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Rubik's Cube Solution and God’s Number Group Theory Keyang He Introduction • Rubik’s Cube: a 3×3×3 cube in which the 26 subcubes on the outside are internally hinged in such a way that rotation is possible in any plane of cubes. It is invented in 1974 by Ernő Rubik. Originally called the "Magic Cube“ Introduction • Rubik’s Group: the group corresponding to possible rotations of a Rubik's cube. The group has order: 8!×37 ×(12! / 2)×211 = 43,252,003,274,489,856,000 • Rubik's Graph: Rubik's graph is the Cayley graph of Rubik's group. Introduction Cont. • The minimum number of turns required to solve the cube from an arbitrary starting position is equal to the graph diameter of Rubik's graph, and is sometimes known as God's number. God’s Number • With about 35 CPU-years of idle computer time donated by Google, a team of researchers has essentially solved every position of the Rubik's Cube, and shown that no position requires more than twenty moves. History of God’s Number Date Lower Bound Upper Bound Gap July, 1981 18 52 34 December, 1990 18 42 24 May, 1992 18 39 21 May, 1992 18 37 19 January, 1995 18 29 11 January, 1995 20 29 9 December, 2005 20 28 8 April, 2006 20 27 7 May, 2007 20 26 6 March, 2008 20 25 5 April, 2008 20 23 3 August, 2008 20 22 2 July, 2010 20 20 0 Method of Proving • Partition the positions into 2,217,093,120 sets of 19,508,428,800 positions each. • Reduce the count of sets we needed to solve to 55,882,296 using symmetry and set covering. • Cannot find optimal solutions to each position, but instead only solutions of length 20 or less. • Write a program that solved a single set in about 20 seconds. • Use about 35 CPU years to find solutions to all of the positions in each of the 55,882,296 sets. Count of Positions • There are positions that require 20 moves. For distances 16 and greater, the number given is just an estimate. Solution to 3×3×3 Rubik’s Cube • Fridrich Method: this method was first developed in the early 1980s combining innovations by a number of cubers. Czech speedcuber Jessica Fridrich is generally given credit for popularizing it by publishing it online in 1997. Fridrich Method • The Cross - This first stage involves solving four edge pieces on one side. It requires 8 or fewer turns to solve. • Solve the First Two Layers (F2L) - In F2L, both the corner piece and edge pieces are solved at the same time, reducing the number of twists needed. There are 42 different cases for F2L, 41 plus the already solved case. • Orientation of Last Layer (OLL) - This stage involves manipulating the top layer so that all the top cubes have the same color on top, even at the expense of incorrect colors on other sides. This stage involves learning a total of 57 algorithms. • Permutation of Last Layer (PLL) - The final stage involves moving the pieces of the top layer while preserving their orientation. There are a total of 21 algorithms for this stage. Solution to n×n×n Rubik’s Cube • Zhihong Qiu’s 8-Step Method: H(p,q,r)=YpZq--Yr-ZqYp-Zq--YrZq Solution to n×n×n Rubik’s Cube A B C D E • A: X1-[xX]X1 • B: Y1-X1-Y1[ (1,1,r) × X]Y1-X1Y1 • D: X1H(1,1,r)X1 • E: -Zp--Y12XrH(p,1,r)Xr-Y12-Zp References • • • • • http://mathworld.wolfram.com/RubiksCube.html http://mathworld.wolfram.com/GodsNumber.html http://rubiks.wikia.com/wiki/Fridrich_Method http://www.mfblog.org/qu-wei-mo-fang/178/#2 http://www.cube20.org/