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Mathematics Behind the Rubik’s Cube
Mathematical Modeling
Bihan Zhang and Trachelle McDonald
C.E. Jordan High School and Pamlico High School
2008
Problem
Explore the mathematics behind Rubik’s cube using simulations in VPythonExplain how permutation relate to the Rubik’s
cubeExplain how group theory relate to the Rubik’s
cube
http://upload.wikimedia.org/wikipedia/commons/6/67/Rubiks_revenge_scrambled.jpg
Outline
History Permutations Operations with Groups Triangle Operations Rubik’s Cube Operations Conclusion
http://www.smh.com.au/ffximage/2007/10/04/cube_narrowweb__300x392,0.jpg
Inventor: Ernö Rubik
http://pics.livejournal.com/sullenfish/pic/0000801h/s640x480
-Born in Budapest, Hungary
-Architect
-Founder of Rubik Studio
History Invented by Ernő Rubik
in 1974
“No arrangement of the
3x3x3 Rubik's Cube
requires more than 20
moves to solve.”
“The Current World
Record is 7.08
Seconds."
http://upload.wikimedia.org/wikipedia/en/thumb/1/1e/Pocket_cube.jpg/200px-Pocket_cube.jpg
http://www.smh.com.au/ffximage/2007/10/04/cube_narrowweb__300x392,0.jpg
Permutations
“A permutation is an arrangement of objects in different orders.”1 2 31 3 22 1 32 3 13 1 2 3 2 1
Permutations
123
123
231
123
312
123
1 1 2 2 3 3
1 2 2 3 3 1
1 3 2 1 3 2
t =
t (1) = 2 t (2) = 3 t (3) = 1
u =
u (1) = 3 u (2) = 1 u (3) = 2
Original
Permuted
Permutations for a Rubik’s Cube
43,252,003,274,489,856,000
3,674,160
107 2!123!8
63!7
http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Solved_2x2x2.jpg/600px-Solved_2x2x2.jpg
http://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik's_cube.svg/480px-Rubik's_cube.svg.png
What is a Group?
A set of elements plus a binary operation A group has the following properties:
Closure 1+2 = 3 Identity element 1+0 = 1 Inverse 1+(-1) = 0Associativity 1+(2+3) = (1+2)+3
Commutative 1+2 = 2+1
Operations with Groups
123
123
231
123
312
123
213
123
321
123
1 = v =
t = w =
u = x =
132
123
1. tx=?
2. t(x(1))=?3. x(1)=14. t(1)=2
5. t(x(2))=?6. x(2)=37. t(3)=1
8. t(x(3))=?9. x(3)=210. t(2)=3
tx=(213)=v
Operations with Groups
123
123
231
123
312
123
213
123
321
123
1 = v =
t = w =
u = x =
132
123
1. xt=?
2. x(t(1))=?3. t(1)=24. x(2)=3
5. x(t(2))=?6. t(2)=37. x(3)=2
8. x(t(3))=?9. t(3)=110. x(1)=1
xt=(321)=w
Operations with Groups
123
123
231
123
312
123
213
123
321
123
1 = v = X 1 t u v w x
1 1 t u v w x
t = w = t t u 1 w x v
u u 1 t x v w
u = x = v v x w 1 u t
w w v x t 1 u
x x w v u t 1
132
123
tx xt
Operations with Groups
123
123
231
123
312
123
213
123
321
123
1 = v = X 1 t u v w x
1 1 t u v w x
t = w = t t u 1 w x v
u u 1 t x v w
u = x = v v x w 1 u t
w w v x t 1 u
x x w v u t 1
132
123
Symmetry Group of Triangles
Identity =
Rotation
Symmetry Group of Triangles
Identity =
Reflection
Symmetry Group of Triangles
Symmetry Group of Triangles
Rubik’s Cube Groups
F = Front B = Back R = Right
L = Left U = Up D = Down
Rubik’s Cube Groups
FF =
FFFF = = I
F = Front
B = Back
L = Left
= F2
R =Right
U = Up
D = Down
Singmaster Notation
FFF = = F3
Our Simulation
Pretty Patterns
Green Mamba
RDRFrfBDrubUDD
Anaconda
LBBDRbFdlRdUfRRu
Christmas Cross
uFFUUlRFFUUFFLru
Conclusion
- Group theory is an integral part of the Rubik’s cube
- It is possible to solve a Rubik’s cube by reversing the operations done
Work Cited
http://cubeland.free.fr/infos/ernorubik.htm Christopher Goudey 2001-2003
http://regentsprep.org/Regents/math/permut/Lperm.htm 1999-2008 http://regentsprep.org
Oswego City School District Regents Exam Prep Center http://www.wikipedia.org http://www.daniweb.com/code/snippet459.html http://www.cs.princeton.edu/courses/archive/fall06/cos402/papers/korfrubik.pdf http://www.dougmair.blogspot.com/ http://match.stanford.edu/bump/newcube.pdf http://www.geometer.org/rubik/group.pdf Joyner, David. Adventures in Group Theory. Baltimore: John Hopkins U P, 2002.
Acknowledgments
Dr. Russell L. Herman Mr. David B. Glasier Mr. Nathaniel Jones Mr. Doug Mair Mr. Ernö Rubik The SVSM Staff
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