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 the 54 pieces by $[54]:=\{1,2,\cdots,54\}$. Given a permutation $\sigma$ of $[54]$, for every $k\in [54]$ we replace the piece numbered by $k$ with […]

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 symmetry of the Rubik’s cube? Is it a thing like, “turn the top face once clockwise, then once counterclockwise”? How are these symmetries counted?

Can RUBIK’s cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?

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?

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 group. My question: How can I use this fact to find possible ways for solving the 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 theory while the cube is just superficially treated. Is there also literature for people already familiar with group theory, who wants to study the […]

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 combinations: $8! \cdot 2^{12} + 3^{12} \cdot 12!$. What I am not able to understand is why do we divide this quantity by […]

Are there positions of the rubik cube which cannot be reached by applying the standard moves starting from the solved cube? If so, how many such positions are there?

Associated to the Rubik’s cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise by $90$ degrees. According to the article the order of the group is: $2^{27} 3^{14} 5^3 7^2 11$. I have […]

I’ve been working on a problem related to the 3x3x3 Rubik’s Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is sliced up into 27 regions and 26 of them are the external pieces of the puzzle (8 corners + […]

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