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There's a natural but nontrivial way to construct a finite subgroup of $G$: to think real Rubik's cube type puzzles. … along the axis of the cube. …

I just came back from a beach which features a large Rubik's cube (2m high). The base of the cube is not visible and the top is not coloured. … I was idly wondering: how can I tell if the patterns appearing on those four sides are actually possible for a real cube? …

The cube is forbidden from ever repeating a position (besides the start position). This guarantees the game is finite. … An example game would be F,F;F,F (using basic rubik's cube notation). Solver wins this game. …

However, it seems that there is not a complete, comprehensive algorithm that uses the Schreier-Sims method to solve the Rubik's Cube. … Is there any simple introduction of using the Schreier-Sims method to solve the Rubik's Cube? …

Is the group of the $3\times 3\times 3 \times 3$ cube a subgroup of a wreath product of another wreath product? In case you don't know about the $3\times 3\times 3$ cube. …

Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? … From "The Mathematics of Rubik's cube" by W. D. Joyner I know that $G$ is generated by two elements and presentations are known, but I have not found one. …

A white Rubik's cube has 6 sides. There are 9 spaces on each side for numbers from 1 to 9, without color. … If we fill each side of a white Rubik's cube with a magic square, how many combinations can be filled on this Rubik's Cube? …

be solved in at most 20 Singmaster moves (under the face-turn metric) from Rubik's Cube move group $(G, *)$, where $*$ is the concatenation operation (summarized here: https://en.wikipedia.org/wiki/Rubik … $n \times n \times n$ cube). …

Hence an equi-$n$-square becomes an object with $n^2$ colored faces that may remind one of the Rubik's cube. … In contrast, Rubik's cube can always be solved with at most $20$ "moves". The lower bound in result 2 is: $4$ shuffles. Hence sharpness of results 2 and 3 is an issue: Question 2. …

Here, God's number $g(n)$ was defined as the smallest number $m$ such that every realizable arrangement of the $n \times n \times n$-cube can be solved within $m$ moves. … Define $g_{k}(n)$ as the smallest number $l$ such that every realizable arrangement of the $n^{k}$-cube can be solved within $l$ moves. Whereas Mr. …

This gap controls the mixing properties of the cube - i.e., how fast the cube can be scrambled. A larger gap may correspond to a faster mixing. … In both the QMBS problems and the Rubik's cube, finding the gap as $n$ go off to infinity corresponds to finding the thermodynamic limit. …

According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. … PS Here is an example from GAP tutorial to handle the standard 3x3x3 Rubik's cube: https://www.math.rwth-aachen.de/homes/GAP/WWW2/Doc/Examples/rubik.html …

Consider higher (NxNxN) Rubik's cube group, with specific set of generators described below. What is important - that there are huge COMMUTING subsets of generators. … The diameter for N-Rubik cube group seems still to be unknown. And that open problem seems which might be tackled by AI methods (to some extent), moreover it can be seen like "toy-model". …

Imagine an $n \times n \times n$ Rubik's cube, where we can transition the state of the cube using Singmaster moves under either the face- or quarter-turn metrics. … How many total cube states are there when $k = {0, 1, 2, 3, 4}$ of the cube faces are solved, and what "distribution" are these states drawn from? …

cube by a (possibly infinite) sequence of twists (without the Solver watching), presents the scrambled cube to the Solver, and the Solver attempts to unscramble it by a sequence of twists. … The cube Let $L$ be a linear order with no greatest element, write its reverse as $-L$. …