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

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 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 Born in Budapest, Hungary Architect Founder of Rubik Studio http://pics.livejournal.com/sullenfish/pic/0000801h/s640x480

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://www.smh.com.au/ffximage/2007/10/04/cube_narrowweb__300x392,0.jpg http://upload.wikimedia.org/wikipedia/en/thumb/1/1e/Pocket_cube.jpg/200px-Pocket_cube.jpg

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

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

Permutations for a Rubik’s Cube 43,252,003,274,489,856,000 3,674,160 http://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik's_cube.svg/480px-Rubik's_cube.svg.png http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Solved_2x2x2.jpg/600px-Solved_2x2x2.jpg

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) = 0 Associativity 1+(2+3) = (1+2)+3 Commutative 1+2 = 2+1

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

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

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

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

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 U = Up D = Down L = Left

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

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