# God's Number for CFOP and Roux

#### GenTheThief

##### Member
I don't know as much as most on this site, but I would have thought 'God's number' is, by definition method absent. It's simply the lowest number of moves to a solution. A computer generated GN solution would/could probably be full of all kinds of whacky stuff that wouldn't fit into 'method' type algs. I may be wrong??
You are entirely right - a true god's number is certainly method absent.

However, everyone already knows that God's number is 20 moves, so now the question falls to categorizing and finding a "god's number" while remaining within a method's restrictions - that is to say, the maximum number of moves necessary to solve a cube while still following the steps of a method. If you read the above messages in the thread, xyzzy quite nicely goes through and finds those numbers.

#### abunickabhi

##### Member
My guess is Roux it depends. (Yes, even if we restrict to face turn metric. I don't care much for STM either. Everything else in this post will use FTM. I'll explain why I think Roux is generally better later.)

"CFOP" and "Roux" are not methods so rigorously defined that you could ask two different people to code implementations of these methods, without further instructions, and end up with solvers that always produce the same solutions. I'm much more familiar with CFOP than with Roux, so I'll use that as an example first.

CFOP is just cross, F2L, OLL, then PLL, right? Right? Wrong.

Very often, you'll have multiple choices of cross colour that take the same number of moves. Which do you choose? Do you pick one arbitrarily, or do you pick one that has the best continuation?

That's a trick question; you don't pick cross colours for their continuations, you pick cross solutions. Do you generate all cross solutions, then pick the one that has the best first pair? What if e.g. white cross has a 5-move solution that leads into 3-move first pair, while yellow cross has a 4-move solution but 7-move first pair?

Moving on to F2L, do you always pick the best pair to solve? What about multislotting? Are algs that exploit free slots allowed? Are adjacent pairs prioritised over diagonal pairs, and if so, to what extent?

OLL and PLL are the mechanical parts involving no degree of freedom, and those aren't really up for dispute.

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You know where this is going already, but just for the sake of formality…

Roux is just FB, SB, CMLL, then LSE, right? Right? Wrong.

FB runs into the same CN-related issue as above.

SB and CMLL have no degree of freedom (assuming we're not using alt solutions for better continuations).

LSE as done by beginners is usually broken into three steps: EO ("4a"), UL UR ("4b"), L4E ("4c"). As done by experts, EO and UL UR are (partially) combined into a single step, EOLR. (EOLR usually involves setting up EO to the arrow case first, but there's also full EOLR which solves 4a and 4b directly.)

If we're to compare methods of similar alg counts, it would only be fair to include EOLR as part of Roux.

(edit As GodCubing pointed out, I misunderstood what EOLR is. This does not materially affect the numbers presented here; EOLR proper uses more algs, but also has slightly lower move count than the variant I described above.)

Also, since we're using FTM as the metric of choice, of course we'll also be using solutions for the individual steps that are optimal in FTM, not STM.

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Let's say we solve the steps individually, without regard for continuations.

CFOP is, in reality, a seven-step method:
1. Cross
2. First pair
3. Second pair
4. Third pair
5. Last pair
6. OLL
7. PLL (and AUF)

Roux is only a six-step method:
1. First block
2. Second block
3. CMLL
4. EOLR setup (either set up to arrow or solve 4a)
5. EOLR
6. 4c

Assuming we allow using free slots, the worst case for each F2L pair is at least 8 moves (corner twisted in place and edge flipped in place), except for the very last pair having a worst case of 9 moves (corner solved, edge flipped in place). That makes F2L take 8+8+8+8+9 = 41 moves. OLL's worst case is 12 moves (OLL 20), while PLL's worst case is 14 moves (V perm at any AUF). This gives a total of 41+12+14 = 67 moves.

I don't know the exact numbers for Roux, but my guesstimates are:
FB: 9
SB: 12
CMLL: 11 (known to be the worst case)
EOLR setup: 8
EOLR: 13
L4E: 8
Total: 61

There's enough slack that I could be slightly wrong in a few of these and Roux's worst case would still be better than CFOP's worst case.

I imagine that if you restrict to purely MU algs for LSE, or if LSE has to be done as 4a-4b-4c, Roux would lose by a significant margin, but for reasons I already stated above, these would be silly restrictions to impose.
A really good explanation.

I believe that god's number should be method independent. Even for a strongly defined method, it is not useful to calculate its upper bound of optimal solve.

#### Manxkiwi

##### Member
I still don't see how you can have a GN within a method. There are basically infinite scrambles, the lowest move count within a method will depend largely on luck, as in, what the scramble gives you. Followed by skill, knowledge and look ahead. I guess it's confusing to say 'GN', it should perhaps be phrased as: fewest moves within a method. Even then there will not be one golden number as it will always be scramble dependant.
Just my thoughts..

#### xyzzy

##### Member
I still don't see how you can have a GN within a method. There are basically infinite scrambles, the lowest move count within a method will depend largely on luck, as in, what the scramble gives you. Followed by skill, knowledge and look ahead. I guess it's confusing to say 'GN', it should perhaps be phrased as: fewest moves within a method. Even then there will not be one golden number as it will always be scramble dependant.
Just my thoughts..
By this reasoning, GN is still "scramble dependent" even if you don't restrict to a method.

I guess the weird thing most non-maths-trained people aren't accustomed to is that GN is defined with nested quantifiers:
GN is an integer $$n$$ such that
for every $$m<n$$,
there exists a scrambled state $$X$$
such that for every move sequence $$S$$ of length $$m$$,
$$X S$$ (i.e. applying the moves to the scramble) is not the solved state.

Three levels of nesting is just not something you see on a day to day basis unless you do mathematics.

If we're considering a method-specific GN, this changes to:
A method's GN is an integer $$n$$ such that
for every $$m<n$$,
there exists a scrambled state $$X$$
such that for every move sequence $$S$$ of length $$m$$,
either $$X S$$ is not the solved state or $$S$$ is not a solution that can result from the given method.

#### Manxkiwi

##### Member
No GN isn't scramble dependant. Gods number is the minimum number of moves to solve a cube from 'any' given state (of scramble).
Or do I have something wrong there?

#### Silky

##### Member
No GN isn't scramble dependant. Gods number is the minimum number of moves to solve a cube from 'any' given state (of scramble).
Or do I have something wrong there?
Maximum number of moves. So your worst case scenario is going to be a 20 move solve ( in HTM, 26 in QTM ). The most famous is the superflip

#### Manxkiwi

##### Member
Ah I see. I have been thinking of it the wrong way around! Thanks for the info.