I was thinking about the methods and substeps that have been invented for solving the cube, and the many factors that are involved with how well and how fast those methods can achieve the solved state. It's not an exact science of course, but it's something.

In essence I wanted to find a quantifiable way to measure the "efficiency" of any given method or substep. Efficiency in the sense of amount of the cube solved, relative to speed (in moves), memorization, and mental effort required. In other words, how much bang you get for your buck.

Now, this

For example, God's algorithm. If it were humanly possible to do, it would be the fastest solving method, but even then, it would not be the most efficient. Why? Because it would require such mental capacity for either memorization or intuitive solving as to be mortally impossible, and therefore is perhaps the most inefficient method of solving the cube because you get very little benefit for the amount of work and resources applied.

On the other end of the spectrum, you have the Turn-Check-If-Solved method, where each step is literally to turn the cube randomly, and after each move, check to see if the cube is in the solved state. While this requires absolutely no mental capacity or skill, the number of moves required on average to achieve a solved state would be ridiculously high, therefore making this method extremely inefficient as well.

And so this formula isn't supposed to be used as a measurement of which method is the best or which method can solve the fastest, because obviously the fastest is God's Algorithm. It's about measuring the efficiency of effort required to achieve a solved state in as few moves as possible.

Here is the formula:

Orientations = Number of cubies correctly oriented by the method/substep

Permutations = Number of cubies correctly permuted by the method/substep

Algorithms = Number of memorized algorithms required (mirrors, etc. included)

AvgMoves = Number of moves on average executed before achieving the desired state

ReplaceAlgs = Number of algorithms that would be required to replace the intuitive step(s)

IntuitiveCubies = Number of cubies solved using intuitive methods

Now to explain the reasoning behind the organization of this formula.

First the numerator of the formula is a measurement of the percentage of the cube solved. Meaning that if a method solves all 20 cubies, then the numerator will be exactly 100. So, for any full method calculation, the numerator is 100.

Now the denominator. The greater the values in the denominator, the less efficient the method/substep is considered. The following values considered negative to efficiency are:

-Algorithms

-Avg Moves

-Intuitive ineffectiveness = (ReplaceAlgs/IntuitiveCubies)

That's the formula. Hopefully I've explained it in a way that makes sense. It could probably be better, but an inexact science is very difficult to "better".

Now there are a few things that need to be estimated, for example the number of algorithms that would replace solving the cross. I estimated about 6 for the purpose of having a numerical base. This gives it an intuitive ineffectiveness of (6/4). Compare that to F2L, which has an intuitive ineffectiveness of (41/8).

Another thing that would need to be estimated is the algorithms that would replace EOLine, or Block Building. In fact, it appears that the first steps of solving the cube in any method do not have any really solid way of measuring the number of algorithms that would replace intuitive execution. Again (again), it's an inexact science.

Now when it comes to measuring substeps, there will be a rather clear decline in efficiency the further into the solve the substep lies. This is because, the more cubies you have that are solved, the more limited you are in solving others efficiently in regards to the mental effort required and the number of moves executed. Therefore the substeps Cross and F2L will be much more efficient than OLL or PLL.

So as a few examples, I'm going to work out the efficiencies of the CFOP method and it's substeps.

100/(78+2(54.8)+(47/12)) = 100/(78+109.6+3.91667) = 100/191.51667 =

20/(0+2(6.5)+(6/4)) = 20/(0+13+1.5) = 20/14.5 =

40/(0+2(26.8)+(41/8)) = 40/(0+53.6+5.125) = 40/58.725 =

20/(57+2(9.7)+0) = 20/(57+19.4) = 20/76.4 =

20/(21+2(11.8)+0) = 20/(21+23.6) = 20/44.6 =

You can even combine substeps and measure their combined efficiency:

40/(78+2(21.5)+0) = 40/(78+43) = 40/121 =

What do you think?

In essence I wanted to find a quantifiable way to measure the "efficiency" of any given method or substep. Efficiency in the sense of amount of the cube solved, relative to speed (in moves), memorization, and mental effort required. In other words, how much bang you get for your buck.

Now, this

*isn't*supposed to measure which methods can achieve the fastest solve, because the more mental effort put forth, the faster it can be solved, but at a certain point, the amount of bang you get for your buck is severely reduced.For example, God's algorithm. If it were humanly possible to do, it would be the fastest solving method, but even then, it would not be the most efficient. Why? Because it would require such mental capacity for either memorization or intuitive solving as to be mortally impossible, and therefore is perhaps the most inefficient method of solving the cube because you get very little benefit for the amount of work and resources applied.

On the other end of the spectrum, you have the Turn-Check-If-Solved method, where each step is literally to turn the cube randomly, and after each move, check to see if the cube is in the solved state. While this requires absolutely no mental capacity or skill, the number of moves required on average to achieve a solved state would be ridiculously high, therefore making this method extremely inefficient as well.

And so this formula isn't supposed to be used as a measurement of which method is the best or which method can solve the fastest, because obviously the fastest is God's Algorithm. It's about measuring the efficiency of effort required to achieve a solved state in as few moves as possible.

Here is the formula:

**2.5*Orientations + 2.5*Permutations**

----------------------------------------------------------------

Algorithms + 2(AvgMoves)+(ReplaceAlgs/IntuitiveCubies)----------------------------------------------------------------

Algorithms + 2(AvgMoves)+(ReplaceAlgs/IntuitiveCubies)

Orientations = Number of cubies correctly oriented by the method/substep

Permutations = Number of cubies correctly permuted by the method/substep

Algorithms = Number of memorized algorithms required (mirrors, etc. included)

AvgMoves = Number of moves on average executed before achieving the desired state

ReplaceAlgs = Number of algorithms that would be required to replace the intuitive step(s)

IntuitiveCubies = Number of cubies solved using intuitive methods

Now to explain the reasoning behind the organization of this formula.

First the numerator of the formula is a measurement of the percentage of the cube solved. Meaning that if a method solves all 20 cubies, then the numerator will be exactly 100. So, for any full method calculation, the numerator is 100.

Now the denominator. The greater the values in the denominator, the less efficient the method/substep is considered. The following values considered negative to efficiency are:

-Algorithms

-Avg Moves

-Intuitive ineffectiveness = (ReplaceAlgs/IntuitiveCubies)

**Algorithms**: Now obviously, the more algorithms required to achieve any desired state, the less efficient the method is. If one method solves 8 cubies with 10 algs, and another solves 8 cubies with 2 algs, then the second method is more efficient (ignoring other variables).**Avg Moves**: Another obvious one. The more moves required on average, the less efficient it is. It's multiplied by 2 because I needed to give the number of moves a good weight in terms of efficiency value (again, this is an inexact science).**Intuitive ineffectiveness**: This one isn't so obvious. There are many substeps that are intuitive, and I needed a way of measuring their effectiveness numerically. So I did the following: I assumed that, the more algorithms that would be required to replace an intuitive step, the more logic/brain-power the step required, therefore it has an increase in inefficient resource usage. But at the same time, the number of cubies that end up solved using this step is important. The more cubies that it solves, the more effective it is, therefore it reduces the ineffectiveness. (An orientation or a permutation is worth 0.5 cubies)That's the formula. Hopefully I've explained it in a way that makes sense. It could probably be better, but an inexact science is very difficult to "better".

Now there are a few things that need to be estimated, for example the number of algorithms that would replace solving the cross. I estimated about 6 for the purpose of having a numerical base. This gives it an intuitive ineffectiveness of (6/4). Compare that to F2L, which has an intuitive ineffectiveness of (41/8).

Another thing that would need to be estimated is the algorithms that would replace EOLine, or Block Building. In fact, it appears that the first steps of solving the cube in any method do not have any really solid way of measuring the number of algorithms that would replace intuitive execution. Again (again), it's an inexact science.

Now when it comes to measuring substeps, there will be a rather clear decline in efficiency the further into the solve the substep lies. This is because, the more cubies you have that are solved, the more limited you are in solving others efficiently in regards to the mental effort required and the number of moves executed. Therefore the substeps Cross and F2L will be much more efficient than OLL or PLL.

So as a few examples, I'm going to work out the efficiencies of the CFOP method and it's substeps.

**CFOP**100/(78+2(54.8)+(47/12)) = 100/(78+109.6+3.91667) = 100/191.51667 =

**0.52215****Cross**20/(0+2(6.5)+(6/4)) = 20/(0+13+1.5) = 20/14.5 =

**1.37931****F2L(intuitive)**40/(0+2(26.8)+(41/8)) = 40/(0+53.6+5.125) = 40/58.725 =

**0.68114****OLL**20/(57+2(9.7)+0) = 20/(57+19.4) = 20/76.4 =

**0.26178****PLL**20/(21+2(11.8)+0) = 20/(21+23.6) = 20/44.6 =

**0.44843**You can even combine substeps and measure their combined efficiency:

**OLL+PLL**40/(78+2(21.5)+0) = 40/(78+43) = 40/121 =

**0.33058**What do you think?

Last edited: Oct 20, 2014