12 Blessing Project

Rubik’s Magic Cube David Blessing Outline of Presentation •  Anatomy of a Rubik’s Cube •  Speed Cubing Method of Solu=on •  Rubik’s Cube Symmetric Group •  Nature of the Group •  Order of the Group •  Subgroups •  God’s Number Anatomy of the Rubik’s Cube The Core of the Cube •  The back bone of a Rubik’s Cube that holds everything together. Center of the Faces •  These aEach to the core via adjustable screws and springs. •  We tune the tension of the cube with the screws. •  They are are immobile in the sense of cube permuta=ons. Edges of the Cube •  First piece we examine for cube permuta=on. •  They are held in place by the spring tension. •  There are 12 in total. Corners of the Cube •  The 8 corners of the cube are responsible for most of its difficulty. Friedrich Method of Solving Bottom Cross •  Typically, this is done from memory. •  No algorithms •  Averages 8 moves. First 2 Layers (F2L) •  Places the boEom corners and middle layer edges at once. •  41 algorithms are involved. •  Averages 28 moves. Orientation of Last Layer (OLL) •  Places all last layer s=ckers face up. •  I use a slight varia=on. •  57 Algorithms Permutation of Last Layer (PLL) •  Places last layer edges and corners in proper place. •  27 Algorithms Rubik’s Cube Group Symmetric Group Generators of the Group •  Cube rota=ons are unnecessary. •  We only need to consider the 6 faces turned 90o clockwise. •  In prac=ce we consider 18 moves, however, they are not part of the genera=ng set. •  U – Upper Face •  D – Down Face •  R – Right Face •  L – Le[ Face •  F – Front Face •  B – Back Face Valid Moves as Cycles Nature of the Group •  Let G be generated by the 6 legal cube moves. •  The set G under func=on composi=on forms the Rubik’s Cube Group. •  G is a normal subgroup of S48 •  Also a subgroup of A48 •  The elements of the group when preformed on a solved cube create a legal state of the cube. •  G is a group: • 
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Associa=ve Closed Iden=ty Inverse •  G is nonabelian. The Order of G •  Permuta=ons of 12 edges: •  12! •  Permuta=ons of 8 corners: •  8! •  Parity of permuta=ons •  Single edge or corner flip. •  Orienta=on of 12 edges: •  211 •  Orienta=on of 8 corners: •  37 •  Orienta=on of last edge and corner dependent on previous ones. The Order of G •  We now mul=ply our results: •  12!*8!*211*37/2 •  43,252,003,274,489,856,000 total legal permuta=ons. •  If we were to simply disassemble the cube and put it back together then, we have 12 =mes as many permuta=ons. •  Also, if we were to put the s=ckers on the cube at random, we would have 48! Permuta=ons. Subgroups of G •  Trivial ones •  <R> •  Not so trivial ones: •  <R2,L2,D2,U2,B2,F2> •  <R-­‐U-­‐R’-­‐U’> •  There are many. •  Subgroups allow you to solve the cube with only select moves. •  It is possible to end up with only top layer incorrect, yet need to change the rest of the cube to solve it. Two Squares Group •  A subgroup of any two adjacent edges. •  Consider the subgroup <U2,R2> •  Has 12 permuta=ons •  6 of order 2 •  6 that form a cyclic subgroup The God Number •  With 43+ quin=llion combina=ons, how many moves is enough to always solve the cube? •  This was recently solved using symmetries and set covering with 35 years of cpu =me. •  They split it up into about 55 million blocks of 19,508,428,800 posi=ons each •  It turns out that 20 moves is sufficient to solve any cube posi=on. •  Another surprise, there are quite a few hardest posi=ons. How many positions are n-­‐moves from solved? Works Cited •  Contemporary Abstract Algebra •  Joseph A. Gallian •  Handbook of Cubik Math •  Alexander Frey and David Singmaster •  www.cubefreak.net •  www.cube20.org •  hEp://ws2.binghamton.edu/fridrich/