in this cube? In a normal cube, the 'face' (i.e. center) tiles are always fixed. In ...
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Factorize a permutation with elements from two permutation groups

Given two permutation groups $G,H$ on the same set $M$, and a permutation $\sigma$ on $M$, how to find a factorization $\sigma = gh$ (but not $\sigma = hg$), where $g \in G$ and $h \in H$? This ...
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Inverse operation reduce the number of moves in solving Rubiks cube?

Suppose we could solve the Rubiks cube(final steps) by using the operation $[R',D']$ a certain number of times where square brackets denote commutator operation. Then, would $[D', R']$ reduce the ...
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Properties of two squares subgroup of Rubiks square group

Is no corner twisted or edge flipped by the action of the two squares group $<F^2,B^2,U^2,D^2,L^2,R^2>$ on the configurations of the Rubik's cube? I think the answer is yes, but think the proof ...
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1answer
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A central 3-cycle in a 4x4x4 Rubik's cube

I am looking for a way to rotate three central squares in a 4x4x4 Rubik's cube while leaving all the other squares intact. See picture. Remark. Of course, if the squares have the same color, they are ...
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Determine the highest order of an element of a Rubik's Cube group

The period of a sequence of moves on a Rubik's Cube is the number of times it must be performed on a solved cube before the cube returns to its solved state. For example, a $90$° clockwise turn on ...
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Is there a sudoku (Latin Square Pattern) state in a Rubik's cube $6\times6\times6?$

Suppose, Initial state Rubik's Cube 6x6x6 444444 444444 444444 444444 444444 444444 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 000000 111111 222222 333333 ...
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How many good position are there?

Two squares on a Rubik's Cube are adjacent if they share a side. A "good" position of a cube is such that no two adjacent squares have the same color. As I try, it's pretty tough to find a good ...
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Best way to memorize a cube by numbers [closed]

Assume that we have a method to memorize a $3$x$3$x$3$ cube only with integer numbers. Take a number among them, call it $a$. Assume that its value can only lie between $m$ and $n$. Then the number $a$...
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Mapping a Rubik's Cube state to a unique natural number

I am writing an implementation of Kociemba's algorithm, and I am trying to generate pruning tables. However, this requires converting the state of the Rubik's Cube to a natural number that can be ...
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Is it possible to make it so that on a $2\times 2 \times 2$ Rubik's cube, no similar colours are on the Same face

On a $2 \times 2 \times 2$ Rubik's cube, is it possible to make it so that each of the similar colours are not on the same face? I've played around with it at the moment, but never had a state where ...
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Rubik's Cube Permutations [duplicate]

I know the formula for finding the number of permutations of the $3$x$3$ cube, which is $$\frac{8! \times 3^8 \times 12! \times 2^{12}}{12}$$ But why must the division by $12$ be a part of the ...
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Grid Puzzle Solvability

I've created a unity game which is sort of like a flat two sided Rubik's Cube. There's a grid (with a dynamic number of rows and columns) and set of tiles numbered from 1 to rows*columns. These tiles ...
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How can I swap opposite corners on the same face of a 2x2 Rubik's cube?

I've been experimenting with using Y-permutations to solve the 2x2 cube as described here. I ended up with the cube solved but for a single face where the opposite corners were swapped and disoriented....
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Period for a Rubik's cube repeated manipulation

A standard Rubik's cube is initially unscrambled (say per picture below, green facing observer) The same manipulation is repeated: rotation of front face by 1/4 turn (say clockwise) rotation of the ...
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1answer
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How many axis of symmetry of the cube are there?

In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there? The teacher gives me the definition: Definition: If we rotate a 3-dimension object around ...
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$N\times N\times N$ Rubik's Cube; maximum number of repetitions of a sequence of moves to return to initial state?

Suppose I have an $N\times N\times N$ Rubik's Cube, and suppose I have a list of moves, say in Singmaster notation. How many times can I expect to be able to repeatedly apply my list of moves before ...