Questions on the mathematics behind the famed toy invented by Ernő Rubik.
Questions tagged [rubiks-cube]
248 questions
47
votes
2 answers
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 if you don't put it together the right way, the…
Mel
- 581
38
votes
3 answers
Can a Rubik's cube be mapped knowing only two sides?
Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a two-dimensional map of a $3\times 3\times 3$ Rubik's cube?
Chris vCB
- 507
37
votes
3 answers
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 unfortunately it was so complex, that I soon got bored as I…
Martin Medro
- 787
37
votes
2 answers
Rubik’s Cube Not a Group?
I read online that
although the $3 \times 3 \times 3$ 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 that must mean that some moves aren’t…
Xodarap
- 6,303
36
votes
2 answers
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 the right face has a period of four; a $180$°…
MD XF
- 363
35
votes
3 answers
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 the cube, and then whenever you want to solve a…
celtschk
- 44,527
31
votes
2 answers
Solving Rubik's cube and other permutation puzzles
I've seen two questions on solving the Rubik's cube but none of the answers have given a complete solution using mainly mathematical techniques. Furthermore, I've not seen a good explanation of general techniques for solving permutation puzzles in…
user21820
- 60,745
28
votes
4 answers
Minimal generating set of Rubik's Cube group
The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $B,U,D,L,R$? If I try this out on an actual…
Simeon
- 967
27
votes
3 answers
Number of moves necessary to solve Rubik's cube by pure chance
Suppose, random moves are made to solve Rubik's cube. A move consists of
a $90$-degree-rotation of some side. The starting position is also random.
What is $E(X)$, where $X$ is the number of moves until the cube is solved ?
How many moves must be…
Peter
- 86,576
21
votes
3 answers
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 questions about Rubik's cube, and what good brings…
Mihai Bujanca
- 827
20
votes
1 answer
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 that can be achieved from the solved cube by only…
Miksu
- 1,394
20
votes
1 answer
Proof that the Rubik’s Cube group is 2-generated
Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube:
Frank Barnes observes that the group of the cube is generated by two moves:
\begin{align*}
\alpha &= L^2 B R D^{-1} L^{-1} &=(RF,RU,RB,UB,LD,LB,LU,BD,DF,FL,RD)& \\
&&\cdot…
lynn
- 3,418
19
votes
2 answers
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 Rubik's Cube as an example of a finite, non-abelian…
Alexander
- 1,281
19
votes
2 answers
Obscuring squares of Rubik's cube
This is a combinatorial question related to Rubik's cube $3\times3\times3$ (and, in the end, its generalizations $n\times n\times n$). I assume that the readers are familiar with this puzzle. Let's recall some basic terminology and conventions to…
Vladimir Reshetnikov
- 32,650
18
votes
3 answers
Can RUBIK's cube be solved using group theory?
Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
scimad.om
- 189