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
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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/