Mathematics of the Rubik's cube

These notes cover enough group theory and graph theory to understand the mathematical description of the Rubik's cube and several other related puzzles.

**Tag(s):**
Mathematics

**Publication date**: 31 Dec 1996

**ISBN-10**:
n/a

**ISBN-13**:
n/a

**Paperback**:
281 pages

**Views**: 17,260

**Type**: N/A

**Publisher**:
n/a

**License**:
n/a

**Post time**: 28 Mar 2007 05:31:13

Mathematics of the Rubik's cube

These notes cover enough group theory and graph theory to understand the mathematical description of the Rubik's cube and several other related puzzles.

From the Introduction:

Groups measure symmetry. No where is this more evident than in the study of symmetry in 2- and 3-dimensional geometric figures. Symmetry, and hence groups, play a key role in the study of crystallography, elementary particle physics, coding theory, and the Rubik' s cube, to name just a few.

This is a book biased towards group theory not the "the cube". To paraphrase the German mathematician David Hilbert, the art of doing group theory is to pick a good example to learn from. The Rubik's cube will be our example. We motivate the study of groups by creating a group-theoretical model of Rubik's cube-like puzzles. Although some solution strategies are discussed (for the Rubik's cube - the 3 x 3 x 3 and 4 x 4 x 4 versions, the "Rubik tetrahedron" or pyraminx, the the "Rubik dodecahedron", or megaminx, the skewb, square 1, the masterball, and the equator puzzle), these are viewed more abstractly than most other books on the subject. We regard a solution strategy merely as a not-too-inefficient algorithm for producing all the elements in the associated group of moves.

The approach here is different than some other group theory presentations (such as Rotman [R]) in that

(a) we stress puzzle-related examples over general theory,

(b) we present some of the basic notions algorithmically (as in [Bu]), and

(c) we tried to keep the level as low as possible for as long as possible (though only the reader can judge if we've keep it low enough long enough or too low too long).

The hope is that by following along and doing as many of the exercises as possible the reader will have fun learning about how groups occur in this area of recreational mathematics. Along the way, we shall also explain the rules of a two person game (the Gordon game) which arises from group theory - a game which arises from a group not a group which arise from a game. See also the superflip game introduced in §4.6 and the Icosian game in §6.2.1.

Groups measure symmetry. No where is this more evident than in the study of symmetry in 2- and 3-dimensional geometric figures. Symmetry, and hence groups, play a key role in the study of crystallography, elementary particle physics, coding theory, and the Rubik' s cube, to name just a few.

This is a book biased towards group theory not the "the cube". To paraphrase the German mathematician David Hilbert, the art of doing group theory is to pick a good example to learn from. The Rubik's cube will be our example. We motivate the study of groups by creating a group-theoretical model of Rubik's cube-like puzzles. Although some solution strategies are discussed (for the Rubik's cube - the 3 x 3 x 3 and 4 x 4 x 4 versions, the "Rubik tetrahedron" or pyraminx, the the "Rubik dodecahedron", or megaminx, the skewb, square 1, the masterball, and the equator puzzle), these are viewed more abstractly than most other books on the subject. We regard a solution strategy merely as a not-too-inefficient algorithm for producing all the elements in the associated group of moves.

The approach here is different than some other group theory presentations (such as Rotman [R]) in that

(a) we stress puzzle-related examples over general theory,

(b) we present some of the basic notions algorithmically (as in [Bu]), and

(c) we tried to keep the level as low as possible for as long as possible (though only the reader can judge if we've keep it low enough long enough or too low too long).

The hope is that by following along and doing as many of the exercises as possible the reader will have fun learning about how groups occur in this area of recreational mathematics. Along the way, we shall also explain the rules of a two person game (the Gordon game) which arises from group theory - a game which arises from a group not a group which arise from a game. See also the superflip game introduced in §4.6 and the Icosian game in §6.2.1.

Tweet

About The Author(s)

W. David Joyner is a professor in the Department of Mathematics at The United States Naval Academy. He received his Ph.D. in Mathematics at the University of Maryland in May 1983. His research areas are error-correcting codes, representation theory and applications, and applications of number theory/group theory to communication theory and cryptography.

Book Categories

Computer Science
15
Introduction to Computer Science
32
Introduction to Computer Programming
52
Algorithms and Data Structures
25
Artificial Intelligence
24
Computer Vision
30
Machine Learning
6
Neural Networks
22
Game Development and Multimedia
25
Data Communication and Networks
5
Coding Theory
16
Computer Security
9
Information Security
35
Cryptography
3
Information Theory
17
Computer Organization and Architecture
22
Operating Systems
2
Image Processing
11
Parallel Computing
4
Concurrent Programming
24
Relational Database
3
Document-oriented Database
14
Data Mining
16
Big Data
17
Data Science
23
Digital Libraries
22
Compiler Design and Construction
26
Functional Programming
11
Logic Programming
26
Object Oriented Programming
21
Formal Methods
70
Software Engineering
3
Agile Software Development
7
Information Systems
5
Geographic Information System (GIS)

Mathematics
67
Mathematics
14
Algebra
1
Abstract Algebra
27
Linear Algebra
3
Number Theory
8
Numerical Methods
2
Precalculus
10
Calculus
3
Differential Equations
5
Category Theory
11
Proofs
19
Discrete Mathematics
24
Theory of Computation
14
Graph Theory
2
Real Analysis
1
Complex Analysis
15
Probability
47
Statistics
7
Game Theory
5
Queueing Theory
13
Operations Research
16
Computer Aided Mathematics

Supporting Fields
21
Web Design and Development
1
Mobile App Design and Development
28
System Administration
2
Cloud Computing
10
Electric Circuits
6
Embedded System
28
Signal Processing
4
Network Science
3
Project Management

Operating System
Programming/Scripting
6
Ada
13
Assembly
34
C / C++
8
Common Lisp
2
Forth
35
Java
13
JavaScript
1
Lua
15
Microsoft .NET
1
Rexx
12
Perl
6
PHP
68
Python
12
R
1
Rebol
13
Ruby
2
Scheme
3
Tcl/Tk

Miscellaneous