"Every position of Rubik's Cube™ can be solved in twenty moves or less.
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.
Every solver of the Cube uses an algorithm, which is a sequence of steps for solving the Cube. One algorithm might use a sequence of moves to solve the top face, then another sequence of moves to position the middle edges, and so on. There are many different algorithms, varying in complexity and number of moves required, but those that can be memorized by a mortal typically require more than forty moves.
One may suppose God would use a much more efficient algorithm, one that always uses the shortest sequence of moves; this is known as God's Algorithm. The number of moves this algorithm would take in the worst case is called God's Number. At long last, God's Number has been shown to be 20.
It took fifteen years after the introduction of the Cube to find the first position that provably requires twenty moves to solve; it is appropriate that fifteen years after that, we prove that twenty moves suffice for all position
A History of God's Number
By 1980, a lower bound of 18 had been established for God's Number by analyzing the number of effectively distinct move sequences of 17 or fewer moves, and finding that there were fewer such sequences than Cube positions. The first upper bound was probably around 80 or so from the algorithm in one of the early solution booklets. This table summarizes the subsequent results.| Date | Lower bound | Upper bound | Gap | Notes and Links |
|---|---|---|---|---|
| July, 1981 | 18 | 52 | 34 | Morwen Thistlethwaite proves 52 moves suffice. |
| April, 1992 | 18 | 42 | 24 | Hans Kloosterman improves this to 42 moves. |
| May, 1992 | 18 | 39 | 21 | Michael Reid shows 39 moves is always sufficient. |
| May, 1992 | 18 | 37 | 19 | Dik Winter lowers this to 37 moves just one day later! |
| January, 1995 | 18 | 29 | 11 | Michael Reid cuts the upper bound to 29 moves by analyzing Kociemba's two-phase algorithm. |
| January, 1995 | 20 | 29 | 9 | Michael Reid proves that the ''superflip'' position (corners correct, edges placed but flipped) requires 20 moves. |
| December, 2005 | 20 | 28 | 8 | Silviu Radu shows that 28 moves is always enough. |
| April, 2006 | 20 | 27 | 7 | Silviu Radu improves his bound to 27 moves. |
| May, 2007 | 20 | 26 | 6 | Dan Kunkle and Gene Cooperman prove 26 moves suffice. |
| March, 2008 | 20 | 25 | 5 | Tomas Rokicki cuts the upper bound to 25 moves. |
| April, 2008 | 20 | 23 | 3 | Tomas Rokicki and John Welborn reduce it to only 23 moves. |
| August, 2008 | 20 | 22 | 2 | Tomas Rokicki and John Welborn continue down to 22 moves. |
| July, 2010 | 20 | 20 | 0 | Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge prove that God's Number for the Cube is exactly 20. |
How We Did It
How did we solve all 43,252,003,274,489,856,000 positions of the Cube?- We partitioned the positions into 2,217,093,120 sets of 19,508,428,800 positions each.
- We reduced the count of sets we needed to solve to 55,882,296 using symmetry and set covering.
- We did not find optimal solutions to each position, but instead only solutions of length 20 or less.
- We wrote a program that solved a single set in about 20 seconds.
- We used about 35 CPU years to find solutions to all of the positions in each of the 55,882,296 sets.
Partitioning
We broke the problem down into 2,217,093,120 smaller problems, each comprising 19,508,428,800 different positions. Each of these subproblems was small enough to fit in the memory of a modern PC, and the way we broke it down (mathematically, using cosets of the group generated by {U,F2,R2,D,B2,L2}, or more concisely, cosets of H) allowed us to solve each set rapidly.Symmetry
If you take a scrambled Cube and turn it upside down, you have not made it any more difficult; it will still take the same number of moves to solve. Instead of solving both of these positions, you can simply solve one, and then turn the solution upside down for the other. There are 24 different ways you can orient the Cube in space, and another factor of two using a mirror, for a total reduction of a factor of about 48 in the number of positions that need solving. Using similar symmetry arguments and by finding a solution to a large "set cover" problem, we were able to reduce the number of sets that needed solving from 2,217,093,120 down to 55,882,296.Good vs. Optimal Solutions
| Random positions | Cosets of H | |
| Optimally | 0.36 | 2,000,000 |
| 20 moves or less | 3,900 | 1,000,000,000 |
Solution rate, in positions/second
Fast Coset Solving Program
Using a combination of mathematical tricks and careful programming, we were able to solve a complete coset of H, either optimally, or with sequences of twenty moves or less, on a single desktop PC, at the rates shown in the table at left.Lots of Computers
Finally, we were able to distribute the 55,882,296 cosets of H among a large number of computers at Google and complete the computation in just a few weeks. Google does not release information on their computer systems, but it would take a good desktop PC (Intel Nehalem, four-core, 2.8GHz) 1.1 billion seconds, or about 35 CPU years, to perform this calculation."What are the Hardest Positions?
| Distance | Count of Positions |
|---|---|
| 0 | 1 |
| 1 | 18 |
| 2 | 243 |
| 3 | 3,240 |
| 4 | 43,239 |
| 5 | 574,908 |
| 6 | 7,618,438 |
| 7 | 100,803,036 |
| 8 | 1,332,343,288 |
| 9 | 17,596,479,795 |
| 10 | 232,248,063,316 |
| 11 | 3,063,288,809,012 |
| 12 | 40,374,425,656,248 |
| 13 | 531,653,418,284,628 |
| 14 | 6,989,320,578,825,358 |
| 15 | 91,365,146,187,124,313 |
| 16 | about 1,100,000,000,000,000,000 |
| 17 | about 12,000,000,000,000,000,000 |
| 18 | about 29,000,000,000,000,000,000 |
| 19 | about 1,500,000,000,000,000,000 |
| 20 | about 300,000,000 |
Taken verbatim from http://www.cube20.org/