1. Mathematical Modeling
Mathematics Behind the Rubik’s Cube
Mathematical Modeling
Bihan Zhang and Trachelle McDonald
C.E. Jordan High School and Pamlico High School
2008

2. Problem
Explore the mathematics behind Rubik’s cube using simulations in VPython
Explain how permutation relate to the Rubik’s cube
Explain how group theory relate to the Rubik’s cube

3. Outline History Permutations Operations with Groups
Triangle Operations
Rubik’s Cube Operations
Conclusion

4. Inventor: Ernö Rubik Born in Budapest, Hungary Architect
Founder of Rubik Studio

5. 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."

6. Permutations
“A permutation  is an arrangement of objects in different orders.”
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1

7. Permutations
Original
Permuted
t =
t (1) = 2
t (2) = 3
t (3) = 1
u =
u (1) = 3
u (2) = 1
u (3) = 2

8. Permutations for a Rubik’s Cube
43,252,003,274,489,856,000
3,674,160

9. What is a Group? A set of elements plus a binary operation
A group has the following properties:
Closure = 3
Identity element = 1
Inverse (-1) = 0
Associativity (2+3) = (1+2)+3
Commutative = 2+1

10. Operations with Groups
tx=?
t(x(1))=?
x(1)=1
t(1)=2
t(x(2))=?
x(2)=3
t(3)=1
1 =
v =
t =
w =
u =
x =
8. t(x(3))=?
9. x(3)=2
10. t(2)=3
tx=(213)=v

11. Operations with Groups
1 =
v =
t =
w =
u =
x =
xt=?
x(t(1))=?
t(1)=2
x(2)=3
x(t(2))=?
t(2)=3
x(3)=2
8. x(t(3))=?
9. t(3)=1
10. x(1)=1
xt=(321)=w

12. Operations with Groups
1 =
v = X 1 t u v w x
t =
w =
u =
x =
tx xt

13. Operations with Groups
1 =
v = X 1 t u v w x
t =
w =
u =
x =

14. Symmetry Group of Triangles
Identity =
Rotation

15. Symmetry Group of Triangles
Identity =
Reflection

16. Symmetry Group of Triangles

17. Symmetry Group of Triangles

18. Rubik’s Cube Groups F = Front B = Back R = Right U = Up D = Down
L = Left

19. Rubik’s Cube Groups FF = = F2 FFF = = F3 FFFF = = I
Singmaster Notation
F = Front
B = Back
L = Left
R =Right
U = Up
D = Down
FFF =
= F3
FFFF =
= I

20. Our Simulation

21. Pretty Patterns Green Mamba RDRFrfBDrubUDD Anaconda LBBDRbFdlRdUfRRu
Christmas Cross
uFFUUlRFFUUFFLru

22. 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

23. Work Cited http://cubeland.free.fr/infos/ernorubik.htm
Christopher Goudey
Oswego City School District Regents Exam Prep Center
Joyner, David. Adventures in Group Theory. Baltimore: John Hopkins U P, 2002.

24. Acknowledgments Dr. Russell L. Herman Mr. David B. Glasier
Mr. Nathaniel Jones
Mr. Doug Mair
Mr. Ernö Rubik
The SVSM Staff