Announcing: New 4x4x4 Brute Force Solver

[rokicki]

What you don't is precisely what you mentioned: execute hash_number modulo some_number = index, then visit database[index] and get your info. At database[index] you have multiple pieces of info, one of which is the FULL 32- bit or 64-bit number. So there is no 1-byte/entry version of a hash table.

For doing an IDA* search, where we just need an admissible heuristic, this approach actually works
*perfectly*; you ensure the single entry contains the minimum distance-to-solved from all values
that collide here, and your table then functions admirably. Further, if you use a fast hash function
(it doesn't have to be cryptographically secure), the elimination of collision detection, etc. cuts the
code down and the performance (with no inessential branching) can be excellent.

You can even use this idea with only one *bit* per entry that says a position is either at a distance
> d for some d, or "no information". I frequently use it with two bits per entry, giving me information
that the position is >d, >d+1, >d+2, and "no information" for a given d. Using 1.6 bits per entry also
works well, in which case you get >d, >d+1, and "no information".

Story time:

Great writeup, Lucas; thanks! That clarifies things tremendously. I think the bulk of this prose should
be on a wiki somewhere for others to profit from.

Everyone on this thread: should we follow the sage advice of Lucas and go with SiGN exclusively, or
should we continue to use whatever we are using? In particular, if we plan to exchange positions
and compare solves it would be nice if we could just copy and paste.

 

[unsolved]

Of course, I can't control what notation people use. But we should have a standard, and SiGN is as good as any. It's simple, and I believe it makes the most sense: r can only stand for one of 2R or Rw, and if we're going to have a standard we need to pick on...

As an outsider, what I am about to offer may be unpopular, but here is my input regarding notation.

The very first cube book that I ever saw in 1980-1981 featured upper case letters for the sides and + for clockwise and - for counterclockwise. Just my opinion, I like this best. How many times have you had to squint and see if a ' was or was not in the middle of a sequences of moves? Then you check over the moves over and over until you see what you missed, etc. But with a + and - after the moves, you get a nice delimeter and your eye does not lose track as easily.

When the 4x4x4 came out, lower case meant slice moves. No problem, I like that.

But 5x5x5 and larger presented problems, of course. Personally, I find E and M and S counterintuitive. I use notation such as {f/r} to indicate a middle slice turn from the front in the direction of the right face. In the opposite direction, it is {r/f}.

I have not given larger cubes any thought, or how to describe multiple slice turns.

Also, I use a right- handed rotational vector notation. R+ and L+ both turn in the same direction, so your L- is my L+. Ditto for top and bottom, your U = my T+ and your D' = my B+, since I use K for the back slice. B = bottom to me, T = top.

Why should r'l and l'r be opposite in direction when you turn them both the same way? In my notation it would be (lr)+ when both turn like r and (lr)- when both spin like r'.

[Ff]+ would be F f and [Tt]- would be U' u'. Slice and face turns use square brackets, pairs of slice turns use round parenthesis, and middle slices use curly braces.

 

[Christopher Mowla]

While most people use SiGN, some people like cmowla will still use old WCA notation. I'm not blaming them, but it makes me sad that as a community we still haven't fully settled on a single notation.

I'm set in my ways! No, I understand the significance of unity (that was the core idea of my single commutator project!), as I made an effort to make my derivation of the "Holy Grail" alg in SiGN notation. I did this because I assumed that in the future, we would all have agreed on SiGN by then. However, for 2 layer wide turns, I used a 2 coefficient in front of the lowercase letter.

Speaking of which,

In addition, R simply meant the outer slice, and r was the simplest multi-slice outer block turn (two slices) -- which was consistent with one common conventions.

It was easy to learn, consistent, and at least *something* we would hopefully all agree on.

Having r = Rw is logically flawed based on the structure of SiGN. r (in SiGN) = 1r = a 1 layer outer block turn. That is, r = 1r = R in SiGN. But in SiGN also wants to make capital letters mean a single slice...so here is a conflict in SiGN, and thus if you omit the "convention" that r = Rw, then SiGN will be consistent.

With the same flawed logic that r = Rw for the sake that r = (R M') on the 3x3x3, then why isn't x = (Rw l') on the 4x4x4 in SiGN? I of course know the answer, but how obvious you think that answer is, I feel it's just as obvious that you shouldn't say that since r = (R M') on the 3x3x3 (which is really equivalent to doing a single slice turn which obviously also has a cube rotation), it's equal to Rw on the 4x4x4.

Since SiGN was claimed to be "simple" and based on "logic", making the exception that 1r = Rw is illogical. Get rid of it, and you will have a good notation. I know it makes things more compact if you keep it, but should that one advantage make it logically inconsistent, and worse, make SiGN cause confusion?

Some day, people would forget r had ever had another meaning. (Slight dramatization, but that's basically how I felt.)

That's never going to happen. r, as I have said before, has no place in SiGN, so omit it (including give a syntax error in alg.cubing.net if someone types r,l,u,d,f,b without a 2 or larger number coefficient, that is, if you eventually make selectable options to use WCA and SiGN separately as is on alg.garron.us), and all of your problems are solved for all time. Until then, I will continue to use WCA whenever I can because there is no mistake about what moves I mean when I post algorithms because almost everyone knows WCA, but I still can't say as many people know SiGN.

In any case, alg.garron.us will start redirecting to alg.cubing.net in a few months (whenever I take a breath from working on the WRC and fix a few alg.cubing.net details first), and all "old WCA" algs will be translated to SiGN. That is, r will be redirected to 2R.

It's good to see that you will be making r become 2R, but I'm sorry to hear that you will be getting rid of alg.garron.us. Oh well, I'm sure for those who prefer alg.garron can always still use CubeTwister on their computers.

EDIT:
I will modify template Alg4 in the wiki to go to alg.cubing.net so that there are not so many redirects because of the 4x4x4 parity algorithm page, but this is of course after you redirect r to 2R. So maybe let me know by pm when this gets done (or you can do it whenever it gets done, if that makes things easier for you).

EDIT2:
And Lucas, what are you going to do about capital M,E,S at alg.cubing.net? It will be a mess unless you make a button like you did in alg.garron.us to use WCA or SiGN.

 

[cuBerBruce]

SiGN notation may be a good idea for a consistent notation across arbitrary size cubes, but for the 4x4x4, it is a bit clumsy, and less consistent with standard mathematical notation. Singmaster notation, in its "pure" form, is actually just a use of mathematical notation (albeit it adds the prime/apostrophe to represent inverses, which alternatively can be expressed as raising to a power of -1). Singmaster extended his notation to 4x4x4 in1982 with the lower case letters representing inner layer turns, and has a long-standing history of being the de facto standard notation. I think SiGN enthusiasts have somewhat of an uphill battle to convince everyone one to use this clumsier notation (as a 4x4x4 notation) in place of the long-standing Singmaster standard.

As far as I know the use of lower case letters for wide turns is a relatively recent addition for 3x3x3. Can anyone show that this convention was in use before 2000?

Using +/- has had some use in cubing, but is also not consistent with standard mathematical notation.

I suppose programs should give the user the choice of notation, because I think a lot of people still prefer the long-standing Singmaster convention as a 4x4x4 notation.

 

[rokicki]

I have no beef with anything you say, Bruce. But for our efforts here in this thread it would be great
if we can agree on a notation and use that consistently. If that notation uses "l" for an inner slice turn,
I'm okay with that, and that seems to be what people are doing.

Long term, big picture, I do plan a 4x4x4 contest, and I need to pick a notation for that, and I doubt
I can make everyone happy there. I personally prefer a notation that extends naturally to bigger cubes
and for this reason I prefer the SiGN notation.

It looks to me like the solver created by unsolved uses single-slice-turn metric (SSTM), and your two
solvers use outer-block-turn metric (OBTM) and block-turn metric (BTM). Is this accurate? In that
case I guess there is no direct comparison between the solvers. I am doing my experiments with OBTM
but plan to also support BTM; I find it hard to be interested in SSTM. Is there a motivation for using
SSTM that I'm not seeing?

 

[cuBerBruce]

I have no beef with anything you say, Bruce. But for our efforts here in this thread it would be great
if we can agree on a notation and use that consistently. If that notation uses "l" for an inner slice turn,
I'm okay with that, and that seems to be what people are doing.

Long term, big picture, I do plan a 4x4x4 contest, and I need to pick a notation for that, and I doubt
I can make everyone happy there. I personally prefer a notation that extends naturally to bigger cubes
and for this reason I prefer the SiGN notation.

It looks to me like the solver created by unsolved uses single-slice-turn metric (SSTM), and your two
solvers use outer-block-turn metric (OBTM) and block-turn metric (BTM). Is this accurate? In that
case I guess there is no direct comparison between the solvers. I am doing my experiments with OBTM
but plan to also support BTM; I find it hard to be interested in SSTM. Is there a motivation for using
SSTM that I'm not seeing?

If you go to the first post of my thread, you'll find 3 versions, one for each of these 3 metrics.

In SSTM, each of the standard moves affect a minimal number of pieces; and the basic moves correspond directly with the Singmaster letters and their powers. If you have multiple turns in a row on the same axis, it's more straightforward to see what the effect on a single piece or single layer is for those turns. These properties are perhaps of more interest to mathematicians than speedcubers. Like on the 15 puzzle, mathematicians seem to be mostly interested in single-tile metric, while the way normal people manipulate the physical puzzle corresponds more with multi-tile metric.

 

[cmhardw]

I think Chris's arguments that the change r /= Rw should be added to the rules for SiGN very convincing. In particular the following is very convincing to me:

Having r = Rw is logically flawed based on the structure of SiGN. r (in SiGN) = 1r = a 1 layer outer block turn. That is, r = 1r = R in SiGN. But in SiGN also wants to make capital letters mean a single slice...so here is a conflict in SiGN, and thus if you omit the "convention" that r = Rw, then SiGN will be consistent.

I would try very hard to use SiGN exclusively if there is consensus to use it as a community standard, but I would prefer that r=Rw be done away with if this happens. Those are my thoughts.

 

[bobthegiraffemonkey]

thoughts on r=(1)r = R

Since lowercase letters are multiple layer turns, why would it make sense to think of the default as a single layer turn? 'r' is a shortcut for '2r', the simplest multiple layer turn. I realise it may be confusing since it conflicts with another notation, but having r=1r=R would be even worse IMO since 1r contradicts itself (it can't be both a single layer turn and a double layer turn).

 

[Lucas Garron]

SiGN notation may be a good idea for a consistent notation across arbitrary size cubes, but for the 4x4x4, it is a bit clumsy, and less consistent with standard mathematical notation. Singmaster notation, in its "pure" form, is actually just a use of mathematical notation (albeit it adds the prime/apostrophe to represent inverses, which alternatively can be expressed as raising to a power of -1). Singmaster extended his notation to 4x4x4 in1982 with the lower case letters representing inner layer turns, and has a long-standing history of being the de facto standard notation. I think SiGN enthusiasts have somewhat of an uphill battle to convince everyone one to use this clumsier notation (as a 4x4x4 notation) in place of the long-standing Singmaster standard.

As far as I know the use of lower case letters for wide turns is a relatively recent addition for 3x3x3. Can anyone show that this convention was in use before 2000?

Using +/- has had some use in cubing, but is also not consistent with standard mathematical notation.

I'm somewhat sympathetic to the historical argument, but what you call "relatively recent" is practically all of modern speedcubing.
I also don't think it's so much of an uphill battle. If it weren't for the 2008 Regs, I'm confident we would already be there.

While I care about the math, too, I don't think we should be too beholden to all arbitrary choices.
For example, alg composition is the reverse order of regular group theory.

I suppose programs should give the user the choice of notation, because I think a lot of people still prefer the long-standing Singmaster convention as a 4x4x4 notation.

Programs should not give the user the choice of notation if it is contradictory (but I'm totally fine with allowing non-conflicting aliases like r=Rw, as on alg.cubing.net).

I would rather use Rw and r=2R than have two contradicting definitions for an indefinite period of time. Perhaps that's a discussion for elsewhere.

 

[qqwref]

How many times have you had to squint and see if a ' was or was not in the middle of a sequences of moves?

This may be a font issue. For the fonts I use, I see 's very clearly. On your computer, do you also have trouble visually seeing the difference between its and it's?

I use notation such as {f/r} to indicate a middle slice turn from the front in the direction of the right face. In the opposite direction, it is {r/f}.

Ewwwwww. For methods/algs that use a lot of these slices, this would get annoying extremely quickly. (For instance: here's a popular Z perm: {f,u}2 U {f,u}2 U {f,u} U2 {f,u}2 U2 {f,u} U2)

Why should r'l and l'r be opposite in direction when you turn them both the same way?

Because R and L are on opposite sides of the cube With SiGN, sure, you could write it 2R' 2L, but you could also write it 2R' 3R' (or 2-3r', or M) and it would be very clear that they go the same direction.

Having r = Rw is logically flawed based on the structure of SiGN. r (in SiGN) = 1r = a 1 layer outer block turn.

You only think it's flawed because your mental definition is wrong. A lack of number does not mean "1 goes here", it means "the move of this type that is furthest from the center of the puzzle". So r = 2r because 1r wouldn't be lowercase, and M = all but the outermost two layers. And in fact, it's perfectly valid to write 2r - but nobody would, because it wastes time.

With the same flawed logic that r = Rw for the sake that r = (R M') on the 3x3x3, then why isn't x = (Rw l') on the 4x4x4 in SiGN? I of course know the answer, but how obvious you think that answer is, I feel it's just as obvious that you shouldn't say that since r = (R M') on the 3x3x3 (which is really equivalent to doing a single slice turn which obviously also has a cube rotation), it's equal to Rw on the 4x4x4.

Huh? x is a rotation of the whole cube, which will never be equivalent to (r 2L'). The idea that "lowercase letter with no number = two layer turn" is in fact a generalization of the 3x3x3 notation you mentioned, whereas the Singmaster notation of a lowercase letter being a slice is not. Again, the notation only doesn't make sense to you because you have the wrong idea of what things mean.

SiGN notation may be a good idea for a consistent notation across arbitrary size cubes, but for the 4x4x4, it is a bit clumsy, and less consistent with standard mathematical notation.

Actually I find SiGN a lot less clumsy for algs or reconstructions which use very few slice moves. And compared to the WCA 4x4x4/5x5x5 notation, which really cannot be extended nicely to bigger cubes, SiGN makes writing algs on larger cubes very easy. WCA 6x6x6+ notation is also pretty clumsy and really makes no accommodation for people who need slice turns or inner block turns, such as anyone working on pretty pattern algs.

And IMO there is no "standard mathematical notation" for what letters go to what turns - Singmaster chose one convention, but math is not known for forcing people to use particular letters. Math does define what the suffixes should be, but both SiGN and WCA notation use an easy-to-type version of the standard (' = ^-1, 2 = ^2).

As far as I know the use of lower case letters for wide turns is a relatively recent addition for 3x3x3. Can anyone show that this convention was in use before 2000?

Maybe this is my young-person bias showing, but I don't consider any non-theoretical writing about cubes before 2003 to be important in deciding what we do now. And besides, this is a new sport, and we can make new rules - almost everything we do now was decided, by the community, since the founding of the WCA.


Anyway, unrelated to all this stuff: the optimal algs people have been finding for ELL cases (such as u2 b' U' b' l D2 f2 D2 l' b U b u2) are extremely interesting, and generally unknown. Keep it up!

 

[cmhardw]

You only think it's flawed because your mental definition is wrong. A lack of number does not mean "1 goes here", it means "the move of this type that is furthest from the center of the puzzle". So r = 2r because 1r wouldn't be lowercase, and M = all but the outermost two layers. And in fact, it's perfectly valid to write 2r - but nobody would, because it wastes time.

That makes sense to me. Lowercase letters denote multiple layer turns, the smallest of which is a turning of two layers. I retract my earlier statement. SiGN notation does appear to be internally consistent.

 

[rokicki]

Here are some distance-14 (OBTM) optimal solves, I believe.

(There could be a bug in my program).

These are for the ELL positions posted earlier.

First line is the position, second is the solution.

Sorry for all the three-wide turns; I haven't yet added code
to "untwist" solutions from my optimal solver (which always
holds the DBL corner fixed).

If there is interest I will post some more. Similarly, if anyone
can confirm these that would be great too.

F2 U' 2L' 2R' U F' L U' F 2L 2R F' U L' F'    
      Uw2 Fw 3Uw2 3Rw' 3Uw' 3Fw Uw2 3Uw2 3Fw' 3Uw 3Rw 3Uw2 Fw' Uw2
2R2 U F2 U B' R B 2R2 B' R' B U' F2 U' 2R2    
      R U' Fw2 3Fw U2 Fw2 R Fw2 U2 Fw2 R 3Fw' U R'
F' R' 2D R U R' 2D' 2U' R U' R' 2U R F     
      F2 Rw2 U R2 U' R2 3Rw2 U R2 U' R2 Rw2 3Rw2 F2
R2 2R2 U2 R2 U' 2R U2 2L 2R U' R2 U 2L' 2R' U2 2R' U' R2 2R2
      F2 Rw2 U' F2 U F2 3Fw2 U' F2 U F2 3Fw2 Rw2 F2

 

[Christopher Mowla]

If there is interest I will post some more. Similarly, if anyone
can confirm these that would be great too.

They are all correct and amazing!
Uw2 Fw 3Uw2 3Rw' 3Uw' 3Fw Uw2 3Uw2 3Fw' 3Uw 3Rw 3Uw2 Fw' Uw2 =
[Uw2 Fw D2 R' B' L: u2](13 btm, 17 ltm, 14 ftm, 20 qtm)

R U' Fw2 3Fw U2 Fw2 R Fw2 U2 Fw2 R 3Fw' U R' =
R U' B Fw2 L2 Fw2 U Fw2 L2 Fw2 U B' U R' (14 btm, 18 ltm, 14 ftm, 20 qtm)

F2 Rw2 U R2 U' R2 3Rw2 U R2 U' R2 Rw2 3Rw2 F2 =
F2 Rw2 U R2 U' M2 U R2 U' R2 l2 F2 (12 btm, 14 ltm, 14 ftm, 24 qtm)

F2 Rw2 U' F2 U F2 3Fw2 U' F2 U F2 3Fw2 Rw2 F2 =
F2 Rw2 U' F2 U S2 U' F2 U S2 Rw2 F2 (12 btm, 16 ltm, 14 ftm, 24 qtm)

Yes there is a position I would like you to find the optimal OBTM (ftm) solution for.

I'm not sure if 18 OBTM is out of reach for your solver, but 18 OBTM is the briefest I have found (and currently exists) for this case (which I have studied extensively).
(Rw2 Uw U S Lw2 u2) r (u2 Lw2 S' U' Uw' Rw2)
(Rw2 Uw' U' S Lw2 u2) r (u2 Lw2 S' U Uw Rw2)

 

[unsolved]

They are all correct and amazing!
Uw2 Fw 3Uw2 3Rw' 3Uw' 3Fw Uw2 3Uw2 3Fw' 3Uw 3Rw 3Uw2 Fw' Uw2 =
[Uw2 Fw D2 R' B' L: u2](13 btm, 17 ltm, 14 ftm, 20 qtm)

R U' Fw2 3Fw U2 Fw2 R Fw2 U2 Fw2 R 3Fw' U R' =
R U' B Fw2 L2 Fw2 U Fw2 L2 Fw2 U B' U R' (14 btm, 18 ltm, 14 ftm, 20 qtm)

F2 Rw2 U R2 U' R2 3Rw2 U R2 U' R2 Rw2 3Rw2 F2 =
F2 Rw2 U R2 U' M2 U R2 U' R2 l2 F2 (12 btm, 14 ltm, 14 ftm, 24 qtm)

F2 Rw2 U' F2 U F2 3Fw2 U' F2 U F2 3Fw2 Rw2 F2 =
F2 Rw2 U' F2 U S2 U' F2 U S2 Rw2 F2 (12 btm, 16 ltm, 14 ftm, 24 qtm)

Yes there is a position I would like you to find the optimal OBTM (ftm) solution for.

I'm not sure if 18 OBTM is out of reach for your solver, but 18 OBTM is the briefest I have found (and currently exists) for this case (which I have studied extensively).
(Rw2 Uw U S Lw2 u2) r (u2 Lw2 S' U' Uw' Rw2)
(Rw2 Uw' U' S Lw2 u2) r (u2 Lw2 S' U Uw Rw2)

I forgot to post your solved scrambles.

Solutions To Scramble = F2 U' l' r' U F' L U' F l r F' U L' F' 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|####|OOOO|   |XXXX|####|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|^^^^|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|^^^^|####|####|OOOO|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|XXXX|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|####|####|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  d2 f  D  F2 U' F2 r2 F2 [inside 5-TFS] ---> U  F2 D' f' d2  @ 0128707314012 nodes with 1010604690894 5-TFS probes   



Solutions To Scramble = r2 U' B2 U' F L F' r2 F L' F' U B2 U r2 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|####|####|OOOO|   |XXXX|XXXX|&&&&|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|OOOO|OOOO|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|XXXX|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  u2 b' U' b' l  D2 f2 D2 [inside 5-TFS] ---> l' b  U  b  u2  @ 0090736443045 nodes with 0750247004859 5-TFS probes   
Solution [002] =  u2 b' U' b' l' D2 f2 D2 [inside 5-TFS] ---> l  b  U  b  u2  @ 0090736464321 nodes with 0750247150267 5-TFS probes   
Solution [003] =  d' r2 d  B2 D2 F' R2 D2 [inside 5-TFS] ---> L2 B  l2 r' u'  @ 0110414034303 nodes with 0885175449290 5-TFS probes   
Solution [004] =  d2 r' U' B2 D' r' f2 r  [inside 5-TFS] ---> D  B2 U  r  d2  @ 0123089215000 nodes with 0972086306647 5-TFS probes   
Solution [005] =  d2 b' U' b' l  D2 f2 D2 [inside 5-TFS] ---> l' b  U  b  d2  @ 0130611366693 nodes with 1023666280309 5-TFS probes   
Solution [006] =  d2 b' U' b' l' D2 f2 D2 [inside 5-TFS] ---> l  b  U  b  d2  @ 0130611387969 nodes with 1023666424889 5-TFS probes   
Solution [007] =  r2 b  U  r  D  l' b2 l  [inside 5-TFS] ---> D' r' U' b' r2  @ 0259062671872 nodes with 1904439161555 5-TFS probes  



Solutions To Scramble = l2 U' B2 U' F L F' l2 F L' F' U B2 U l2 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|####|####|OOOO|   |XXXX|&&&&|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|OOOO|OOOO|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|XXXX|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  u2 f  U' f  r  D2 b2 D2 [inside 5-TFS] ---> r' f' U  f' u2  @ 0088691387205 nodes with 0736229771312 5-TFS probes   
Solution [002] =  u2 f  U' f  r' D2 b2 D2 [inside 5-TFS] ---> r  f' U  f' u2  @ 0088691410749 nodes with 0736229934429 5-TFS probes   
Solution [003] =  d2 l  U' B2 D' l  b2 l' [inside 5-TFS] ---> D  B2 U  l' d2  @ 0124135640777 nodes with 0979264921448 5-TFS probes   
Solution [004] =  d2 f  U' f  r  D2 b2 D2 [inside 5-TFS] ---> r' f' U  f' d2  @ 0128566310853 nodes with 1009645389864 5-TFS probes   
Solution [005] =  d2 f  U' f  r' D2 b2 D2 [inside 5-TFS] ---> r  f' U  f' d2  @ 0128566334397 nodes with 1009645555635 5-TFS probes   



Solutions To Scramble = r2 U l r U' F R' U F' l' r' F U' R F' r2 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|^^^^|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|####|####|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|XXXX|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|&&&&|####|####|XXXX|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|####|&&&&|&&&&|   |^^^^|^^^^|####|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  u2 f' D  B2 U' B2 r2 B2 [inside 5-TFS] ---> U  B2 D' f  u2  @ 0089361202383 nodes with 0740808355065 5-TFS probes   
Solution [002] =  d2 f' D  B2 U' B2 r2 B2 [inside 5-TFS] ---> U  B2 D' f  d2  @ 0129236126031 nodes with 1014223718305 5-TFS probes   
Solution [003] =  l2 f' U  R2 U' L2 b2 L2 [inside 5-TFS] ---> U  R2 U' f  l2  @ 0297890262609 nodes with 2170654243169 5-TFS probes   
Solution [004] =  l2 f' U  R2 D' B2 r2 B2 [inside 5-TFS] ---> D  R2 U' f  l2  @ 0297890471079 nodes with 2170655663948 5-TFS probes   
Solution [005] =  l2 f' U' b2 D' B2 r2 B2 [inside 5-TFS] ---> D  b2 U  f  l2  @ 0297922559391 nodes with 2170875725249 5-TFS probes   
Solution [006] =  l2 f' L2 U  R2 U' b2 U  [inside 5-TFS] ---> R2 U' L2 f  l2  @ 0298295606800 nodes with 2173433925721 5-TFS probes   
Solution [007] =  l2 f' L2 D  B2 U' l2 U  [inside 5-TFS] ---> B2 D' L2 f  l2  @ 0298301859982 nodes with 2173476807387 5-TFS probes   



Solutions To Scramble = R2 r' F' L R F r' F' R' F r2 F' L' F R2 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|OOOO|####|   |OOOO|OOOO|####|OOOO|   |XXXX|XXXX|####|XXXX|
---------------------   ---------------------   ---------------------
|XXXX|####|####|^^^^|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|&&&&|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|####|&&&&|&&&&|   |^^^^|####|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  r  F2 r  U  b' L  B' L' [inside 5-TFS] ---> l  B' U2 f  l2  @ 0227762813531 nodes with 1689809785046 5-TFS probes   
Solution [002] =  r  B' R  L' B  r  B' L  [inside 5-TFS] ---> B  r2 B' R' B   @ 0231706464436 nodes with 1716850086155 5-TFS probes   
Solution [003] =  r' u2 r  u2 b' l  U  F2 [inside 5-TFS] ---> F  r  B  L2 R2  @ 0235278692736 nodes with 1741344225356 5-TFS probes   
Solution [004] =  r' F' R' L  F  r' F' R  [inside 5-TFS] ---> F  r2 F' L' F   @ 0241864066531 nodes with 1786499735871 5-TFS probes   
Solution [005] =  L  F' L' b2 L  B  L' b' [inside 5-TFS] ---> L  B' F  L' b'  @ 0308623479380 nodes with 2244260809189 5-TFS probes   



Solutions To Scramble = r2 U F2 U B' R B r2 B' R' B U' F2 U' r2 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|&&&&|OOOO|   |XXXX|####|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|XXXX|####|####|^^^^|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|OOOO|&&&&|   |^^^^|^^^^|####|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  R2 f' D  F2 f' u2 l2 u2 [inside 5-TFS] ---> f  F2 D' f  R2  @ 0214056276801 nodes with 1595827407974 5-TFS probes   
Solution [002] =  R2 f' D  F2 f' d2 l2 d2 [inside 5-TFS] ---> f  F2 D' f  R2  @ 0214056279072 nodes with 1595827423505 5-TFS probes   
Solution [003] =  R2 f' D  b' B2 u2 r2 u2 [inside 5-TFS] ---> B2 b  D' f  R2  @ 0214059607653 nodes with 1595850227820 5-TFS probes   
Solution [004] =  R2 f' D  b' B2 d2 r2 d2 [inside 5-TFS] ---> B2 b  D' f  R2  @ 0214059609924 nodes with 1595850243100 5-TFS probes   
Solution [005] =  R2 f' D  B2 d2 b' r2 b  [inside 5-TFS] ---> d2 B2 D' f  R2  @ 0214061510578 nodes with 1595863321622 5-TFS probes   
Solution [006] =  R2 f' D' F2 u2 b' l2 b  [inside 5-TFS] ---> u2 F2 D  f  R2  @ 0214071048247 nodes with 1595928696941 5-TFS probes   
Solution [007] =  R2 f' D' F2 b' u2 l2 u2 [inside 5-TFS] ---> b  F2 D  f  R2  @ 0214071539037 nodes with 1595932069181 5-TFS probes   
Solution [008] =  R2 f' D' F2 b' d2 l2 d2 [inside 5-TFS] ---> b  F2 D  f  R2  @ 0214071541308 nodes with 1595932084901 5-TFS probes   
Solution [009] =  R2 f' D' f' B2 u2 r2 u2 [inside 5-TFS] ---> B2 f  D  f  R2  @ 0214072943817 nodes with 1595941686238 5-TFS probes   
Solution [010] =  R2 f' D' f' B2 d2 r2 d2 [inside 5-TFS] ---> B2 f  D  f  R2  @ 0214072946088 nodes with 1595941701785 5-TFS probes   
Solution [011] =  R2 f' D' B2 u2 f' r2 f  [inside 5-TFS] ---> u2 B2 D  f  R2  @ 0214076638621 nodes with 1595967056929 5-TFS probes   
Solution [012] =  F  L  f2 D' B  r2 d2 r2 [inside 5-TFS] ---> B' D  f2 L' F'  @ 0348708026223 nodes with 2519103241100 5-TFS probes   
Solution [013] =  F  L  f2 D' B  l2 d2 l2 [inside 5-TFS] ---> B' D  f2 L' F'  @ 0348708028575 nodes with 2519103256699 5-TFS probes   



Solutions To Scramble = L U F L' U' L U L F' L2 U L U L' U' L U' L' 

TOP                     FRONT                   RIGHT
---------------------   ---------------------   ---------------------
|####|####|####|####|   |^^^^|&&&&|&&&&|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------
|####|####|####|####|   |OOOO|OOOO|OOOO|OOOO|   |XXXX|XXXX|XXXX|XXXX|
---------------------   ---------------------   ---------------------


BOTTOM                  BACK                    LEFT
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|OOOO|OOOO|^^^^|   |OOOO|^^^^|^^^^|&&&&|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------
|~~~~|~~~~|~~~~|~~~~|   |&&&&|&&&&|&&&&|&&&&|   |^^^^|^^^^|^^^^|^^^^|
---------------------   ---------------------   ---------------------



Solution [001] =  U  B2 L2 F  B2 U' F  R2 B' [inside 5-TFS] ---> D  F' R2 B  F'  @ 0001119126581 nodes with 0017853193809 5-TFS probes   
Solution [002] =  U' R  L' B2 R  D' L  B2 R' [inside 5-TFS] ---> U  L2 R' F2 L2  @ 0001371388154 nodes with 0019582866730 5-TFS probes   
Solution [003] =  U' F2 L2 F2 B' U  B' R2 F  [inside 5-TFS] ---> D' B  R2 B  F'  @ 0002020120411 nodes with 0024031075167 5-TFS probes   
Solution [004] =  U2 F  R' B  U2 F' L  F' B2 [inside 5-TFS] ---> D2 B2 L' B' F   @ 0002951816961 nodes with 0030420382639 5-TFS probes   
Solution [005] =  U2 F  B' R  F2 D2 F2 B  R' [inside 5-TFS] ---> B  U2 F' L  B'  @ 0002982652928 nodes with 0030631752492 5-TFS probes   
Solution [006] =  U2 F  B' R' B2 D2 F' B2 R  [inside 5-TFS] ---> F' U2 B  L' F   @ 0002983217161 nodes with 0030635615750 5-TFS probes   
Solution [007] =  U2 B' R  F' U2 B  L' F2 B  [inside 5-TFS] ---> D2 F2 L  B' F   @ 0003265166068 nodes with 0032568908988 5-TFS probes   
Solution [008] =  R2 F  R  B' R' F' R' D2 L' [inside 5-TFS] ---> F' L  D2 R' B   @ 0009056298908 nodes with 0072277389472 5-TFS probes   
Solution [009] =  R2 F  R' D2 L  B' L' D2 R' [inside 5-TFS] ---> B' R' F' R  B   @ 0009059132519 nodes with 0072296853649 5-TFS probes   
Solution [010] =  R2 B' R  D2 L' F  L  D2 R  [inside 5-TFS] ---> F  R  B  R' F'  @ 0009373437574 nodes with 0074452290522 5-TFS probes   
Solution [011] =  R2 B' R' F  R  B  R  D2 L  [inside 5-TFS] ---> B  L' D2 R  F'  @ 0009381341449 nodes with 0074506439427 5-TFS probes   
Solution [012] =  L2 U2 D  R' L  F2 R  D' L  [inside 5-TFS] ---> B2 R' U  R  B2  @ 0011473651735 nodes with 0088853541481 5-TFS probes   
Solution [013] =  L2 U2 D' R  L' B2 R' D  L' [inside 5-TFS] ---> F2 R  U' R' F2  @ 0011478354296 nodes with 0088885796272 5-TFS probes   
Solution [014] =  L2 D  R  U2 R' L  F2 D' L  [inside 5-TFS] ---> B2 R' U  R  B2  @ 0011558045923 nodes with 0089432096854 5-TFS probes   
Solution [015] =  L2 D' R' U2 R  L' B2 D  L' [inside 5-TFS] ---> F2 R  U' R' F2  @ 0011632890326 nodes with 0089945242796 5-TFS probes   
Solution [016] =  L2 F2 R  L2 U' R  B2 L' D  [inside 5-TFS] ---> R' B2 L  R' U   @ 0011968390258 nodes with 0092245688140 5-TFS probes   
Solution [017] =  L2 B2 R' L2 U  R' F2 L  D' [inside 5-TFS] ---> R  F2 L' R  U'  @ 0012195816719 nodes with 0093805095305 5-TFS probes   
Solution [018] =  F  R  B' R' F' R' D2 L' F' [inside 5-TFS] ---> L  D2 R' B  R2  @ 0012755309015 nodes with 0097641491153 5-TFS probes   
Solution [019] =  F  R' D2 L  B' L' D2 R' B' [inside 5-TFS] ---> R' F' R  B  R2  @ 0012793133387 nodes with 0097900948987 5-TFS probes   
Solution [020] =  F  R' L2 D2 L  B' L' D2 R2 [inside 5-TFS] ---> F' R' F  L2 F'  @ 0012809096856 nodes with 0098010381424 5-TFS probes   
Solution [021] =  F  L2 F' R  F  R2 D2 L  B  [inside 5-TFS] ---> L' D2 L2 R  F'  @ 0013110831019 nodes with 0100079073946 5-TFS probes   
Solution [022] =  F  B' R2 F  D' B  R2 F' U  [inside 5-TFS] ---> B2 F' L2 B2 U'  @ 0013255306225 nodes with 0101069575163 5-TFS probes   
Solution [023] =  F  B' R2 B' D  F' R2 B  U' [inside 5-TFS] ---> B  F2 L2 F2 U   @ 0013257071300 nodes with 0101081687503 5-TFS probes   
Solution [024] =  F' U  D2 R2 D' F  D  R2 U2 [inside 5-TFS] ---> B  U  B' D2 F   @ 0013353083421 nodes with 0101740007997 5-TFS probes   
Solution [025] =  F' D2 B  U' B' U2 R2 D' F' [inside 5-TFS] ---> D  R2 D2 U' F   @ 0013784940047 nodes with 0104701441777 5-TFS probes   
Solution [026] =  F' L  B' U2 F  R' F  B2 D2 [inside 5-TFS] ---> B2 R  B  F' U2  @ 0014111006466 nodes with 0106937420192 5-TFS probes   
Solution [027] =  F' B  L  B2 D2 F  B2 L' F  [inside 5-TFS] ---> U2 B' R  F' U2  @ 0014315314555 nodes with 0108338340723 5-TFS probes   
Solution [028] =  F' B  L' F2 D2 F2 B' L  B' [inside 5-TFS] ---> U2 F  R' B  U2  @ 0014319216425 nodes with 0108365054083 5-TFS probes   
Solution [029] =  F2 R  U  R' F2 L  D' R  B2 [inside 5-TFS] ---> L  R' D  U2 L2  @ 0014922710151 nodes with 0112502821614 5-TFS probes   
Solution [030] =  F2 R  U  R' F2 L  D' B2 R' [inside 5-TFS] ---> L  U2 R  D  L2  @ 0014922710294 nodes with 0112502822550 5-TFS probes   
Solution [031] =  F2 R  U  R' F2 L  D' B2 L  [inside 5-TFS] ---> R' U2 R  D  L2  @ 0014922710296 nodes with 0112502822551 5-TFS probes   
Solution [032] =  B  U' D2 R2 D  B' D' R2 U2 [inside 5-TFS] ---> F' U' F  D2 B'  @ 0015679560453 nodes with 0117692608142 5-TFS probes   
Solution [033] =  B  D2 F' U  F  U2 R2 D  B  [inside 5-TFS] ---> D' R2 D2 U  B'  @ 0016014245689 nodes with 0119987577690 5-TFS probes   
Solution [034] =  B  L' F  U2 B' R  F2 B' D2 [inside 5-TFS] ---> F2 R' B  F' U2  @ 0016398204444 nodes with 0122620173492 5-TFS probes   
Solution [035] =  B' R  D2 L' F  L  D2 R  F  [inside 5-TFS] ---> R  B  R' F' R2  @ 0016988624704 nodes with 0126668356753 5-TFS probes   
Solution [036] =  B' R  L2 D2 L' F  L  D2 R2 [inside 5-TFS] ---> B  R  B' L2 B   @ 0017004358800 nodes with 0126776299331 5-TFS probes   
Solution [037] =  B' R' F  R  B  R  D2 L  B  [inside 5-TFS] ---> L' D2 R  F' R2  @ 0017094129355 nodes with 0127392092332 5-TFS probes   
Solution [038] =  B' L2 B  R' B' R2 D2 L' F' [inside 5-TFS] ---> L  D2 L2 R' B   @ 0017403264353 nodes with 0129512031676 5-TFS probes   
Solution [039] =  B2 R' U' R  B2 L' D  R' F2 [inside 5-TFS] ---> L' R  D' U2 L2  @ 0017964864801 nodes with 0133362816543 5-TFS probes   
Solution [040] =  B2 R' U' R  B2 L' D  F2 R  [inside 5-TFS] ---> L' U2 R' D' L2  @ 0017964864892 nodes with 0133362817287 5-TFS probes   
Solution [041] =  B2 R' U' R  B2 L' D  F2 L' [inside 5-TFS] ---> R  U2 R' D' L2  @ 0017964864896 nodes with 0133362817359 5-TFS probes

That makes sense to me. Lowercase letters denote multiple layer turns, the smallest of which is a turning of two layers. I retract my earlier statement. SiGN notation does appear to be internally consistent.

A Move Notation Proposal: Lodino = Logigcal Direction Notation

Background: In reviewing the various notations used by cubers,
and offering a fresh perspective as an outsider (I am no speed
cuber and can't even solve a 5x5x5 without constantly referencing
cmowla's numerous PDFs on the subject), I decided to start with a
tabula rasa.

A summary first for those who can't read this in its entirety
without falling asleep.

Goals for a hypothetical new system:

1. A notation system that reads in such a way that the cubies
to which a formula applies is obvious, with an intuitive
direction indicator.

2. The system should be telescoping and inclusive of all
cubes from 2x2x2 through NxNxN (with N currently at 11 max)
with no need to add terms as the size of the cube grows.

3. No dependence on the knowledge of mathematical notation
or symbology to ease the burden on the newly initiated.

4. System should resolve the present ambiguity concerning the
ambiguous rotation notation governing the middle slices for
all NxNxN cubes with odd N >= 5.

5. The system should be neither cumbersome for the human
to learn, nor difficult for the programmer to implement.

With these goals in mind, the system below accomplishes all
of the above.


========================================================

Logical Direction Notation (Lodino) has the following
good features.

1. Easy to understand direction notation. It eliminates
the arbitrary prime mark (') assignment altogether,
and the unitiated can understand which way to turn the
cube at once. Older systems may even have a right-handed
bias in that clockwise turns (no prime notation) are more
easily performed with a right hand on the R and U faces
than a left hand on the L and D faces.

2. Rotation notation is also "global" and not "relative."
There is a rotation indicator for every direction, not
merely the absence of one for the arbitrary unprimed move.
In this way, lodino supports positive reinforcement with
a standard number of descriptors per move of the same
cube size.

3. Moves are of the form: cubie_subset.direction

4. Lodino corrects the defect where larger cubes have
some move designations with no bearing or resemblance
to smaller cubes. All moves on larger cubes look like
those already seen before on smaller cubes.

5. Lodino corrects larger cube notations that complicate
situations where:

A. Multiple slices with or without the center slice
are involved in the move.
B. Slice turns + face turns are required for a move.
C. Moves involve only the center slice, which, by definition,
cannot have a rotation indicator that is unambiguous.

6. Lodino smooths out the 4x4x4 move description defect
where 2 slices in motion being turned at once have
opposing direction indicators. For example, ud' moves
the up and down slices in the same direction, yet the
move notation is inconsistent and not natural.

========================================================

Sample of Moves

========================================================

2X2X2 and larger

U.L = turn the top towards the left = U presently
U.R = turn the top towards the right = U' presently
U.2 = turn the top twice

F.R = F
F.L = F'
F.2 = F2

R.B = R
R.F = R'
R.2 = R2

D.R = D
D.L = D'
D.2 = D2

B.L = B
B.R = B'
B.2 = B2

L.F = L
L.B = L'
L.2 = L2

========================================================

4x4x4 single slice turns

Only time to use lower case for the direction is when
all moves are slices.

u.l = turn the top slice towards the left = u presently
u.r = turn the top slice towards the right = u' presently
u.2 = turn the top slice twice

f.r = f
f.l = f'
f.2 = f2

r.b = r
r.f = r'
r.2 = r2

d.r = d
d.l = d'
d.2 = d2

b.l = b
b.r = b'
b.2 = b2

l.f = l
l.b = l'
l.2 = l2

4x4x4 double slice turns

ud.l = ud'
ud.r = u'd

fb.r = fb'
fb.l = f'b

rl.b = rl'
rl.f = r'l

4x4x4 face and slice turns

Uu.L = U u
Uu.R = U' u'
Uu.2 = U2 u2

Ff.R = F f
Ff.L = F' f'

Rr.B = R r
Rr.F = R' r'

Dd.R = D d
Dd.L = D' d'

Bb.L = B b
Bb.R = B' b'

Ll.F = L l
Ll.B = L' l'

========================================================

5x5x5 middle slice turns

Enclose the moves in round parenthesis to demarcate a middle edge.

(uf) = turn the middle slice towards the front face from the top face.
When stand alone, it is both a cube selector and a direction selector.

(fu) = the opposite direction of (uf). Move from the front towards
the top

(uf).2 = move the slice twice.
(fu).2 = same as (uf).2


l.(fu) = adjoin the l slice to the middle slice and turn it from f to u.

lr.(uf) = turn 3 slices at once; l, the middle, and r, all from u to f.
This is essentially like moving R and L' on separate moves.

L.(uf).2 = turn the L face and the middle edge at the same time, from u to f.

d.(fr).2 = Turn the d and middle slices twice, in the direction of f to r.


u.(fr) = adjoin the middle slice to the top slice and it them from f to r.


========================================================

NxNxN very large cubes with multiple slice turns for N > 5

"Build" the moves from the perspective of looking either from UFR with the
middle edge rotating vertically or from DBL, likewise. Pad moves to the
left or right of the middle edge in the order in which they appear.
Enclose the moves in round parenthesis to demarcate a middle edge.

Add a padded 0 for any ommited slice not in a series of consecutive slices.
Repeat the same slice letter as many times as needed if it is in relative
motion. Always use all N terms when describing a move sequence.

There should never be more than N/2 terms in motion for a move sequence.

11x11x11 examples:

F00ff(u.r)00000
0000f(u.r)bb000

000ll(u.f)0r00R

 

[Carrot]

========================================================

NxNxN very large cubes with multiple slice turns for N > 5

"Build" the moves from the perspective of looking either from UFR with the
middle edge rotating vertically or from DBL, likewise. Pad moves to the
left or right of the middle edge in the order in which they appear.
Enclose the moves in round parenthesis to demarcate a middle edge.

Add a padded 0 for any ommited slice not in a series of consecutive slices.
Repeat the same slice letter as many times as needed if it is in relative
motion. Always use all N terms when describing a move sequence.

There should never be more than N/2 terms in motion for a move sequence.

11x11x11 examples:

F00ff(u.r)00000
0000f(u.r)bb000

000ll(u.f)0r00R

I prefer this:
5F 3F' F
5F 4F' 5B' 3B

or this:
5f 4f F
5f 5b' 4b'


assuming f is understood as short for 2f. (reason why I would write F and not f)

one could however however argue for this notation (working for nxnxn for n<22):
145f = F 4f 5f = 5F 3F' F
124f = F f 4f = 4F 3F' 2F

and this (but this would give problems with consistency as 4F=4f=/=4F, so bad idea, let's keep 4F=/=4f):
145F = F 4f 5f = 5F 3F' F
124F = F f 4f = 4F 3F' 2F

so in your cases you could write them as:
F00ff(u.r)00000 = 145f
0000f(u.r)bb000 = 5f 45b'

and since this shouldn't ruin the consistency too much of what things mean, you can write things like(completely made up conjugated commutator, I doubt it's for ANY use except it shouldn't scramble too much):
[f: [24u 5b, 2L' U 2L]] = f 24u 5b 2L' u 2L 5b' 24u' 2L' U' 2L f'

 

[unsolved]

I prefer this:
5F 3F' F
5F 4F' 5B' 3B

or this:
5f 4f F
5f 5b' 4b'


assuming f is understood as short for 2f. (reason why I would write F and not f)

one could however however argue for this notation (working for nxnxn for n<22):
145f = F 4f 5f = 5F 3F' F
124f = F f 4f = 4F 3F' 2F

and this (but this would give problems with consistency as 4F=4f=/=4F, so bad idea, let's keep 4F=/=4f):
145F = F 4f 5f = 5F 3F' F
124F = F f 4f = 4F 3F' 2F

so in your cases you could write them as:
F00ff(u.r)00000 = 145f
0000f(u.r)bb000 = 5f 45b'

and since this shouldn't ruin the consistency too much of what things mean, you can write things like(completely made up conjugated commutator, I doubt it's for ANY use except it shouldn't scramble too much):
[f: [24u 5b, 2L' U 2L]] = f 24u 5b 2L' u 2L 5b' 24u' 2L' U' 2L f'

I don't have any experience with these larger cubes, so I defer to you. The idea was to toss something out there that made sense. I gave it a fair amount of thought.

In other news, the first implementation of the massively parallel version of OO_4x4x4 was not a success, but for the opposite reason you might suspect. It is too parallel!

I've never seen my 32-threaded computer (16 physical cores) get pegged at 100% before, but then again, the software thinks there are 1.3 million cores (2.6 million threads).

 

[Herbert Kociemba]

A Move Notation Proposal: Lodino = Logigcal Direction Notation

I must admit that I do not like this notation at all. Why we can't just accept these simple rules:

1. A letter *without* a number as prefix means a *single* slice move, upper cases like U , F etc. mean the outer slice, lower cases like u , f etc mean the adjacent slice.

2. With a number prefix, with upper case letters we denote a single slice move like 3U, 3F etc. With lower case letters we denote outer block turns, like 3u, 3f etc.

 

[Carrot]

I must admit that I do not like this notation at all. Why we can't just accept these simple rules:

1. A letter *without* a number as prefix means a *single* slice move, upper cases like U , F etc. mean the outer slice, lower cases like u , f etc mean the adjacent slice.

2. With a number prefix, with upper case letters we denote a single slice move like 3U, 3F etc. With lower case letters we denote outer block turns, like 3u, 3f etc.

Why not:

• lowercase letter: single slice turns
• uppercase letter: outer block turns
• prefixes: defines "how deep" the turn should be (3f = 3rd layer single slice turn of the f-face in a clockwise direction etc.)
• postfixes: defines the direction of the turn
• special case: since f=F we denote all f's as F. We can then denote all 2f's as f for more compactness.

Your notation gives unconsistency in when uppercase and lower cases means what, it shouldn't. Why should U=OBT and U=SS, u=SS => 3U=SS, 3u=OBT? that makes no sense.

 

[rokicki]

Here's another six optimal OTBM solves; some are 14, some are 15. Still all
from the ELL list.

2R2 U' B2 U' F L F' 2R2 F L' F' U B2 U 2R2
        Rw2 U 3Rw 3Uw' Fw Rw2 Uw2 3Uw2 Rw2 Fw' 3Uw 3Rw' U' Rw2
R2 2R' F' L R F 2R' F' R' F 2R2 F' L' F R2
        U F 3Fw' Rw 3Fw U' 3Fw' Rw2 3Rw U' 3Rw' F Rw F' 3Fw
2L2 2R2 U 2R2 F2 2R2 F2 U' 2L2 2U2 2R2 2U2
        F2 3Fw2 U Uw2 3Fw2 R2 Fw2 U2 Uw2 3Fw2 R2 Fw2 U F2 3Fw2
2L2 2R2 U' 2R2 F2 2R2 F2 U 2L2 2U2 2R2 2U2
        F2 3Fw2 U' Uw2 F2 R2 Fw2 U2 Uw2 F2 R2 Fw2 U' F2 3Fw2
R2 D' F2 2F' U 2F U2 2F' U F2 U' 2F U2 2F' U' 2F D R2
        Fw2 R2 F' Fw Rw2 F 3Rw U' Rw2 U Uw' R2 Uw 3Rw' Fw
2L2 2R2 U' 2L' 2R U2 F2 2L 2R F2 D2 F2 2L2 F2 2L2 D2 2R2 U' 2L2 2R2
        Fw2 U2 Rw2 3Uw R2 3Fw2 Rw2 3Uw2 R2 Rw2 3Uw' Fw2 U2 3Fw2 Rw2

I may not be able to solve all of them with my current program.

Working on the double-edge flip; I've found no solution through
length 14 in the outer block metric and am now looking on a length
15 solution.

I'm not sure if 18 OBTM is out of reach for your solver, but 18 OBTM is the briefest I have found (and currently exists) for this case (which I have studied extensively).
(Rw2 Uw U S Lw2 u2) r (u2 Lw2 S' U' Uw' Rw2)
(Rw2 Uw' U' S Lw2 u2) r (u2 Lw2 S' U Uw Rw2)

That one was easy; 13 outer block turns:

Rw2 Uw U S Lw2 2U2 2R 2U2 Lw2 S' U' Uw' Rw2
    Rw2 U2 R2 Rw' U2 3Uw2 R' Rw U2 3Uw2 Rw' U2 Rw2

And another ELL just popped out.

2R2 U' 2R2 F2 2R2 F2 U2 F2 2R2 F2 2R2 U' 2R2
     R2 Fw2 Rw2 U' R2 F2 R2 U2 R2 F2 R2 U' Rw2 Fw2 R2

 

[Christopher Mowla]

That one was easy; 13 outer block turns:

Rw2 Uw U S Lw2 2U2 2R 2U2 Lw2 S' U' Uw' Rw2
    Rw2 U2 R2 Rw' U2 3Uw2 R' Rw U2 3Uw2 Rw' U2 Rw2

You just lowered the BHTM for this case by 1 move, the BQTM by 1 move, and the OBTM by 5 moves! It also works on the nxnxn! How did I miss this?!
Rw2 U2 R r' E2 r E2 Rw' U2 Rw2

Now, I'm wondering...

Single Dedge Flip (Current minimum, 19 OBTM)
Rw2 B r B U2 B' r' B U2 Lw' B2 r' B2 Lw B2 Rw2
Rw2 B' r B' U2 B r' B' U2 Lw' B2 r' B2 Lw B2 Rw2

Double Parity (Current minimum, 21 OBTM)
Rw2 U B2 U' r' U B2 U' Rw r U B2 U r' U' B2 U' Rw

Adjacent Double Parity (Current minimum, 22 OBTM)
r2 F Rw' F' r2 U2 r' U2 r' U2 r' U2 F Rw F' r2

Adjacent checkerboard case (Current minimum, 22 OBTM)
Rw' F2 M2 F U2 l U2 l2 U2 l U2 l F' M2 F2 Rw

Adjacent hourglass case (Current minimum, 22 OBTM)
(Lw' U2 M2 U' x) r' U2 r' U2 r2 U2 r' U2 (x' U M2 U2 Lw)


Lastly, algs for all 6 unique "oriented" 3-cycle and 2-cycle last layer cases:

Spoiler: 3-cycle and a 2-cycle cases

In 3 dedges:

Case 1
R2 Lw F2 Lw U' R2 U' l U2 l' U' R2 U Lw' U2 l U2 F2 Lw' R2 (23)

Case 2
Lw F2 U2 l' U2 Lw U' R U l U2 l' U R' U Lw' F2 Lw' (21)

Case 3
R B Rw' S2 B' U2 r U2 r' B Rw' U2 r U2 Rw S2 Rw B' R' (24)

In 4 dedges:

Case 1
B2 U l Rw U' r F2 r' F2 U l' U2 r' U2 Rw' U' B2 (22)
B2 U l Rw U' Rw F2 r' F2 R' U Rw' F2 r' F2 l' U' B2 (22)

Case 2 (I doubt this is alg is close to optimal!)
F2 l U' D2 r U2 R r' B Rw' U2 r U2 Rw B' R' U' D2 l' F2 (25)

Case 3
F2 U Rw2 F R F' r F2 r' F' R' F' Rw' F2 r' F2 Rw' U' F2 (22)