God’s Number: Why 20 Moves Is Always Enough

“God’s number” is the most moves you would ever need to solve a 3x3 cube if you always knew the shortest way. For the standard cube it is 20.

What it means

Pick any of the 43 quintillion positions. There is always a way to solve it in 20 face turns or fewer, counting half turns like R2 as one move (the half-turn metric). Most positions need 17 or 18 moves. Only a tiny fraction need the full 20.

How it was proven

In July 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson and John Dethridge announced the proof. They split all positions into about 2.2 billion groups, used clever symmetry tricks to skip many of them, and ran a fast solver on the rest. The computing time, about 35 CPU-years, was donated by Google. Every position was covered, and none needed more than 20 moves (cube20.org).

It had been known since 1995 that some positions need at least 20 moves, when Michael Reid showed that the superflip (every edge flipped in place) can’t be done in fewer. The 2010 work showed that nothing needs more.

Quarter turns: 26

If you count half turns as two moves (the quarter-turn metric), God’s number is 26. That was proven in 2014 by Rokicki and Davidson (cube20.org).

Does our solver find 20-move solutions?

Our cube solver uses Kociemba’s two-phase method, which finds a short solution very fast rather than the very shortest. In our tests on 1,000 random cubes it averaged about 20 moves, very close to the best possible. Finding the guaranteed shortest solution for a position can take much longer, which is why fast solvers settle for “nearly perfect”.

Why “God’s number”?

The idea is that an all-knowing being would always use the shortest solution. The worst case for that perfect solver is the number. People call the perfect method “God’s algorithm”.

Questions people ask

What is God’s number for a Rubik’s cube?

20 moves in the half-turn metric, and 26 in the quarter-turn metric.

When was God’s number proven?

July 2010, by Tomas Rokicki, Herbert Kociemba, Morley Davidson and John Dethridge, using computer time donated by Google.

What position needs 20 moves?

The best known is the superflip, where every edge is flipped in its place. Many other positions also need 20.