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Page 1 of 1http://kociemba.org/homex.htm
Cube Explorer 5.01
If you like Cube Explorer you can show your appreciation.
Please note: Cube Explorer is not derived from, is not associated with and is not endorsed or
sponsered by the owner of the RUBIK'S CUBE Trademark. This owner is Seven Towns Limited,
the manufacturer and worldwide distributor of the RUBIK'S CUBE three dimensional puzzle and
provider of an electronic version of the puzzle via its official web site.
Cube Explorer is a non-commercial, educational product. It is freeware and the result of scientific
research.
© 2013 Herbert Kociemba
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Page 1 of 2http://kociemba.org/twophase.htm
The Two-Phase-Algorithm
The following description is intended to give you a basic idea of how the algorithm works.
The 6 different faces of the Cube are called U(p), D(own), R(ight), L(eft), F(ront) and B(ack). While
U denotes an Up Face quarter turn of 90 degrees clockwise, U2 denotes a 180 degrees turn and
U' denotes a quarter turn of 90 degrees counter-clockwise. A sequence like U D R' D2 of Cube
moves is called a maneuver.
If you turn the faces of a solved cube and do not use the moves R, R', L, L', F, F', B and B' you will
only generate a subset of all possible cubes. This subset is denoted by G1 = <U,D,R2,L2,F2,B2>.
In this subset, the orientations of the corners and edges cannot be changed. That is, the
orientation of an edge or corner at a certain location is always the same. And the four edges in the
UD-slice (between the U-face and D-face) stay isolated in that slice.
In phase 1, the algorithm looks for maneuvers which will transform a scrambled cube to G1. That
is, the orientations of corners and edges have to be constrained and the edges of the UD-slice
have to be transferred into that slice. In this abstract space, a move just transforms a triple (x,y,z)
into another triple (x',y',z'). All cubes of G1 have the same triple (x0,y0,z0) and this is the goal
state of phase 1.
To find this goal state the program uses a search algorithm which is called iterative deepening A*
with a lowerbound heuristic function (IDA*). In the case of the Cube, this means that it iterates
through all maneuvers of increasing length. The heuristic function h1(x,y,z) estimates for each
cube state (x,y,z) the number of moves that are necessary to reach the goal state. It is essential
that the function never overestimates this number. In Cube Explorer 2, it gives the exact number of
moves which are necessary to reach the goal state in Phase 1. The heuristic allows pruning while
generating the maneuvers, which is essential if you do not want to wait a very, very long time
before the goal state is reached. The heuristic function h1 is a memory based lookup table and
allows pruning up to 12 moves in advance.
In phase 2 the algorithm restores the cube in the subgroup G1, using only moves of this subgroup.
It restores the permutation of the 8 corners, the permutation of the 8 edges of the U-face and D-
face and the permutation of the 4 UD-slice edges. The heuristic function h2(a,b,c) only estimates
the number of moves that are necessary to reach the goal state, because there are too many
different elements in G1.
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Page 2 of 2http://kociemba.org/twophase.htm
The algorithm does not stop when a first solution is found but continues to search for shorter
solutions by carrying out phase 2 from suboptimal solutions of phase 1. For example, if the first
solution has 10 moves in phase 1 followed by 12 moves in phase 2, the second solution could
have 11 moves in phase 1 and only 5 moves in phase 2. The length of the phase 1 maneuvers
increase and the length of the phase 2 maneuvers decrease. If the phase 2 length reaches zero,
the solution is optimal and the algorithm stops.
In the current implementation the Two-Phase-Algorithm does not look for some solutions that are
optimal overall, those that must cross into and back out of phase 2. This increases the speed
considerably. Use the Optimal Solver, if you want to prove some maneuver to be optimal.
11/29/13 5:17 PMThe Performance of the Two-Phase-Algorithm
Page 1 of 4http://kociemba.org/performance.htm
Two-Phase-Algorithm and God's Algorithm:
God's number is 20
The algorithm which gives an optimal solution in the sense that there is no shorter solution is called God's
algorithm. There are cube positions (for example the superflip which flips all 12 edges), which are known to
have a shortest maneuver length of 20 moves to be solved. After 30 years it has finally been shown in July
2010, that all cube positions can be solved within 20 moves or less.
God's Number is 20
An overview is given on the webpage http://cube20.org/.
Our proof was recently published in SIAM Journal on Discrete Mathematics (Volume 27, Issue 2).
A tried to make a few webpages, which give more detailed information about the proof without being too
technical. Some subgroups and their cosets play an important role here, and if you have some basic
knowlege of group theory and combinatorics you should be able to follow the train of thoughts.
1. The subgroup H and its cosets
2. The subgroup Q and its cosets
3. Solving a reduced set cover problem
4. A fast coset solver
The Two-Phase-Algorithm gives near optimal solutions
I generated 1 million random cubes on a 3 GHz Pentium 4 PC, trying to find a cube which was not solvable
within 20 moves with the Two-Phase-Algorithm.
But the Two-Phase-Algorithm solved all generated random positions within 20 moves. More precisely, it
solved about 30000 random cubes per hour and the final distribution of the maneuver length was:
13: 4, 14: 18, 15: 81, 16: 609, 17: 3893, 18: 23411, 19: 141366, 20: 830618.
Nevertheless the computation of God's number showed, that there some very rare positions which are quite
hard to solve within 20 moves with the two-phase algorithm, for example the cube generated by the
maneuver
L R2 U2 B' D2 L D2 F' U' R U2 L' F' D' R' B2 D2 R' U' F2 . On my machine this one takes 16 minutes (in
triple search mode).
11/29/13 5:17 PMThe Performance of the Two-Phase-Algorithm
Page 2 of 4http://kociemba.org/performance.htm
Optimal Cube Solver
In 2009 I used Cube Explorer 4.64s and an Intel Core i7 920 CPU machine (Vista 64 bit with 6 GB of RAM)
to solve 100000 random positions optimally in parallel on 8 cores (4 physical and 4 virtual cores).
On average the program solved about 7000 cubes/day ! In this sense Cube Explorer's implementation of
God's Algorithm does a decent job.. The average optimal solving length is ~17.7 . If you want to download
the 100000 optimally solved cubes for some reason, you can do this here.
In January 2010 Tomas Rokicki even solved 1 million random cubes optimally and got the following
distribution, which is very close to the distribution I got:
