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This is related to this question, but different. I'm not asking for the least number of moves needed, nor for pointers on how to use fewer moves.

In Singmeister's notes (an 80's book!) there is mention of M. Thistlewaite's algorithm, which could solve the cube from any position in a guaranteed 45 moves. Of course we know now that an all-seeing cuber needs at most 20 moves, but what I am asking is about a method that can be applied by a human and which guarantees as few moves as possible regardless of the initial position.

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  • $\begingroup$ Just so that I fully understand what you're asking: are you asking for the method that guarantees a solve in the fewest number of moves, while still being reasonable for a human to use? (So for example, a method that would involve a lookup of a table of a few million configurations, while feasible for a computer, would not do the trick here?) $\endgroup$
    – Dennis Meng
    Oct 4, 2014 at 5:58
  • $\begingroup$ @Dennis: Correct. $\endgroup$
    – Yrodro
    Oct 4, 2014 at 6:00
  • $\begingroup$ "An all seeking cuber needs at most 20 moves" That seems pretty low to me, after all the record for minimum moves in WCA is 20 $\endgroup$
    – For I In Range
    Dec 31, 2014 at 20:06
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    $\begingroup$ The meaning of "all seeking cuber" is a computer. $\endgroup$
    – Anonymous
    Mar 20, 2017 at 9:04
  • $\begingroup$ @ForIInRange by the half-turn metric, "God's Number" has been proved by exhaustion to be 20 (26 by quarter-turn metric): cube20.org $\endgroup$
    – Nick T
    Aug 7, 2017 at 21:13

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I don't think that anyone has put a lot of effort in developing such a method. The problem on such a method is, that it would be require a lot of searching for pieces, the algorithms are hard to recognize and so on. At least nobody have proven otherwise.

I'm into speedsolving (solving the cube as fast as possible). Basically we happily choose methods with a 10 more moves, over one with crappy recognition, bad (slow to execute) algorithms, ...

For instance the top guys in speedcubing average about 8 seconds on average on a cube, but still use a method with 50 moves. I doubt that times like that would be possible with a method averaging 35 moves or so.

But back to the question: There are a few methods out there, that require fewer moves than the popular ones. I can't give you numbers for the worst case, nobody has made any efforts in calculating these yet. But I can give you average numbers of turns.

  • Snyder 2: 40 moves on average, the developer claims to have ideas for a 30 move method, but never explored it.

  • Heise Method: 40 moves on average, more famous than Snyder method. You will probably find much more documentation on this method.

Big parts of these methods are only ideas though. Parts of these methods depend on building random blocks. There are no algorithms for these, all you need is lots of brain power, experience and random tricks. A beginner or even an intermediate cuber you will get nowhere near those 40 moves on average.

This may be a bit off topic. But in speedsoving competition there is also a event called Fewest Moves. Each competitor gets a piece of paper with instruction on how to scramble the cube, and the competitor has 1 hour to find a short solution to this scramble. The competitor with the shortest solution wins. In this 1 hour you can scramble and solve as many cubes as you want, try different tricks, and so on.

Good people find sub 30 solutions quite regularly. Tomoaki Okayama from Japan even found a 20 move solution once.

The main method people use at a fewest move event depend also a lot on intuitive block building. And use lots of 'tricks' someone can't use in a regular solve. For instance someone can find a skeleton (e.g. all pieces solved except 3 corners) and then solve the missing pieces somewhere at a suitable place in the middle of the solve.

But good fmc solver can also find quite short solutions on a linear solve (where you can't undo moves). You might wanna check out this video: FMC Head to Head

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    $\begingroup$ Actually Heise method can easily be done in below 40 moves, especially with 1 hour. Averaging around 30 isn't too difficult with a little extra algorithms to assist in some cases. Averaging sub-30 is harder and requires combinations of method. $\endgroup$
    – Ariana
    Aug 22, 2016 at 16:34
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Although not answering the question, asking for a human method, no cube position requires 21+ moves. Every position on the cube can be solved in 20 moves or less;

http://www.cube20.org/

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    $\begingroup$ comment instead. $\endgroup$
    – Anonymous
    Mar 20, 2017 at 9:05

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