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Announcing: New 4x4x4 Brute Force Solver

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OO_4x4x4 Version 1.2.8 is available

Pretty cool with all those 15 move dedge flips, but as for the display wouldn't having two characters instead of four between make the squares look more, well square? And also just so you know the de facto standard is having green on front and white on top.
Omnia Obtorquebantur Version 1.2.8 With Leaf Node + Massive Tree Branch Pruning Without Parallel Search

http://lightningcloudcomputing.com/OO_4x4x4.zip



I placed the new version online. There is a known bug where it will miss an optimal line since it over-pruned one critical position too early. Other than that, it is very fast AFTER DEPTH 6 (explained below).

Features:

  1. New Pruning Data Collector + Branch Pruning Algorithm
  2. Bi-Directional Searching from the scramble to the TFS databases, and the databases back towards the current position being solved
  3. Leaf Nodes are pruned to prevent duplicate positions, and Branches are pruned (entire lines of play, reducing the tree tremendously) based on a new experimental algorithm
  4. New graphics, as requested, shows Top over Front, and Left-Front-Right-Back, then Bottom under Front

This new version, while still with an embedded bug, will prove very valuable should enough of the cubing population test it and submit the solves it does find. After examining enough of them, I will be able to track it down and make the algorithm water-tight. Some of you will notice that during the pass of either depth-5 or depth-6, the search seems to slow down. It is because it is performing a massive RAM-based sort of data that it is collecting as it gathers Branch Pruning Data. What it is doing is a little lengthy to explain, but the short version is the program is trying to determine what branches of play, not what moves, are getting nowhere fast. If subsequent branches (varying moves in length) show up in later searches, a quick poll against the TFS databases will give a pass/fail regarding taking that branch to the next level of search. It's pretty complicated, but once I get this nailed down, OO_4x4x4 will be able to search with blinding speed.

Depth 7- 8- and 9- will fly by at much faster than expected speeds. Don't forget, if you are probing the 5-TFS database, you are really searching up to ply 14 .

The single dedge flip, by luck or just plain matching all the specs of the algorithm, is able to be solved in just under 15 minutes on a "normal" 3.1 GHz Westmere. Some of you will have even faster systems, so please test it out.

Features Still To Come:

1. Fix that bug!
2. Fixed-Core Multiprocessing for 2-, 4-, 8-, 10-, 12-, and 16-core implementations.
3. Massively Parallel Bitboard version for those with >= 20 cores
4. Solving the 7-TFS database on my 192 GB machine at work :)

As always, any feedback is appreciated.
 
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Bi-Directional Searching from the scramble to the TFS databases, and the databases back towards the current position being solved
This sounds scary. The solved state contains matching color centers on each face while the scrambled position in general does not. So you can't use the same database in both directions, unless perhaps if you are using a supercube model and then properly accounting for which pieces are indistinguishable in each direction of the search.

EDIT: I would like you to keep a link to a known good version of the program in the initial post of thread. Your experimental versions with bugs aren't of that much interest to me.

EDIT 2:
This new version, while still with an embedded bug, will prove very valuable should enough of the cubing population test it and submit the solves it does find. After examining enough of them, I will be able to track it down and make the algorithm water-tight. Some of you will notice that during the pass of either depth-5 or depth-6, the search seems to slow down. It is because it is performing a massive RAM-based sort of data that it is collecting as it gathers Branch Pruning Data. What it is doing is a little lengthy to explain, but the short version is the program is trying to determine what branches of play, not what moves, are getting nowhere fast. If subsequent branches (varying moves in length) show up in later searches, a quick poll against the TFS databases will give a pass/fail regarding taking that branch to the next level of search. It's pretty complicated, but once I get this nailed down, OO_4x4x4 will be able to search with blinding speed.
It almost sounds here that you are guessing whether or not you can prune a given branch. In a proper optimal solver, you never prune a branch based upon a guess that it is safe to do so. You always make certain it is safe to do so.
 
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that just seems to scramble the cube

edit: jus downloaded it and found a 14 move bldparity

U2 r2 U2 B2 r' U2 B2 r2 B2 r' U2 r2 B2 r'

any chance of qtm mode?
That is an interesting algo! What was the scramble you used to get there? How long did it take to reach 14 moves?

This sounds scary. The solved state contains matching color centers on each face while the scrambled position in general does not. So you can't use the same database in both directions, unless perhaps if you are using a supercube model and then properly accounting for which pieces are indistinguishable in each direction of the search.

EDIT: I would like you to keep a link to a known good version of the program in the initial post of thread. Your experimental versions with bugs aren't of that much interest to me.
I have a 96-color move generator, so I am safe. I have tested this very thoroughly. I just spawn 6 threads, one for each face, and look for independent face-matching. I relinquish thread control to the main core so that the additional cores do not overburden the computer. It is a tame version of what can be done with a system with 24 CPU cores/threads, one for each possible rotation state, but that is in a #define statement I can change and bring on full force when on the proper hardware.

I just can't halt progress and only release squeaky-clean versions, not when there is little-to-no adverse effects and much to gain with a significant amount of input from the cubers. If this branch-pruning algorithm turns out to be water-tight, it will allow for huge strides to be taken. Depth-6 and Depth-7 both examine fewer nodes than at Depth-5, and Depth-8 is not much larger than Depth-5.

I would prefer that others download this version and run a few scrambles up and report back which ones don't find optimal solves, and which ones do. Without a fair amount of documenting the ones that work and fail, I'll never track it down on my own.
 
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Watching thread with great interest. Looking forward to seeing what else turns up.
Me too!

By the way, you probably already did this (or someone did), but if we conjugate that alg differently, we have a (13,9) (but it affects the entire r slice, and of course it only works on the 4x4x4).
f2 E r E2 r' E2 r E' f2
 
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It almost sounds here that you are guessing whether or not you can prune a given branch. In a proper optimal solver, you never prune a branch based upon a guess that it is safe to do so. You always make certain it is safe to do so.
To me it is an obvious, well-behaved algorithm with one or maybe two small and very unlikely scenarios that are cropping up that spoil it. I am trying to generate a fail condition with random moves, and the thing runs all day and behaves without flaw. There is some fluke, such as the move generator creating a position it already created, or something like that, which is causing the over-pruning condition. Once it is located, it will be overly-obvious what was missed, but until then, it is a stumper.

Kind of like how I used to bet people I could finish 3 full mugs of beer before they could finish a shot of whatever they wanted to have. I'd bet them $50 and win it every time. Once I say how I do it, it is obvious. Until then, you are most likely perplexed and would probably accept the wager. I never lost it in over 100 demonstrations.
 
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I worry about the double-direction pruning, but the new view looks pretty good. Much easier to see what the case would look like on a real cube.

Here's the problem - you may have two algs which look like they do the same thing on the 4x4x4, but their inverses do NOT do the same thing. An example: (a) U r U l' U' r' U l U2 and (b) F l F r' F' l' F r F2. This happens because of indistinguishable centers. Now, as long as you test positions that do not change centers, you can go in both directions... BUT when you try positions with centers out of place you may find that your algs do not quite work. If you want to make sure your new approach is valid, I suggest testing it on some of the positions you had earlier with 4 or 6 centers wrong, or on a position you get from doing 12ish random moves on a solved cube.
 
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l2 U2 l' B2 l' B2 U2 D2 l F2 l' F2 D2 l

This one is only 14 moves :) But it seems very bad...
Bad for... speed solving? Or is the move sequence bad? Was this generated by my solver?

I worry about the double-direction pruning
It is a bi-directional search, but the pruning is only directed towards the solve.

Here's what the program is doing.

1. You enter a scramble. There is a single move generator that makes moves on the scrambled cube.

2. At terminal depths, it stops to check for a solved state. If it is solved, done. If not solved:

3. Probe the 6-TFS database. Is a 6-turn through 1-turn solution found? Yes, done, no, continue.

4. Add the move sequences in move group chunks and axis rotator chunks to the pruning information collector.

5. Start 6 threads, one for each face. Go to the largest TFS database in RAM. Visit each position in the TFS.
Each thread applys a different set of moves to the TFS. Does the move land in the 1- through "MAX - 1" TFS?
Then that move makes NO PROGRESS toward the scramble solver, skip it, and try another one. Otherwise...

6. Compare the position against positions being processed by the forward solver. Did the two ends meet? Reverse
the retro-solver moves, create a PV from the forward solver, the reverse retro solver, and the TFS databases.
That whole line is the solution, done. Otherwise...

7. Are we (MAX_TFS_DATABASE + 1) moves or more from the root? (At least 5, 6, or 7, depending on your RAM and elections at startup).
No, go to step 9.

8. Consult the pruning collector. Apply the "chunking algorithm" and determine if the BRANCH, not the MOVE,
is getting closer to moves coming off of the retro solver. Branches can be in the early move generation stage,
the middle, or the end. There are no hard-coded boundaries. If there is a solution that will happen in
N turns, then consult the pruning data collector for moves 1 to (N/2) + 1. An algorithm determines if there
is either a "progress", "scramble", or "not much progress" happening. Only the "progress" branches are
allowed to continue. Occasionally the algorithm makes a "no decision." <--- this is new code, attempting to fix over-aggressive pruning.

9. Generate more forward moves and reverse moves, then go to step 2. When back at step 5, there is no need to start more threads,
just use the existing ones to process more moves.

UPDATE: I am getting better results now after I changed 2 lines of code!



I am hoping the solutions below are all correct and there are no faster ones available. Anyone care to "spot check" a few?

Code:
Solution [001] =  R  F  u2 R2 u2 r2 [inside 5-TFS] ---> R2 u2 r2 F' R'  @ 0000226702773 nodes with 0001051015719 5-TFS probes          
Solution [002] =  R  F  u2 R2 u2 l2 [inside 5-TFS] ---> R2 u2 l2 F' R'  @ 0000226702776 nodes with 0001051015723 5-TFS probes          
Solution [003] =  R  F  u2 F2 R2 u2 [inside 5-TFS] ---> R2 F2 u2 F' R'  @ 0000226712628 nodes with 0001051058090 5-TFS probes          
Solution [004] =  R  F  u2 F2 L2 d2 [inside 5-TFS] ---> L2 F2 u2 F' R'  @ 0000226712883 nodes with 0001051059194 5-TFS probes          
Solution [005] =  R  F  u2 B2 R2 u2 [inside 5-TFS] ---> R2 B2 u2 F' R'  @ 0000226719702 nodes with 0001051087397 5-TFS probes          
Solution [006] =  R  F  u2 B2 L2 d2 [inside 5-TFS] ---> L2 B2 u2 F' R'  @ 0000226719957 nodes with 0001051088528 5-TFS probes          
Solution [007] =  R  F  d2 R2 d2 R2 [inside 5-TFS] ---> r2 d2 r2 F' R'  @ 0000226766679 nodes with 0001051284391 5-TFS probes          
Solution [008] =  R  F  d2 R2 d2 r2 [inside 5-TFS] ---> R2 d2 r2 F' R'  @ 0000226766682 nodes with 0001051284407 5-TFS probes          
Solution [009] =  R  F  d2 R2 d2 l2 [inside 5-TFS] ---> R2 d2 l2 F' R'  @ 0000226766685 nodes with 0001051284415 5-TFS probes          
Solution [010] =  R  F  d2 F2 R2 d2 [inside 5-TFS] ---> R2 F2 d2 F' R'  @ 0000226776459 nodes with 0001051324598 5-TFS probes          
Solution [011] =  R  F  d2 F2 L2 u2 [inside 5-TFS] ---> L2 F2 d2 F' R'  @ 0000226776708 nodes with 0001051325689 5-TFS probes          
Solution [012] =  R  F  d2 B2 R2 d2 [inside 5-TFS] ---> R2 B2 d2 F' R'  @ 0000226783533 nodes with 0001051354683 5-TFS probes          
Solution [013] =  R  F  d2 B2 L2 u2 [inside 5-TFS] ---> L2 B2 d2 F' R'  @ 0000226783782 nodes with 0001051355840 5-TFS probes          
Solution [014] =  R  F  r2 u2 R2 u2 [inside 5-TFS] ---> R2 u2 r2 F' R'  @ 0000226971324 nodes with 0001052143552 5-TFS probes          
Solution [015] =  R  F  r2 u2 R2 r2 [inside 5-TFS] ---> u2 R2 u2 F' R'  @ 0000226971333 nodes with 0001052143602 5-TFS probes          
Solution [016] =  R  F  r2 u2 L2 d2 [inside 5-TFS] ---> L2 u2 r2 F' R'  @ 0000226971579 nodes with 0001052144772 5-TFS probes          
Solution [017] =  R  F  r2 d2 R2 d2 [inside 5-TFS] ---> R2 d2 r2 F' R'  @ 0000226973595 nodes with 0001052153139 5-TFS probes          
Solution [018] =  R  F  r2 d2 R2 r2 [inside 5-TFS] ---> d2 R2 d2 F' R'  @ 0000226973601 nodes with 0001052153155 5-TFS probes          
Solution [019] =  R  F  r2 d2 L2 u2 [inside 5-TFS] ---> L2 d2 r2 F' R'  @ 0000226973844 nodes with 0001052154079 5-TFS probes          
Solution [020] =  R  F  l2 u2 R2 u2 [inside 5-TFS] ---> R2 u2 l2 F' R'  @ 0000227037420 nodes with 0001052422228 5-TFS probes          
Solution [021] =  R  F  l2 u2 R2 l2 [inside 5-TFS] ---> u2 R2 u2 F' R'  @ 0000227037432 nodes with 0001052422236 5-TFS probes          
Solution [022] =  R  F  l2 u2 L2 d2 [inside 5-TFS] ---> L2 u2 l2 F' R'  @ 0000227037675 nodes with 0001052423173 5-TFS probes          
Solution [023] =  R  F  l2 d2 R2 d2 [inside 5-TFS] ---> R2 d2 l2 F' R'  @ 0000227039691 nodes with 0001052431559 5-TFS probes          
Solution [024] =  R  F  l2 d2 R2 l2 [inside 5-TFS] ---> d2 R2 d2 F' R'  @ 0000227039700 nodes with 0001052431565 5-TFS probes          
Solution [025] =  R  F  l2 d2 L2 u2 [inside 5-TFS] ---> L2 d2 l2 F' R'  @ 0000227039940 nodes with 0001052432554 5-TFS probes          
Solution [026] =  R  B  u2 r2 U2 u2 [inside 5-TFS] ---> r2 U2 r2 B' R'  @ 0000232681128 nodes with 0001076114491 5-TFS probes          
Solution [027] =  R  B  u2 r2 U2 r2 [inside 5-TFS] ---> U2 r2 u2 B' R'  @ 0000232681140 nodes with 0001076114525 5-TFS probes          
Solution [028] =  R  B  u2 r2 D2 l2 [inside 5-TFS] ---> D2 r2 u2 B' R'  @ 0000232681386 nodes with 0001076115666 5-TFS probes          
Solution [029] =  R  B  u2 l2 U2 u2 [inside 5-TFS] ---> l2 U2 l2 B' R'  @ 0000232683477 nodes with 0001076124404 5-TFS probes          
Solution [030] =  R  B  u2 l2 U2 l2 [inside 5-TFS] ---> U2 l2 u2 B' R'  @ 0000232683492 nodes with 0001076124470 5-TFS probes          
Solution [031] =  R  B  u2 l2 D2 r2 [inside 5-TFS] ---> D2 l2 u2 B' R'  @ 0000232683732 nodes with 0001076125519 5-TFS probes          
Solution [032] =  R  B  d2 r2 U2 d2 [inside 5-TFS] ---> r2 U2 r2 B' R'  @ 0000232744959 nodes with 0001076384035 5-TFS probes          
Solution [033] =  R  B  d2 r2 U2 r2 [inside 5-TFS] ---> U2 r2 d2 B' R'  @ 0000232744968 nodes with 0001076384044 5-TFS probes          
Solution [034] =  R  B  d2 r2 D2 l2 [inside 5-TFS] ---> D2 r2 d2 B' R'  @ 0000232745214 nodes with 0001076385088 5-TFS probes          
Solution [035] =  R  B  d2 l2 U2 d2 [inside 5-TFS] ---> l2 U2 l2 B' R'  @ 0000232747308 nodes with 0001076393420 5-TFS probes          
Solution [036] =  R  B  d2 l2 U2 l2 [inside 5-TFS] ---> U2 l2 d2 B' R'  @ 0000232747320 nodes with 0001076393457 5-TFS probes          
Solution [037] =  R  B  d2 l2 D2 r2 [inside 5-TFS] ---> D2 l2 d2 B' R'  @ 0000232747560 nodes with 0001076394289 5-TFS probes          
Solution [038] =  R  B  r2 U2 r2 U2 [inside 5-TFS] ---> u2 r2 u2 B' R'  @ 0000232944759 nodes with 0001077224817 5-TFS probes          
Solution [039] =  R  B  r2 U2 r2 u2 [inside 5-TFS] ---> U2 r2 u2 B' R'  @ 0000232944762 nodes with 0001077224819 5-TFS probes          
Solution [040] =  R  B  r2 U2 r2 d2 [inside 5-TFS] ---> U2 r2 d2 B' R'  @ 0000232944765 nodes with 0001077224826 5-TFS probes          
Solution [041] =  R  B  r2 F2 U2 r2 [inside 5-TFS] ---> U2 F2 r2 B' R'  @ 0000232958367 nodes with 0001077283616 5-TFS probes          
Solution [042] =  R  B  r2 F2 D2 l2 [inside 5-TFS] ---> D2 F2 r2 B' R'  @ 0000232958613 nodes with 0001077284560 5-TFS probes          
Solution [043] =  R  B  r2 B2 U2 r2 [inside 5-TFS] ---> U2 B2 r2 B' R'  @ 0000232965441 nodes with 0001077313080 5-TFS probes          
Solution [044] =  R  B  r2 B2 D2 l2 [inside 5-TFS] ---> D2 B2 r2 B' R'  @ 0000232965687 nodes with 0001077314071 5-TFS probes          
Solution [045] =  R  B  l2 U2 l2 U2 [inside 5-TFS] ---> u2 l2 u2 B' R'  @ 0000233010939 nodes with 0001077502656 5-TFS probes          
Solution [046] =  R  B  l2 U2 l2 u2 [inside 5-TFS] ---> U2 l2 u2 B' R'  @ 0000233010942 nodes with 0001077502660 5-TFS probes          
Solution [047] =  R  B  l2 U2 l2 d2 [inside 5-TFS] ---> U2 l2 d2 B' R'  @ 0000233010945 nodes with 0001077502673 5-TFS probes          
Solution [048] =  R  B  l2 F2 U2 l2 [inside 5-TFS] ---> U2 F2 l2 B' R'  @ 0000233022198 nodes with 0001077549893 5-TFS probes          
Solution [049] =  R  B  l2 F2 D2 r2 [inside 5-TFS] ---> D2 F2 l2 B' R'  @ 0000233022438 nodes with 0001077550884 5-TFS probes          
Solution [050] =  R  B  l2 B2 U2 l2 [inside 5-TFS] ---> U2 B2 l2 B' R'  @ 0000233029272 nodes with 0001077579541 5-TFS probes          
Solution [051] =  R  B  l2 B2 D2 r2 [inside 5-TFS] ---> D2 B2 l2 B' R'  @ 0000233029512 nodes with 0001077580568 5-TFS probes          
Solution [052] =  F' R' u2 R2 F2 u2 [inside 5-TFS] ---> F2 R2 u2 R  F   @ 0000456220647 nodes with 0002014315744 5-TFS probes          
Solution [053] =  F' R' u2 R2 B2 d2 [inside 5-TFS] ---> B2 R2 u2 R  F   @ 0000456220902 nodes with 0002014316786 5-TFS probes          
Solution [054] =  F' R' u2 L2 F2 u2 [inside 5-TFS] ---> F2 L2 u2 R  F   @ 0000456227478 nodes with 0002014344879 5-TFS probes          
Solution [055] =  F' R' u2 L2 B2 d2 [inside 5-TFS] ---> B2 L2 u2 R  F   @ 0000456227733 nodes with 0002014345813 5-TFS probes          
Solution [056] =  F' R' u2 F2 u2 F2 [inside 5-TFS] ---> f2 u2 f2 R  F   @ 0000456229782 nodes with 0002014354359 5-TFS probes          
Solution [057] =  F' R' u2 F2 u2 f2 [inside 5-TFS] ---> F2 u2 f2 R  F   @ 0000456229785 nodes with 0002014354363 5-TFS probes          
Solution [058] =  F' R' u2 F2 u2 b2 [inside 5-TFS] ---> F2 u2 b2 R  F   @ 0000456229788 nodes with 0002014354372 5-TFS probes          
Solution [059] =  F' R' d2 R2 F2 d2 [inside 5-TFS] ---> F2 R2 d2 R  F   @ 0000456284478 nodes with 0002014580316 5-TFS probes          
Solution [060] =  F' R' d2 R2 B2 u2 [inside 5-TFS] ---> B2 R2 d2 R  F   @ 0000456284727 nodes with 0002014581313 5-TFS probes          
Solution [061] =  F' R' d2 L2 F2 d2 [inside 5-TFS] ---> F2 L2 d2 R  F   @ 0000456291309 nodes with 0002014609311 5-TFS probes          
Solution [062] =  F' R' d2 L2 B2 u2 [inside 5-TFS] ---> B2 L2 d2 R  F   @ 0000456291558 nodes with 0002014610497 5-TFS probes          
Solution [063] =  F' R' d2 F2 d2 F2 [inside 5-TFS] ---> f2 d2 f2 R  F   @ 0000456293691 nodes with 0002014619726 5-TFS probes          
Solution [064] =  F' R' d2 F2 d2 f2 [inside 5-TFS] ---> F2 d2 f2 R  F   @ 0000456293694 nodes with 0002014619747 5-TFS probes          
Solution [065] =  F' R' d2 F2 d2 b2 [inside 5-TFS] ---> F2 d2 b2 R  F   @ 0000456293697 nodes with 0002014619771 5-TFS probes          
Solution [066] =  F' R' f2 u2 F2 u2 [inside 5-TFS] ---> F2 u2 f2 R  F   @ 0000456680538 nodes with 0002016240014 5-TFS probes          
Solution [067] =  F' R' f2 u2 F2 f2 [inside 5-TFS] ---> u2 F2 u2 R  F   @ 0000456680559 nodes with 0002016240088 5-TFS probes          
Solution [068] =  F' R' f2 u2 B2 d2 [inside 5-TFS] ---> B2 u2 f2 R  F   @ 0000456680793 nodes with 0002016240962 5-TFS probes          
Solution [069] =  F' R' f2 d2 F2 d2 [inside 5-TFS] ---> F2 d2 f2 R  F   @ 0000456682809 nodes with 0002016249584 5-TFS probes          
Solution [070] =  F' R' f2 d2 F2 f2 [inside 5-TFS] ---> d2 F2 d2 R  F   @ 0000456682827 nodes with 0002016249612 5-TFS probes          
Solution [071] =  F' R' f2 d2 B2 u2 [inside 5-TFS] ---> B2 d2 f2 R  F   @ 0000456683058 nodes with 0002016250483 5-TFS probes          
Solution [072] =  F' R' b2 u2 F2 u2 [inside 5-TFS] ---> F2 u2 b2 R  F   @ 0000456746634 nodes with 0002016518597 5-TFS probes          
Solution [073] =  F' R' b2 u2 F2 b2 [inside 5-TFS] ---> u2 F2 u2 R  F   @ 0000456746658 nodes with 0002016518674 5-TFS probes          
Solution [074] =  F' R' b2 u2 B2 d2 [inside 5-TFS] ---> B2 u2 b2 R  F   @ 0000456746889 nodes with 0002016519577 5-TFS probes          
Solution [075] =  F' R' b2 d2 F2 d2 [inside 5-TFS] ---> F2 d2 b2 R  F   @ 0000456748905 nodes with 0002016528206 5-TFS probes          
Solution [076] =  F' R' b2 d2 F2 b2 [inside 5-TFS] ---> d2 F2 d2 R  F   @ 0000456748926 nodes with 0002016528295 5-TFS probes          
Solution [077] =  F' R' b2 d2 B2 u2 [inside 5-TFS] ---> B2 d2 b2 R  F   @ 0000456749154 nodes with 0002016529245 5-TFS probes          
Solution [078] =  F' L' u2 f2 U2 u2 [inside 5-TFS] ---> f2 U2 f2 L  F   @ 0000462016644 nodes with 0002038668170 5-TFS probes          
Solution [079] =  F' L' u2 f2 U2 f2 [inside 5-TFS] ---> U2 f2 u2 L  F   @ 0000462016668 nodes with 0002038668234 5-TFS probes          
Solution [080] =  F' L' u2 f2 D2 b2 [inside 5-TFS] ---> D2 f2 u2 L  F   @ 0000462016914 nodes with 0002038669347 5-TFS probes          
Solution [081] =  F' L' u2 b2 U2 u2 [inside 5-TFS] ---> b2 U2 b2 L  F   @ 0000462018993 nodes with 0002038678296 5-TFS probes          
Solution [082] =  F' L' u2 b2 U2 b2 [inside 5-TFS] ---> U2 b2 u2 L  F   @ 0000462019020 nodes with 0002038678371 5-TFS probes          
Solution [083] =  F' L' u2 b2 D2 f2 [inside 5-TFS] ---> D2 b2 u2 L  F   @ 0000462019260 nodes with 0002038679525 5-TFS probes          
Solution [084] =  F' L' d2 f2 U2 d2 [inside 5-TFS] ---> f2 U2 f2 L  F   @ 0000462080475 nodes with 0002038937634 5-TFS probes          
Solution [085] =  F' L' d2 f2 U2 f2 [inside 5-TFS] ---> U2 f2 d2 L  F   @ 0000462080496 nodes with 0002038937781 5-TFS probes          
Solution [086] =  F' L' d2 f2 D2 b2 [inside 5-TFS] ---> D2 f2 d2 L  F   @ 0000462080742 nodes with 0002038938805 5-TFS probes          
Solution [087] =  F' L' d2 b2 U2 d2 [inside 5-TFS] ---> b2 U2 b2 L  F   @ 0000462082824 nodes with 0002038947580 5-TFS probes          
Solution [088] =  F' L' d2 b2 U2 b2 [inside 5-TFS] ---> U2 b2 d2 L  F   @ 0000462082848 nodes with 0002038947666 5-TFS probes          
Solution [089] =  F' L' d2 b2 D2 f2 [inside 5-TFS] ---> D2 b2 d2 L  F   @ 0000462083088 nodes with 0002038948558 5-TFS probes          
Solution [090] =  F' L' f2 U2 f2 U2 [inside 5-TFS] ---> u2 f2 u2 L  F   @ 0000462271005 nodes with 0002039734524 5-TFS probes          
Solution [091] =  F' L' f2 U2 f2 u2 [inside 5-TFS] ---> U2 f2 u2 L  F   @ 0000462271008 nodes with 0002039734534 5-TFS probes          
Solution [092] =  F' L' f2 U2 f2 d2 [inside 5-TFS] ---> U2 f2 d2 L  F   @ 0000462271011 nodes with 0002039734539 5-TFS probes          
Solution [093] =  F' L' f2 R2 U2 f2 [inside 5-TFS] ---> U2 R2 f2 L  F   @ 0000462279963 nodes with 0002039771443 5-TFS probes          
Solution [094] =  F' L' f2 R2 D2 b2 [inside 5-TFS] ---> D2 R2 f2 L  F   @ 0000462280209 nodes with 0002039772336 5-TFS probes          
Solution [095] =  F' L' f2 L2 U2 f2 [inside 5-TFS] ---> U2 L2 f2 L  F   @ 0000462287037 nodes with 0002039801428 5-TFS probes          
Solution [096] =  F' L' f2 L2 D2 b2 [inside 5-TFS] ---> D2 L2 f2 L  F   @ 0000462287283 nodes with 0002039802353 5-TFS probes          
Solution [097] =  F' L' b2 U2 b2 U2 [inside 5-TFS] ---> u2 b2 u2 L  F   @ 0000462337185 nodes with 0002040007770 5-TFS probes          
Solution [098] =  F' L' b2 U2 b2 u2 [inside 5-TFS] ---> U2 b2 u2 L  F   @ 0000462337188 nodes with 0002040007791 5-TFS probes          
Solution [099] =  F' L' b2 U2 b2 d2 [inside 5-TFS] ---> U2 b2 d2 L  F   @ 0000462337191 nodes with 0002040007792 5-TFS probes          
Solution [100] =  F' L' b2 R2 U2 b2 [inside 5-TFS] ---> U2 R2 b2 L  F   @ 0000462346062 nodes with 0002040046228 5-TFS probes          
Solution [101] =  F' L' b2 R2 D2 f2 [inside 5-TFS] ---> D2 R2 b2 L  F   @ 0000462346302 nodes with 0002040047129 5-TFS probes          
Solution [102] =  F' L' b2 L2 U2 b2 [inside 5-TFS] ---> U2 L2 b2 L  F   @ 0000462353136 nodes with 0002040075398 5-TFS probes          
Solution [103] =  F' L' b2 L2 D2 f2 [inside 5-TFS] ---> D2 L2 b2 L  F   @ 0000462353376 nodes with 0002040076394 5-TFS probes
 
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Five new optimal distance-16 (OBTM) solutions popped up this morning.
They are numbers 1004 1005 1028 3002 3003 on

http://tomas.rokicki.com/ell4x4.html

My initial attempt at other metrics (SSTM and BTM) work but are too
slow to be useful; I will have to continue working on them.
wouldn't it be faster to not use any Lw/Bw/Dw moves?

edit: cool, z perm on 4x4 is 2 moves shorter than 3x3

r2 U r2 D' f2 l2 U' l2 D f2
 
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wouldn't it be faster to not use any Lw/Bw/Dw moves?

edit: cool, z perm on 4x4 is 2 moves shorter than 3x3

r2 U r2 D' f2 l2 U' l2 D f2
Hmm, remember, I'm computing OBTM sequences---outer block turns only, which
includes the Lw/Bw/Dw moves. I presume you mean for speed solvers it would be
faster to execute without these moves, and this is assuredly true; OBTM-optimal
sequences are not necessarily the fastest for speedsolving. (I'm also stopping at
the first solution, instead of finding all solutions, but I may rerun some of these
finding all solutions.)

Your Z-perm 10-mover; is this new? How did you find this? I confirm that this is
optimal in single-slice-turn metric.

Of course for the 3x3 there are well-known seven-turn sequences in the slice turn
metric for Z-perm; these are optimal in the BTM on the 4x4 as well.

Best solution in OBTM for zperm is 12 moves.
 
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Hmm, remember, I'm computing OBTM sequences---outer block turns only, which
includes the Lw/Bw/Dw moves. I presume you mean for speed solvers it would be
faster to execute without these moves, and this is assuredly true; OBTM-optimal
sequences are not necessarily the fastest for speedsolving. (I'm also stopping at
the first solution, instead of finding all solutions, but I may rerun some of these
finding all solutions.)

Your Z-perm 10-mover; is this new? How did you find this? I confirm that this is
optimal in single-slice-turn metric.

Of course for the 3x3 there are well-known seven-turn sequences in the slice turn
metric for Z-perm; these are optimal in the BTM on the 4x4 as well.

Best solution in OBTM for zperm is 12 moves.
well I thought you wouldn't have needed Lw/Bw/Dw because they are just the same as Rw/Fw/Uw with a rotation.

found the Z perm with unsolveds solver, I don't know if its been found before.
 
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well I thought you wouldn't have needed Lw/Bw/Dw because they are just the same as Rw/Fw/Uw with a rotation.
Ahh, I see. My twisted solves as they come out of the solver don't use Lw/Bw/Dw because they keep
the DBL corner fixed, but they *do* use moves like 3Lw and 3Bw which are unusual.

My "untwister" considers all the various reorientations of the sequences possible by converting Uw to Dw
and the like, and picks out those sequences that have a short text description (preferring U over
3Dw for instance). It *only* shows the ones that come up with short descriptions.

To make them speed-solving friendly, we probably want to revise the metric used to pick the solutions to
reflect how fast a typical speedsolver can manipulate the cube. If we could come up with such a simple
metric, that would be greatly useful. Different people can do different moves at different speeds, but
certainly there's some common denominator we can use. At the start maybe we can just assign a
"time" to each possible move (including slice moves) and see what happens with that. For the BTM on the
4x4x4 there are 54 different moves; that's a lot of numbers.
 
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To make them speed-solving friendly, we probably want to revise the metric used to pick the solutions to
reflect how fast a typical speedsolver can manipulate the cube. If we could come up with such a simple
metric, that would be greatly useful. Different people can do different moves at different speeds, but
certainly there's some common denominator we can use. At the start maybe we can just assign a
"time" to each possible move (including slice moves) and see what happens with that. For the BTM on the
4x4x4 there are 54 different moves; that's a lot of numbers.
I know this kind of thing has been discussed before somewhere, but I don't think it ever got as far as trying to assign specific numbers to each move. The problem is that not only does it depend on the person, but it also depends on the cube itself. It is harder to perform slice turns quickly on some cubes, while others might not have as fast outer layers. And depending on tensions, performing a lot of D moves (say, the common <R,U,D> E-perm) may or may not be easy.
 
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I know this kind of thing has been discussed before somewhere, but I don't think it ever got as far as trying to assign specific numbers to each move. The problem is that not only does it depend on the person, but it also depends on the cube itself. It is harder to perform slice turns quickly on some cubes, while others might not have as fast outer layers. And depending on tensions, performing a lot of D moves (say, the common <R,U,D> E-perm) may or may not be easy.
I think it's even more complicated as you really need to consider sequences of moves. Some sequences called triggers (e.g. R' U R') can be particularly fast to perform. Likewise some sequences can be awkward and slow. A regrip (usually to perform a whole cube rotation) can result in a significant delay even though it doesn't change the intrinsic state of the cube, but if it can set up the use of nice triggers, the cost of a regrip may be made up for. I think basically, you may need to take into account how executing moves affects the state of the fingers. I'm clearly not the expert on this, though.

EDIT (appending):
I tried the following scramble on OO4x4x4 version 1.2.3 and 1.2.8:

L l r' F f b' R r l' F2 f2 b2 L l U u d' R2 r2 l2

My solver solved this in around 1 second to finish a depth 8 search (yes, corners pruning really helps on this one). OO4x4x4 (using 5-TFS) took a few minutes to search up to a "depths 6-11" and still did not find the solution. Even though this position can be solved back to the original cube orientation in 11 moves, it might be that OO4x4x4 didn't find it due to its restricting to no more than 3 consecutive turns on an axis. The rule that you shouldn't need more than 3 turns in a row on the same axis is based upon an assumption that the orientation of the cube shouldn't matter. OO4x4x4 is using this rule, but it still apparently considering the orientation of the cube to matter.

On version 1.2.8, it took until the iteration for "depths 10-15" where the solutions found were initially at depth 14, and then later on depth 13 solutions were found. These solutions had obvious cancellations between the tree search part and the TFS part. It looks to me that these cancellations are what allows the program to actually find these solutions, since after removing these moves that cancel out, you end up with a solution containing 4 consecutive moves on an axis - necessary in this case to get an optimal solution that leaves the cube in the original pre-scramble orientation.

The output from Version 1.2.8 is in the spoiler.

Code:
  Completed searching 0000000000000001  nodes through DEPTHS 00-05:     LEAF NODES: 0000000000000000 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000000000010: 1-TFS probes 0000000000000000: 2-TFS probes 0000000000000000: 3-TFS probes 0000000000000000
  TFS Database misses 0000000000000011: 4-TFS probes 0000000000000001: 5-TFS probes 0000000000000009: 6-TFS probes 0000000000000000

  Completed searching 0000000000000036  nodes through DEPTHS 01-06:     LEAF NODES: 0000000000000000 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000000000077: 1-TFS probes 0000000000000000: 2-TFS probes 0000000000000000: 3-TFS probes 0000000000000000
  TFS Database misses 0000000000000096: 4-TFS probes 0000000000000006: 5-TFS probes 0000000000000071: 6-TFS probes 0000000000000000

  Completed searching 0000000000001026  nodes through DEPTHS 02-07:     LEAF NODES: 0000000000000000 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000000004483: 1-TFS probes 0000000000000000: 2-TFS probes 0000000000000000: 3-TFS probes 0000000000000016
  TFS Database misses 0000000000005216: 4-TFS probes 0000000000000222: 5-TFS probes 0000000000004245: 6-TFS probes 0000000000000000

  Completed searching 0000000000028836  nodes through DEPTHS 03-08:     LEAF NODES: 0000000000000000 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000000131255: 1-TFS probes 0000000000000000: 2-TFS probes 0000000000000008: 3-TFS probes 0000000000000308
  TFS Database misses 0000000000152192: 4-TFS probes 0000000000007092: 5-TFS probes 0000000000123847: 6-TFS probes 0000000000000000

  Completed searching 0000000000810648  nodes through DEPTHS 04-09:     LEAF NODES: 0000000000000243 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000003734589: 1-TFS probes 0000000000000005: 2-TFS probes 0000000000000269: 3-TFS probes 0000000000007848
  TFS Database misses 0000000004321908: 4-TFS probes 0000000000201370: 5-TFS probes 0000000003525097: 6-TFS probes 0000000000000000

  Completed searching 0000000022791456  nodes through DEPTHS 05-10:     LEAF NODES: 0000000000006318 cut BRANCHES: 0000000000000000 pruned
  TFS Database probes 0000000105060036: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223766
  TFS Database misses 0000000121593540: 4-TFS probes 0000000005669000: 5-TFS probes 0000000099159181: 6-TFS probes 0000000000000000

  Completed searching 0000000000000000  nodes through DEPTHS 06-11:     LEAF NODES: 0000000000006318 cut BRANCHES: 0000000000000677 pruned
  TFS Database probes 0000000105060036: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223766
  TFS Database misses 0000000121593540: 4-TFS probes 0000000005669000: 5-TFS probes 0000000099159181: 6-TFS probes 0000000000000000

  Completed searching 0000000000000000  nodes through DEPTHS 07-12:     LEAF NODES: 0000000000006327 cut BRANCHES: 0000000000032308 pruned
  TFS Database probes 0000000105060036: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223766
  TFS Database misses 0000000121593540: 4-TFS probes 0000000005669000: 5-TFS probes 0000000099159181: 6-TFS probes 0000000000000000

  Completed searching 0000000000000000  nodes through DEPTHS 08-13:     LEAF NODES: 0000000000006696 cut BRANCHES: 0000000001161567 pruned
  TFS Database probes 0000000105060036: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223766
  TFS Database misses 0000000121593540: 4-TFS probes 0000000005669000: 5-TFS probes 0000000099159181: 6-TFS probes 0000000000000000

  Completed searching 0000000000000000  nodes through DEPTHS 09-14:     LEAF NODES: 0000000000018531 cut BRANCHES: 0000000036078657 pruned
  TFS Database probes 0000000105060036: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223766
  TFS Database misses 0000000121593540: 4-TFS probes 0000000005669000: 5-TFS probes 0000000099159181: 6-TFS probes 0000000000000000

  Completed searching 0000000000000822  nodes through DEPTHS 10-15:     LEAF NODES: 0000000000345645 cut BRANCHES: 0000001054370361 pruned
  TFS Database probes 0000000105063957: 1-TFS probes 0000000000000251: 2-TFS probes 0000000000007838: 3-TFS probes 0000000000223774
  TFS Database misses 0000000121598034: 4-TFS probes 0000000005669198: 5-TFS probes 0000000099162896: 6-TFS probes 0000000000000000

 ?Solution [001] =  L2 U' F' f' R2 B' U2 u2 d2 R' U' [inside 4-TFS] ---> U  R  D  R'  @ 0000000009908 nodes with 0000005671622 4-TFS probes          e!
  Solution [002] =  L2 U' F' f' R2 B' U2 u2 d2 R' U2 [inside 4-TFS] ---> U2 R  D  R'  @ 0000000009909 nodes with 0000005671623 4-TFS probes
  Solution [003] =  L2 U' F' f' R2 B' U2 u2 d2 R' u  [inside 4-TFS] ---> u' R  D  R'  @ 0000000009910 nodes with 0000005671624 4-TFS probes
  Solution [004] =  L2 U' F' f' R2 B' U2 u2 d2 R' u' [inside 4-TFS] ---> u  R  D  R'  @ 0000000009911 nodes with 0000005671625 4-TFS probes
  Solution [005] =  L2 U' F' f' R2 B' U2 u2 d2 R' u2 [inside 4-TFS] ---> u2 R  D  R'  @ 0000000009912 nodes with 0000005671626 4-TFS probes
  Solution [006] =  L2 U' F' f' R2 B' U2 u2 d2 R' d  [inside 4-TFS] ---> d' R  D  R'  @ 0000000009913 nodes with 0000005671627 4-TFS probes
  Solution [007] =  L2 U' F' f' R2 B' U2 u2 d2 R' d' [inside 4-TFS] ---> d  R  D  R'  @ 0000000009914 nodes with 0000005671628 4-TFS probes
  Solution [008] =  L2 U' F' f' R2 B' U2 u2 d2 R' d2 [inside 4-TFS] ---> d2 R  D  R'  @ 0000000009915 nodes with 0000005671629 4-TFS probes
  Solution [009] =  L2 U' F' f' R2 B' U2 u2 d2 R' D  [inside 4-TFS] ---> D' R  D  R'  @ 0000000009916 nodes with 0000005671630 4-TFS probes
  Solution [010] =  L2 U' F' f' R2 B' U2 u2 d2 R' D' [inside 4-TFS] ---> D  R  D  R'  @ 0000000009917 nodes with 0000005671631 4-TFS probes
  Solution [011] =  L2 U' F' f' R2 B' U2 u2 d2 R' D2 [inside 4-TFS] ---> D2 R  D  R'  @ 0000000009918 nodes with 0000005671632 4-TFS probes
  Solution [012] =  L2 U' F' f' R2 B' U2 u2 d2 R' r  [inside 4-TFS] ---> r' R  D  R'  @ 0000000009919 nodes with 0000005671633 4-TFS probes
  Solution [013] =  L2 U' F' f' R2 B' U2 u2 d2 R' r' [inside 4-TFS] ---> r  R  D  R'  @ 0000000009920 nodes with 0000005671634 4-TFS probes
  Solution [014] =  L2 U' F' f' R2 B' U2 u2 d2 R' r2 [inside 4-TFS] ---> r2 R  D  R'  @ 0000000009921 nodes with 0000005671635 4-TFS probes
  Solution [015] =  L2 U' F' f' R2 B' U2 u2 d2 R' l  [inside 4-TFS] ---> l' R  D  R'  @ 0000000009922 nodes with 0000005671636 4-TFS probes
  Solution [016] =  L2 U' F' f' R2 B' U2 u2 d2 R' l' [inside 4-TFS] ---> l  R  D  R'  @ 0000000009923 nodes with 0000005671637 4-TFS probes
  Solution [017] =  L2 U' F' f' R2 B' U2 u2 d2 R' l2 [inside 4-TFS] ---> l2 R  D  R'  @ 0000000009924 nodes with 0000005671638 4-TFS probes
  Solution [018] =  L2 U' F' f' R2 B' U2 u2 d2 R' L  [inside 4-TFS] ---> L' R  D  R'  @ 0000000009925 nodes with 0000005671639 4-TFS probes
  Solution [019] =  L2 U' F' f' R2 B' U2 u2 d2 R' L' [inside 4-TFS] ---> L  R  D  R'  @ 0000000009926 nodes with 0000005671640 4-TFS probes
  Solution [020] =  L2 U' F' f' R2 B' U2 u2 d2 R' L2 [inside 4-TFS] ---> L2 R  D  R'  @ 0000000009927 nodes with 0000005671641 4-TFS probes
  Solution [021] =  L2 U' F' f' R2 B' U2 u2 d2 R' F  [inside 4-TFS] ---> F' R  D  R'  @ 0000000009928 nodes with 0000005671642 4-TFS probes
  Solution [022] =  L2 U' F' f' R2 B' U2 u2 d2 R' F' [inside 4-TFS] ---> F  R  D  R'  @ 0000000009929 nodes with 0000005671643 4-TFS probes
  Solution [023] =  L2 U' F' f' R2 B' U2 u2 d2 R' F2 [inside 4-TFS] ---> F2 R  D  R'  @ 0000000009930 nodes with 0000005671644 4-TFS probes
  Solution [024] =  L2 U' F' f' R2 B' U2 u2 d2 R' f  [inside 4-TFS] ---> f' R  D  R'  @ 0000000009931 nodes with 0000005671645 4-TFS probes
  Solution [025] =  L2 U' F' f' R2 B' U2 u2 d2 R' f' [inside 4-TFS] ---> f  R  D  R'  @ 0000000009932 nodes with 0000005671646 4-TFS probes
  Solution [026] =  L2 U' F' f' R2 B' U2 u2 d2 R' f2 [inside 4-TFS] ---> f2 R  D  R'  @ 0000000009933 nodes with 0000005671647 4-TFS probes
  Solution [027] =  L2 U' F' f' R2 B' U2 u2 d2 R' b  [inside 4-TFS] ---> b' R  D  R'  @ 0000000009934 nodes with 0000005671649 4-TFS probes
  Solution [028] =  L2 U' F' f' R2 B' U2 u2 d2 R' b' [inside 4-TFS] ---> b  R  D  R'  @ 0000000009935 nodes with 0000005671650 4-TFS probes
  Solution [029] =  L2 U' F' f' R2 B' U2 u2 d2 R' b2 [inside 4-TFS] ---> b2 R  D  R'  @ 0000000009936 nodes with 0000005671651 4-TFS probes
  Solution [030] =  L2 U' F' f' R2 B' U2 u2 d2 R' B  [inside 4-TFS] ---> B' R  D  R'  @ 0000000009937 nodes with 0000005671652 4-TFS probes
  Solution [031] =  L2 U' F' f' R2 B' U2 u2 d2 R' B' [inside 4-TFS] ---> B  R  D  R'  @ 0000000009938 nodes with 0000005671653 4-TFS probes
  Solution [032] =  L2 U' F' f' R2 B' U2 u2 d2 R' B2 [inside 4-TFS] ---> B2 R  D  R'  @ 0000000009939 nodes with 0000005671654 4-TFS probes
  Solution [033] =  L2 U' F' f' R2 B' U2 u2 D  R  U  [inside 4-TFS] ---> U' R' d2 R'  @ 0000000009940 nodes with 0000005671655 4-TFS probes
  Solution [034] =  L2 U' F' f' R2 B' U2 u2 D  R  U' [inside 4-TFS] ---> U  R' d2 R'  @ 0000000009941 nodes with 0000005671656 4-TFS probes
  Solution [035] =  L2 U' F' f' R2 B' U2 u2 D  R  U2 [inside 4-TFS] ---> U2 R' d2 R'  @ 0000000009942 nodes with 0000005671657 4-TFS probes
  Solution [036] =  L2 U' F' f' R2 B' U2 u2 D  R  u  [inside 4-TFS] ---> u' R' d2 R'  @ 0000000009943 nodes with 0000005671658 4-TFS probes
  Solution [037] =  L2 U' F' f' R2 B' U2 u2 D  R  u' [inside 4-TFS] ---> u  R' d2 R'  @ 0000000009944 nodes with 0000005671659 4-TFS probes
  Solution [038] =  L2 U' F' f' R2 B' U2 u2 D  R  u2 [inside 4-TFS] ---> u2 R' d2 R'  @ 0000000009945 nodes with 0000005671660 4-TFS probes
  Solution [039] =  L2 U' F' f' R2 B' U2 u2 D  R  d  [inside 4-TFS] ---> d' R' d2 R'  @ 0000000009946 nodes with 0000005671661 4-TFS probes
  Solution [040] =  L2 U' F' f' R2 B' U2 u2 D  R  d' [inside 4-TFS] ---> d  R' d2 R'  @ 0000000009947 nodes with 0000005671662 4-TFS probes
  Solution [041] =  L2 U' F' f' R2 B' U2 u2 D  R  d2 [inside 4-TFS] ---> d2 R' d2 R'  @ 0000000009948 nodes with 0000005671663 4-TFS probes
  Solution [042] =  L2 U' F' f' R2 B' U2 u2 D  R  D  [inside 4-TFS] ---> D' R' d2 R'  @ 0000000009949 nodes with 0000005671664 4-TFS probes
  Solution [043] =  L2 U' F' f' R2 B' U2 u2 D  R  D' [inside 4-TFS] ---> D  R' d2 R'  @ 0000000009950 nodes with 0000005671665 4-TFS probes
  Solution [044] =  L2 U' F' f' R2 B' U2 u2 D  R  D2 [inside 4-TFS] ---> D2 R' d2 R'  @ 0000000009951 nodes with 0000005671667 4-TFS probes
  Solution [045] =  L2 U' F' f' R2 B' U2 u2 D  R  r  [inside 4-TFS] ---> r' R' d2 R'  @ 0000000009952 nodes with 0000005671668 4-TFS probes
  Solution [046] =  L2 U' F' f' R2 B' U2 u2 D  R  r' [inside 4-TFS] ---> r  R' d2 R'  @ 0000000009953 nodes with 0000005671669 4-TFS probes
  Solution [047] =  L2 U' F' f' R2 B' U2 u2 D  R  r2 [inside 4-TFS] ---> r2 R' d2 R'  @ 0000000009954 nodes with 0000005671670 4-TFS probes
  Solution [048] =  L2 U' F' f' R2 B' U2 u2 D  R  l  [inside 4-TFS] ---> l' R' d2 R'  @ 0000000009955 nodes with 0000005671671 4-TFS probes
  Solution [049] =  L2 U' F' f' R2 B' U2 u2 D  R  l' [inside 4-TFS] ---> l  R' d2 R'  @ 0000000009956 nodes with 0000005671672 4-TFS probes
  Solution [050] =  L2 U' F' f' R2 B' U2 u2 D  R  l2 [inside 4-TFS] ---> l2 R' d2 R'  @ 0000000009957 nodes with 0000005671673 4-TFS probes
  Solution [051] =  L2 U' F' f' R2 B' U2 u2 D  R  L  [inside 4-TFS] ---> L' R' d2 R'  @ 0000000009958 nodes with 0000005671674 4-TFS probes
  Solution [052] =  L2 U' F' f' R2 B' U2 u2 D  R  L' [inside 4-TFS] ---> L  R' d2 R'  @ 0000000009959 nodes with 0000005671675 4-TFS probes
  Solution [053] =  L2 U' F' f' R2 B' U2 u2 D  R  L2 [inside 4-TFS] ---> L2 R' d2 R'  @ 0000000009960 nodes with 0000005671676 4-TFS probes
  Solution [054] =  L2 U' F' f' R2 B' U2 u2 D  R  F  [inside 4-TFS] ---> F' R' d2 R'  @ 0000000009961 nodes with 0000005671677 4-TFS probes
  Solution [055] =  L2 U' F' f' R2 B' U2 u2 D  R  F' [inside 4-TFS] ---> F  R' d2 R'  @ 0000000009962 nodes with 0000005671678 4-TFS probes
  Solution [056] =  L2 U' F' f' R2 B' U2 u2 D  R  F2 [inside 4-TFS] ---> F2 R' d2 R'  @ 0000000009963 nodes with 0000005671679 4-TFS probes
  Solution [057] =  L2 U' F' f' R2 B' U2 u2 D  R  f  [inside 4-TFS] ---> f' R' d2 R'  @ 0000000009964 nodes with 0000005671680 4-TFS probes
  Solution [058] =  L2 U' F' f' R2 B' U2 u2 D  R  f' [inside 4-TFS] ---> f  R' d2 R'  @ 0000000009965 nodes with 0000005671681 4-TFS probes
  Solution [059] =  L2 U' F' f' R2 B' U2 u2 D  R  f2 [inside 4-TFS] ---> f2 R' d2 R'  @ 0000000009966 nodes with 0000005671682 4-TFS probes
  Solution [060] =  L2 U' F' f' R2 B' U2 u2 D  R  b  [inside 4-TFS] ---> b' R' d2 R'  @ 0000000009967 nodes with 0000005671683 4-TFS probes
  Solution [061] =  L2 U' F' f' R2 B' U2 u2 D  R  b' [inside 4-TFS] ---> b  R' d2 R'  @ 0000000009968 nodes with 0000005671684 4-TFS probes
  Solution [062] =  L2 U' F' f' R2 B' U2 u2 D  R  b2 [inside 4-TFS] ---> b2 R' d2 R'  @ 0000000009969 nodes with 0000005671686 4-TFS probes
  Solution [063] =  L2 U' F' f' R2 B' U2 u2 D  R  B  [inside 4-TFS] ---> B' R' d2 R'  @ 0000000009970 nodes with 0000005671687 4-TFS probes
  Solution [064] =  L2 U' F' f' R2 B' U2 u2 D  R  B' [inside 4-TFS] ---> B  R' d2 R'  @ 0000000009971 nodes with 0000005671688 4-TFS probes
  Solution [065] =  L2 U' F' f' R2 B' U2 u2 D  R  B2 [inside 4-TFS] ---> B2 R' d2 R'  @ 0000000009972 nodes with 0000005671689 4-TFS probes
  Solution [066] =  L2 U' F' f' R2 B' U2 u2 D  R' U  [inside 4-TFS] ---> U' R  d2 R'  @ 0000000009973 nodes with 0000005671690 4-TFS probes
  Solution [067] =  L2 U' F' f' R2 B' U2 u2 D  R' U' [inside 4-TFS] ---> U  R  d2 R'  @ 0000000009974 nodes with 0000005671691 4-TFS probes
  Solution [068] =  L2 U' F' f' R2 B' U2 u2 D  R' U2 [inside 4-TFS] ---> U2 R  d2 R'  @ 0000000009975 nodes with 0000005671692 4-TFS probes
  Solution [069] =  L2 U' F' f' R2 B' U2 u2 D  R' u  [inside 4-TFS] ---> u' R  d2 R'  @ 0000000009976 nodes with 0000005671693 4-TFS probes
  Solution [070] =  L2 U' F' f' R2 B' U2 u2 D  R' u' [inside 4-TFS] ---> u  R  d2 R'  @ 0000000009977 nodes with 0000005671694 4-TFS probes
  Solution [071] =  L2 U' F' f' R2 B' U2 u2 D  R' u2 [inside 4-TFS] ---> u2 R  d2 R'  @ 0000000009978 nodes with 0000005671695 4-TFS probes
  Solution [072] =  L2 U' F' f' R2 B' U2 u2 D  R' d  [inside 4-TFS] ---> d' R  d2 R'  @ 0000000009979 nodes with 0000005671696 4-TFS probes
  Solution [073] =  L2 U' F' f' R2 B' U2 u2 D  R' d' [inside 4-TFS] ---> d  R  d2 R'  @ 0000000009980 nodes with 0000005671697 4-TFS probes
  Solution [074] =  L2 U' F' f' R2 B' U2 u2 D  R' d2 [inside 4-TFS] ---> d2 R  d2 R'  @ 0000000009981 nodes with 0000005671698 4-TFS probes
  Solution [075] =  L2 U' F' f' R2 B' U2 u2 D  R' D  [inside 4-TFS] ---> D' R  d2 R'  @ 0000000009982 nodes with 0000005671699 4-TFS probes
  Solution [076] =  L2 U' F' f' R2 B' U2 u2 D  R' D' [inside 4-TFS] ---> D  R  d2 R'  @ 0000000009983 nodes with 0000005671700 4-TFS probes
  Solution [077] =  L2 U' F' f' R2 B' U2 u2 D  R' D2 [inside 4-TFS] ---> D2 R  d2 R'  @ 0000000009984 nodes with 0000005671702 4-TFS probes
  Solution [078] =  L2 U' F' f' R2 B' U2 u2 D  R' r  [inside 4-TFS] ---> r' R  d2 R'  @ 0000000009985 nodes with 0000005671703 4-TFS probes
  Solution [079] =  L2 U' F' f' R2 B' U2 u2 D  R' r' [inside 4-TFS] ---> r  R  d2 R'  @ 0000000009986 nodes with 0000005671704 4-TFS probes
  Solution [080] =  L2 U' F' f' R2 B' U2 u2 D  R' r2 [inside 4-TFS] ---> r2 R  d2 R'  @ 0000000009987 nodes with 0000005671705 4-TFS probes
  Solution [081] =  L2 U' F' f' R2 B' U2 u2 D  R' l  [inside 4-TFS] ---> l' R  d2 R'  @ 0000000009988 nodes with 0000005671706 4-TFS probes
  Solution [082] =  L2 U' F' f' R2 B' U2 u2 D  R' l' [inside 4-TFS] ---> l  R  d2 R'  @ 0000000009989 nodes with 0000005671707 4-TFS probes
  Solution [083] =  L2 U' F' f' R2 B' U2 u2 D  R' l2 [inside 4-TFS] ---> l2 R  d2 R'  @ 0000000009990 nodes with 0000005671708 4-TFS probes
  Solution [084] =  L2 U' F' f' R2 B' U2 u2 D  R' L  [inside 4-TFS] ---> L' R  d2 R'  @ 0000000009991 nodes with 0000005671709 4-TFS probes
  Solution [085] =  L2 U' F' f' R2 B' U2 u2 D  R' L' [inside 4-TFS] ---> L  R  d2 R'  @ 0000000009992 nodes with 0000005671710 4-TFS probes
  Solution [086] =  L2 U' F' f' R2 B' U2 u2 D  R' L2 [inside 4-TFS] ---> L2 R  d2 R'  @ 0000000009993 nodes with 0000005671711 4-TFS probes
  Solution [087] =  L2 U' F' f' R2 B' U2 u2 D  R' F  [inside 4-TFS] ---> F' R  d2 R'  @ 0000000009994 nodes with 0000005671712 4-TFS probes
  Solution [088] =  L2 U' F' f' R2 B' U2 u2 D  R' F' [inside 4-TFS] ---> F  R  d2 R'  @ 0000000009995 nodes with 0000005671713 4-TFS probes
  Solution [089] =  L2 U' F' f' R2 B' U2 u2 D  R' F2 [inside 4-TFS] ---> F2 R  d2 R'  @ 0000000009996 nodes with 0000005671714 4-TFS probes
  Solution [090] =  L2 U' F' f' R2 B' U2 u2 D  R' f  [inside 4-TFS] ---> f' R  d2 R'  @ 0000000009997 nodes with 0000005671715 4-TFS probes
  Solution [091] =  L2 U' F' f' R2 B' U2 u2 D  R' f' [inside 4-TFS] ---> f  R  d2 R'  @ 0000000009998 nodes with 0000005671716 4-TFS probes
  Solution [092] =  L2 U' F' f' R2 B' U2 u2 D  R' f2 [inside 4-TFS] ---> f2 R  d2 R'  @ 0000000009999 nodes with 0000005671717 4-TFS probes
  Solution [093] =  L2 U' F' f' R2 B' U2 u2 D  R' b  [inside 4-TFS] ---> b' R  d2 R'  @ 0000000010000 nodes with 0000005671718 4-TFS probes
  Solution [094] =  L2 U' F' f' R2 B' U2 u2 D  R' b' [inside 4-TFS] ---> b  R  d2 R'  @ 0000000010001 nodes with 0000005671719 4-TFS probes
  Solution [095] =  L2 U' F' f' R2 B' U2 u2 D  R' b2 [inside 4-TFS] ---> b2 R  d2 R'  @ 0000000010002 nodes with 0000005671720 4-TFS probes
  Solution [096] =  L2 U' F' f' R2 B' U2 u2 D  R' B  [inside 4-TFS] ---> B' R  d2 R'  @ 0000000010003 nodes with 0000005671721 4-TFS probes
  Solution [097] =  L2 U' F' f' R2 B' U2 u2 D  R' B' [inside 4-TFS] ---> B  R  d2 R'  @ 0000000010004 nodes with 0000005671722 4-TFS probes
  Solution [098] =  L2 U' F' f' R2 B' U2 u2 D  R' B2 [inside 4-TFS] ---> B2 R  d2 R'  @ 0000000010005 nodes with 0000005671723 4-TFS probes
  Solution [099] =  L2 U' F' f' R2 B' U2 u2 D  R2 u  [inside 4-TFS] ---> u' R2 d2 R'  @ 0000000010009 nodes with 0000005671724 4-TFS probes
  Solution [100] =  L2 U' F' f' R2 B' U2 u2 D  R2 u' [inside 4-TFS] ---> u  R2 d2 R'  @ 0000000010010 nodes with 0000005671725 4-TFS probes
  Solution [101] =  L2 U' F' f' R2 B' U2 u2 D  R2 u2 [inside 4-TFS] ---> u2 R2 d2 R'  @ 0000000010011 nodes with 0000005671726 4-TFS probes
  Solution [102] =  L2 U' F' f' R2 B' U2 u2 D  R2 d  [inside 4-TFS] ---> d' R2 d2 R'  @ 0000000010012 nodes with 0000005671727 4-TFS probes
  Solution [103] =  L2 U' F' f' R2 B' U2 u2 D  R2 d' [inside 4-TFS] ---> d  R2 d2 R'  @ 0000000010013 nodes with 0000005671728 4-TFS probes
  Solution [104] =  L2 U' F' f' R2 B' U2 u2 D  R2 d2 [inside 4-TFS] ---> d2 R2 d2 R'  @ 0000000010014 nodes with 0000005671729 4-TFS probes
  Solution [105] =  L2 U' F' f' R2 B' U2 u2 D  R2 r  [inside 4-TFS] ---> r' R2 d2 R'  @ 0000000010018 nodes with 0000005671730 4-TFS probes
  Solution [106] =  L2 U' F' f' R2 B' U2 u2 D  R2 r' [inside 4-TFS] ---> r  R2 d2 R'  @ 0000000010019 nodes with 0000005671731 4-TFS probes
  Solution [107] =  L2 U' F' f' R2 B' U2 u2 D  R2 r2 [inside 4-TFS] ---> r2 R2 d2 R'  @ 0000000010020 nodes with 0000005671732 4-TFS probes
  Solution [108] =  L2 U' F' f' R2 B' U2 u2 D  R2 l  [inside 4-TFS] ---> l' R2 d2 R'  @ 0000000010021 nodes with 0000005671733 4-TFS probes
  Solution [109] =  L2 U' F' f' R2 B' U2 u2 D  R2 l' [inside 4-TFS] ---> l  R2 d2 R'  @ 0000000010022 nodes with 0000005671735 4-TFS probes
  Solution [110] =  L2 U' F' f' R2 B' U2 u2 D  R2 l2 [inside 4-TFS] ---> l2 R2 d2 R'  @ 0000000010023 nodes with 0000005671736 4-TFS probes
  Solution [111] =  L2 U' F' f' R2 B' U2 u2 D  R2 L  [inside 4-TFS] ---> L' R2 d2 R'  @ 0000000010024 nodes with 0000005671737 4-TFS probes
  Solution [112] =  L2 U' F' f' R2 B' U2 u2 D  R2 L' [inside 4-TFS] ---> L  R2 d2 R'  @ 0000000010025 nodes with 0000005671738 4-TFS probes
  Solution [113] =  L2 U' F' f' R2 B' U2 u2 D  R2 L2 [inside 4-TFS] ---> L2 R2 d2 R'  @ 0000000010026 nodes with 0000005671740 4-TFS probes
  Solution [114] =  L2 U' F' f' R2 B' U2 u2 D  R2 F  [inside 4-TFS] ---> F' R2 d2 R'  @ 0000000010027 nodes with 0000005671741 4-TFS probes
  Solution [115] =  L2 U' F' f' R2 B' U2 u2 D  R2 F' [inside 4-TFS] ---> F  R2 d2 R'  @ 0000000010028 nodes with 0000005671742 4-TFS probes
  Solution [116] =  L2 U' F' f' R2 B' U2 u2 D  R2 F2 [inside 4-TFS] ---> F2 R2 d2 R'  @ 0000000010029 nodes with 0000005671743 4-TFS probes
  Solution [117] =  L2 U' F' f' R2 B' U2 u2 D  R2 f  [inside 4-TFS] ---> f' R2 d2 R'  @ 0000000010030 nodes with 0000005671744 4-TFS probes
  Solution [118] =  L2 U' F' f' R2 B' U2 u2 D  R2 f' [inside 4-TFS] ---> f  R2 d2 R'  @ 0000000010031 nodes with 0000005671745 4-TFS probes
  Solution [119] =  L2 U' F' f' R2 B' U2 u2 D  R2 b  [inside 4-TFS] ---> b' R2 d2 R'  @ 0000000010033 nodes with 0000005671746 4-TFS probes
  Solution [120] =  L2 U' F' f' R2 B' U2 u2 D  R2 b' [inside 4-TFS] ---> b  R2 d2 R'  @ 0000000010034 nodes with 0000005671747 4-TFS probes
  Solution [121] =  L2 U' F' f' R2 B' U2 u2 D  R2 B  [inside 4-TFS] ---> B' R2 d2 R'  @ 0000000010036 nodes with 0000005671748 4-TFS probes
  Solution [122] =  L2 U' F' f' R2 B' U2 u2 D  R2 B' [inside 4-TFS] ---> B  R2 d2 R'  @ 0000000010037 nodes with 0000005671749 4-TFS probes
  Solution [123] =  L2 U' F' f' R2 B' U2 u2 D  R2 B2 [inside 4-TFS] ---> B2 R2 d2 R'  @ 0000000010038 nodes with 0000005671751 4-TFS probes
  Solution [124] =  L2 U' F' f' R2 B' U2 d2 D  R  U  [inside 4-TFS] ---> U' R' u2 R'  @ 0000000010039 nodes with 0000005671752 4-TFS probes
  Solution [125] =  L2 U' F' f' R2 B' U2 d2 D  R  U' [inside 4-TFS] ---> U  R' u2 R'  @ 0000000010040 nodes with 0000005671753 4-TFS probes
  Solution [126] =  L2 U' F' f' R2 B' U2 d2 D  R  U2 [inside 4-TFS] ---> U2 R' u2 R'  @ 0000000010041 nodes with 0000005671754 4-TFS probes
  Solution [127] =  L2 U' F' f' R2 B' U2 d2 D  R  u  [inside 4-TFS] ---> u' R' u2 R'  @ 0000000010042 nodes with 0000005671755 4-TFS probes
  Solution [128] =  L2 U' F' f' R2 B' U2 d2 D  R  u' [inside 4-TFS] ---> u  R' u2 R'  @ 0000000010043 nodes with 0000005671756 4-TFS probes
  Solution [129] =  L2 U' F' f' R2 B' U2 d2 D  R  u2 [inside 4-TFS] ---> u2 R' u2 R'  @ 0000000010044 nodes with 0000005671757 4-TFS probes
  Solution [130] =  L2 U' F' f' R2 B' U2 d2 D  R  d  [inside 4-TFS] ---> d' R' u2 R'  @ 0000000010045 nodes with 0000005671758 4-TFS probes
  Solution [131] =  L2 U' F' f' R2 B' U2 d2 D  R  d' [inside 4-TFS] ---> d  R' u2 R'  @ 0000000010046 nodes with 0000005671759 4-TFS probes
  Solution [132] =  L2 U' F' f' R2 B' U2 d2 D  R  d2 [inside 4-TFS] ---> d2 R' u2 R'  @ 0000000010047 nodes with 0000005671760 4-TFS probes
  Solution [133] =  L2 U' F' f' R2 B' U2 d2 D  R  D  [inside 4-TFS] ---> D' R' u2 R'  @ 0000000010048 nodes with 0000005671761 4-TFS probes
  Solution [134] =  L2 U' F' f' R2 B' U2 d2 D  R  D' [inside 4-TFS] ---> D  R' u2 R'  @ 0000000010049 nodes with 0000005671762 4-TFS probes
  Solution [135] =  L2 U' F' f' R2 B' U2 d2 D  R  D2 [inside 4-TFS] ---> D2 R' u2 R'  @ 0000000010050 nodes with 0000005671763 4-TFS probes
  Solution [136] =  L2 U' F' f' R2 B' U2 d2 D  R  r  [inside 4-TFS] ---> r' R' u2 R'  @ 0000000010051 nodes with 0000005671764 4-TFS probes
  Solution [137] =  L2 U' F' f' R2 B' U2 d2 D  R  r' [inside 4-TFS] ---> r  R' u2 R'  @ 0000000010052 nodes with 0000005671765 4-TFS probes
  Solution [138] =  L2 U' F' f' R2 B' U2 d2 D  R  r2 [inside 4-TFS] ---> r2 R' u2 R'  @ 0000000010053 nodes with 0000005671767 4-TFS probes
  Solution [139] =  L2 U' F' f' R2 B' U2 d2 D  R  l  [inside 4-TFS] ---> l' R' u2 R'  @ 0000000010054 nodes with 0000005671768 4-TFS probes
  Solution [140] =  L2 U' F' f' R2 B' U2 d2 D  R  l' [inside 4-TFS] ---> l  R' u2 R'  @ 0000000010055 nodes with 0000005671769 4-TFS probes
  Solution [141] =  L2 U' F' f' R2 B' U2 d2 D  R  l2 [inside 4-TFS] ---> l2 R' u2 R'  @ 0000000010056 nodes with 0000005671770 4-TFS probes
  Solution [142] =  L2 U' F' f' R2 B' U2 d2 D  R  L  [inside 4-TFS] ---> L' R' u2 R'  @ 0000000010057 nodes with 0000005671771 4-TFS probes
  Solution [143] =  L2 U' F' f' R2 B' U2 d2 D  R  L' [inside 4-TFS] ---> L  R' u2 R'  @ 0000000010058 nodes with 0000005671772 4-TFS probes
  Solution [144] =  L2 U' F' f' R2 B' U2 d2 D  R  L2 [inside 4-TFS] ---> L2 R' u2 R'  @ 0000000010059 nodes with 0000005671773 4-TFS probes
  Solution [145] =  L2 U' F' f' R2 B' U2 d2 D  R  F  [inside 4-TFS] ---> F' R' u2 R'  @ 0000000010060 nodes with 0000005671774 4-TFS probes
  Solution [146] =  L2 U' F' f' R2 B' U2 d2 D  R  F' [inside 4-TFS] ---> F  R' u2 R'  @ 0000000010061 nodes with 0000005671775 4-TFS probes
  Solution [147] =  L2 U' F' f' R2 B' U2 d2 D  R  F2 [inside 4-TFS] ---> F2 R' u2 R'  @ 0000000010062 nodes with 0000005671776 4-TFS probes
  Solution [148] =  L2 U' F' f' R2 B' U2 d2 D  R  f  [inside 4-TFS] ---> f' R' u2 R'  @ 0000000010063 nodes with 0000005671777 4-TFS probes
  Solution [149] =  L2 U' F' f' R2 B' U2 d2 D  R  f' [inside 4-TFS] ---> f  R' u2 R'  @ 0000000010064 nodes with 0000005671778 4-TFS probes
  Solution [150] =  L2 U' F' f' R2 B' U2 d2 D  R  f2 [inside 4-TFS] ---> f2 R' u2 R'  @ 0000000010065 nodes with 0000005671779 4-TFS probes
  Solution [151] =  L2 U' F' f' R2 B' U2 d2 D  R  b  [inside 4-TFS] ---> b' R' u2 R'  @ 0000000010066 nodes with 0000005671780 4-TFS probes
  Solution [152] =  L2 U' F' f' R2 B' U2 d2 D  R  b' [inside 4-TFS] ---> b  R' u2 R'  @ 0000000010067 nodes with 0000005671781 4-TFS probes
  Solution [153] =  L2 U' F' f' R2 B' U2 d2 D  R  b2 [inside 4-TFS] ---> b2 R' u2 R'  @ 0000000010068 nodes with 0000005671782 4-TFS probes
  Solution [154] =  L2 U' F' f' R2 B' U2 d2 D  R  B  [inside 4-TFS] ---> B' R' u2 R'  @ 0000000010069 nodes with 0000005671784 4-TFS probes
  Solution [155] =  L2 U' F' f' R2 B' U2 d2 D  R  B' [inside 4-TFS] ---> B  R' u2 R'  @ 0000000010070 nodes with 0000005671785 4-TFS probes
  Solution [156] =  L2 U' F' f' R2 B' U2 d2 D  R  B2 [inside 4-TFS] ---> B2 R' u2 R'  @ 0000000010071 nodes with 0000005671786 4-TFS probes
  Solution [157] =  L2 U' F' f' R2 B' U2 d2 D  R' U  [inside 4-TFS] ---> U' R  u2 R'  @ 0000000010072 nodes with 0000005671787 4-TFS probes
  Solution [158] =  L2 U' F' f' R2 B' U2 d2 D  R' U' [inside 4-TFS] ---> U  R  u2 R'  @ 0000000010073 nodes with 0000005671788 4-TFS probes
  Solution [159] =  L2 U' F' f' R2 B' U2 d2 D  R' U2 [inside 4-TFS] ---> U2 R  u2 R'  @ 0000000010074 nodes with 0000005671790 4-TFS probes
  Solution [160] =  L2 U' F' f' R2 B' U2 d2 D  R' u  [inside 4-TFS] ---> u' R  u2 R'  @ 0000000010075 nodes with 0000005671791 4-TFS probes
  Solution [161] =  L2 U' F' f' R2 B' U2 d2 D  R' u' [inside 4-TFS] ---> u  R  u2 R'  @ 0000000010076 nodes with 0000005671792 4-TFS probes
  Solution [162] =  L2 U' F' f' R2 B' U2 d2 D  R' u2 [inside 4-TFS] ---> u2 R  u2 R'  @ 0000000010077 nodes with 0000005671793 4-TFS probes
  Solution [163] =  L2 U' F' f' R2 B' U2 d2 D  R' d  [inside 4-TFS] ---> d' R  u2 R'  @ 0000000010078 nodes with 0000005671794 4-TFS probes
  Solution [164] =  L2 U' F' f' R2 B' U2 d2 D  R' d' [inside 4-TFS] ---> d  R  u2 R'  @ 0000000010079 nodes with 0000005671795 4-TFS probes
  Solution [165] =  L2 U' F' f' R2 B' U2 d2 D  R' d2 [inside 4-TFS] ---> d2 R  u2 R'  @ 0000000010080 nodes with 0000005671796 4-TFS probes
  Solution [166] =  L2 U' F' f' R2 B' U2 d2 D  R' D  [inside 4-TFS] ---> D' R  u2 R'  @ 0000000010081 nodes with 0000005671797 4-TFS probes
  Solution [167] =  L2 U' F' f' R2 B' U2 d2 D  R' D' [inside 4-TFS] ---> D  R  u2 R'  @ 0000000010082 nodes with 0000005671798 4-TFS probes
  Solution [168] =  L2 U' F' f' R2 B' U2 d2 D  R' D2 [inside 4-TFS] ---> D2 R  u2 R'  @ 0000000010083 nodes with 0000005671799 4-TFS probes
  Solution [169] =  L2 U' F' f' R2 B' U2 d2 D  R' r  [inside 4-TFS] ---> r' R  u2 R'  @ 0000000010084 nodes with 0000005671800 4-TFS probes
  Solution [170] =  L2 U' F' f' R2 B' U2 d2 D  R' r' [inside 4-TFS] ---> r  R  u2 R'  @ 0000000010085 nodes with 0000005671801 4-TFS probes
  Solution [171] =  L2 U' F' f' R2 B' U2 d2 D  R' r2 [inside 4-TFS] ---> r2 R  u2 R'  @ 0000000010086 nodes with 0000005671802 4-TFS probes
  Solution [172] =  L2 U' F' f' R2 B' U2 d2 D  R' l  [inside 4-TFS] ---> l' R  u2 R'  @ 0000000010087 nodes with 0000005671803 4-TFS probes
  Solution [173] =  L2 U' F' f' R2 B' U2 d2 D  R' l' [inside 4-TFS] ---> l  R  u2 R'  @ 0000000010088 nodes with 0000005671804 4-TFS probes
  Solution [174] =  L2 U' F' f' R2 B' U2 d2 D  R' l2 [inside 4-TFS] ---> l2 R  u2 R'  @ 0000000010089 nodes with 0000005671805 4-TFS probes
  Solution [175] =  L2 U' F' f' R2 B' U2 d2 D  R' L  [inside 4-TFS] ---> L' R  u2 R'  @ 0000000010090 nodes with 0000005671806 4-TFS probes
  Solution [176] =  L2 U' F' f' R2 B' U2 d2 D  R' L' [inside 4-TFS] ---> L  R  u2 R'  @ 0000000010091 nodes with 0000005671807 4-TFS probes
  Solution [177] =  L2 U' F' f' R2 B' U2 d2 D  R' L2 [inside 4-TFS] ---> L2 R  u2 R'  @ 0000000010092 nodes with 0000005671808 4-TFS probes
  Solution [178] =  L2 U' F' f' R2 B' U2 d2 D  R' F  [inside 4-TFS] ---> F' R  u2 R'  @ 0000000010093 nodes with 0000005671809 4-TFS probes
  Solution [179] =  L2 U' F' f' R2 B' U2 d2 D  R' F' [inside 4-TFS] ---> F  R  u2 R'  @ 0000000010094 nodes with 0000005671810 4-TFS probes
  Solution [180] =  L2 U' F' f' R2 B' U2 d2 D  R' F2 [inside 4-TFS] ---> F2 R  u2 R'  @ 0000000010095 nodes with 0000005671811 4-TFS probes
  Solution [181] =  L2 U' F' f' R2 B' U2 d2 D  R' f  [inside 4-TFS] ---> f' R  u2 R'  @ 0000000010096 nodes with 0000005671812 4-TFS probes
  Solution [182] =  L2 U' F' f' R2 B' U2 d2 D  R' f' [inside 4-TFS] ---> f  R  u2 R'  @ 0000000010097 nodes with 0000005671813 4-TFS probes
  Solution [183] =  L2 U' F' f' R2 B' U2 d2 D  R' f2 [inside 4-TFS] ---> f2 R  u2 R'  @ 0000000010098 nodes with 0000005671814 4-TFS probes
  Solution [184] =  L2 U' F' f' R2 B' U2 d2 D  R' b  [inside 4-TFS] ---> b' R  u2 R'  @ 0000000010099 nodes with 0000005671815 4-TFS probes
  Solution [185] =  L2 U' F' f' R2 B' U2 d2 D  R' b' [inside 4-TFS] ---> b  R  u2 R'  @ 0000000010100 nodes with 0000005671816 4-TFS probes
  Solution [186] =  L2 U' F' f' R2 B' U2 d2 D  R' b2 [inside 4-TFS] ---> b2 R  u2 R'  @ 0000000010101 nodes with 0000005671817 4-TFS probes
  Solution [187] =  L2 U' F' f' R2 B' U2 d2 D  R' B  [inside 4-TFS] ---> B' R  u2 R'  @ 0000000010102 nodes with 0000005671818 4-TFS probes
  Solution [188] =  L2 U' F' f' R2 B' U2 d2 D  R' B' [inside 4-TFS] ---> B  R  u2 R'  @ 0000000010103 nodes with 0000005671819 4-TFS probes
  Solution [189] =  L2 U' F' f' R2 B' U2 d2 D  R' B2 [inside 4-TFS] ---> B2 R  u2 R'  @ 0000000010104 nodes with 0000005671820 4-TFS probes
  Solution [190] =  L2 U' F' f' R2 B' U2 d2 D  R2 u  [inside 4-TFS] ---> u' R2 u2 R'  @ 0000000010108 nodes with 0000005671821 4-TFS probes
  Solution [191] =  L2 U' F' f' R2 B' U2 d2 D  R2 u' [inside 4-TFS] ---> u  R2 u2 R'  @ 0000000010109 nodes with 0000005671822 4-TFS probes
  Solution [192] =  L2 U' F' f' R2 B' U2 d2 D  R2 u2 [inside 4-TFS] ---> u2 R2 u2 R'  @ 0000000010110 nodes with 0000005671823 4-TFS probes
  Solution [193] =  L2 U' F' f' R2 B' U2 d2 D  R2 d  [inside 4-TFS] ---> d' R2 u2 R'  @ 0000000010111 nodes with 0000005671824 4-TFS probes
  Solution [194] =  L2 U' F' f' R2 B' U2 d2 D  R2 d' [inside 4-TFS] ---> d  R2 u2 R'  @ 0000000010112 nodes with 0000005671825 4-TFS probes
  Solution [195] =  L2 U' F' f' R2 B' U2 d2 D  R2 d2 [inside 4-TFS] ---> d2 R2 u2 R'  @ 0000000010113 nodes with 0000005671826 4-TFS probes
  Solution [196] =  L2 U' F' f' R2 B' U2 d2 D  R2 r  [inside 4-TFS] ---> r' R2 u2 R'  @ 0000000010117 nodes with 0000005671827 4-TFS probes
  Solution [197] =  L2 U' F' f' R2 B' U2 d2 D  R2 r' [inside 4-TFS] ---> r  R2 u2 R'  @ 0000000010118 nodes with 0000005671828 4-TFS probes
  Solution [198] =  L2 U' F' f' R2 B' U2 d2 D  R2 r2 [inside 4-TFS] ---> r2 R2 u2 R'  @ 0000000010119 nodes with 0000005671829 4-TFS probes
  Solution [199] =  L2 U' F' f' R2 B' U2 d2 D  R2 l  [inside 4-TFS] ---> l' R2 u2 R'  @ 0000000010120 nodes with 0000005671830 4-TFS probes
  Solution [200] =  L2 U' F' f' R2 B' U2 d2 D  R2 l' [inside 4-TFS] ---> l  R2 u2 R'  @ 0000000010121 nodes with 0000005671832 4-TFS probes
  Solution [201] =  L2 U' F' f' R2 B' U2 d2 D  R2 l2 [inside 4-TFS] ---> l2 R2 u2 R'  @ 0000000010122 nodes with 0000005671833 4-TFS probes
  Solution [202] =  L2 U' F' f' R2 B' U2 d2 D  R2 L  [inside 4-TFS] ---> L' R2 u2 R'  @ 0000000010123 nodes with 0000005671834 4-TFS probes
  Solution [203] =  L2 U' F' f' R2 B' U2 d2 D  R2 L' [inside 4-TFS] ---> L  R2 u2 R'  @ 0000000010124 nodes with 0000005671835 4-TFS probes
  Solution [204] =  L2 U' F' f' R2 B' U2 d2 D  R2 L2 [inside 4-TFS] ---> L2 R2 u2 R'  @ 0000000010125 nodes with 0000005671836 4-TFS probes
  Solution [205] =  L2 U' F' f' R2 B' U2 d2 D  R2 F  [inside 4-TFS] ---> F' R2 u2 R'  @ 0000000010126 nodes with 0000005671837 4-TFS probes
  Solution [206] =  L2 U' F' f' R2 B' U2 d2 D  R2 F' [inside 4-TFS] ---> F  R2 u2 R'  @ 0000000010127 nodes with 0000005671840 4-TFS probes
  Solution [207] =  L2 U' F' f' R2 B' U2 d2 D  R2 F2 [inside 4-TFS] ---> F2 R2 u2 R'  @ 0000000010128 nodes with 0000005671841 4-TFS probes
  Solution [208] =  L2 U' F' f' R2 B' U2 d2 D  R2 f  [inside 4-TFS] ---> f' R2 u2 R'  @ 0000000010129 nodes with 0000005671842 4-TFS probes
  Solution [209] =  L2 U' F' f' R2 B' U2 d2 D  R2 f' [inside 4-TFS] ---> f  R2 u2 R'  @ 0000000010130 nodes with 0000005671843 4-TFS probes
  Solution [210] =  L2 U' F' f' R2 B' U2 d2 D  R2 b  [inside 4-TFS] ---> b' R2 u2 R'  @ 0000000010132 nodes with 0000005671844 4-TFS probes
  Solution [211] =  L2 U' F' f' R2 B' U2 d2 D  R2 b' [inside 4-TFS] ---> b  R2 u2 R'  @ 0000000010133 nodes with 0000005671845 4-TFS probes
  Solution [212] =  L2 U' F' f' R2 B' U2 d2 D  R2 B  [inside 4-TFS] ---> B' R2 u2 R'  @ 0000000010135 nodes with 0000005671846 4-TFS probes
  Solution [213] =  L2 U' F' f' R2 B' U2 d2 D  R2 B' [inside 4-TFS] ---> B  R2 u2 R'  @ 0000000010136 nodes with 0000005671847 4-TFS probes
  Solution [214] =  L2 U' F' f' R2 B' U2 d2 D  R2 B2 [inside 4-TFS] ---> B2 R2 u2 R'  @ 0000000010137 nodes with 0000005671848 4-TFS probes
  Solution [215] =  L2 U' F' f' R2 B' u2 d2 D  R' U  [inside 4-TFS] ---> U' R  U2 R'  @ 0000000010252 nodes with 0000005671876 4-TFS probes
  Solution [216] =  L2 U' F' f' R2 B' u2 d2 D  R' U' [inside 4-TFS] ---> U  R  U2 R'  @ 0000000010253 nodes with 0000005671877 4-TFS probes
  Solution [217] =  L2 U' F' f' R2 B' u2 d2 D  R' U2 [inside 4-TFS] ---> U2 R  U2 R'  @ 0000000010254 nodes with 0000005671878 4-TFS probes
  Solution [218] =  L2 U' F' f' R2 B' u2 d2 D  R' u  [inside 4-TFS] ---> u' R  U2 R'  @ 0000000010255 nodes with 0000005671879 4-TFS probes
  Solution [219] =  L2 U' F' f' R2 B' u2 d2 D  R' u' [inside 4-TFS] ---> u  R  U2 R'  @ 0000000010256 nodes with 0000005671880 4-TFS probes
  Solution [220] =  L2 U' F' f' R2 B' u2 d2 D  R' u2 [inside 4-TFS] ---> u2 R  U2 R'  @ 0000000010257 nodes with 0000005671881 4-TFS probes
  Solution [221] =  L2 U' F' f' R2 B' u2 d2 D  R' d  [inside 4-TFS] ---> d' R  U2 R'  @ 0000000010258 nodes with 0000005671882 4-TFS probes
  Solution [222] =  L2 U' F' f' R2 B' u2 d2 D  R' d' [inside 4-TFS] ---> d  R  U2 R'  @ 0000000010259 nodes with 0000005671883 4-TFS probes
  Solution [223] =  L2 U' F' f' R2 B' u2 d2 D  R' d2 [inside 4-TFS] ---> d2 R  U2 R'  @ 0000000010260 nodes with 0000005671884 4-TFS probes
  Solution [224] =  L2 U' F' f' R2 B' u2 d2 D  R' D  [inside 4-TFS] ---> D' R  U2 R'  @ 0000000010261 nodes with 0000005671885 4-TFS probes
  Solution [225] =  L2 U' F' f' R2 B' u2 d2 D  R' D' [inside 4-TFS] ---> D  R  U2 R'  @ 0000000010262 nodes with 0000005671886 4-TFS probes
  Solution [226] =  L2 U' F' f' R2 B' u2 d2 D  R' D2 [inside 4-TFS] ---> D2 R  U2 R'  @ 0000000010263 nodes with 0000005671887 4-TFS probes
  Solution [227] =  L2 U' F' f' R2 B' u2 d2 D  R' r  [inside 4-TFS] ---> r' R  U2 R'  @ 0000000010264 nodes with 0000005671888 4-TFS probes
  Solution [228] =  L2 U' F' f' R2 B' u2 d2 D  R' r' [inside 4-TFS] ---> r  R  U2 R'  @ 0000000010265 nodes with 0000005671889 4-TFS probes
  Solution [229] =  L2 U' F' f' R2 B' u2 d2 D  R' r2 [inside 4-TFS] ---> r2 R  U2 R'  @ 0000000010266 nodes with 0000005671890 4-TFS probes
  Solution [230] =  L2 U' F' f' R2 B' u2 d2 D  R' l  [inside 4-TFS] ---> l' R  U2 R'  @ 0000000010267 nodes with 0000005671891 4-TFS probes
  Solution [231] =  L2 U' F' f' R2 B' u2 d2 D  R' l' [inside 4-TFS] ---> l  R  U2 R'  @ 0000000010268 nodes with 0000005671892 4-TFS probes
  Solution [232] =  L2 U' F' f' R2 B' u2 d2 D  R' l2 [inside 4-TFS] ---> l2 R  U2 R'  @ 0000000010269 nodes with 0000005671893 4-TFS probes
  Solution [233] =  L2 U' F' f' R2 B' u2 d2 D  R' L  [inside 4-TFS] ---> L' R  U2 R'  @ 0000000010270 nodes with 0000005671894 4-TFS probes
  Solution [234] =  L2 U' F' f' R2 B' u2 d2 D  R' L' [inside 4-TFS] ---> L  R  U2 R'  @ 0000000010271 nodes with 0000005671895 4-TFS probes
  Solution [235] =  L2 U' F' f' R2 B' u2 d2 D  R' L2 [inside 4-TFS] ---> L2 R  U2 R'  @ 0000000010272 nodes with 0000005671896 4-TFS probes
  Solution [236] =  L2 U' F' f' R2 B' u2 d2 D  R' F  [inside 4-TFS] ---> F' R  U2 R'  @ 0000000010273 nodes with 0000005671897 4-TFS probes
  Solution [237] =  L2 U' F' f' R2 B' u2 d2 D  R' F' [inside 4-TFS] ---> F  R  U2 R'  @ 0000000010274 nodes with 0000005671898 4-TFS probes
  Solution [238] =  L2 U' F' f' R2 B' u2 d2 D  R' F2 [inside 4-TFS] ---> F2 R  U2 R'  @ 0000000010275 nodes with 0000005671899 4-TFS probes
  Solution [239] =  L2 U' F' f' R2 B' u2 d2 D  R' f  [inside 4-TFS] ---> f' R  U2 R'  @ 0000000010276 nodes with 0000005671900 4-TFS probes
  Solution [240] =  L2 U' F' f' R2 B' u2 d2 D  R' f' [inside 4-TFS] ---> f  R  U2 R'  @ 0000000010277 nodes with 0000005671901 4-TFS probes
  Solution [241] =  L2 U' F' f' R2 B' u2 d2 D  R' f2 [inside 4-TFS] ---> f2 R  U2 R'  @ 0000000010278 nodes with 0000005671902 4-TFS probes
  Solution [242] =  L2 U' F' f' R2 B' u2 d2 D  R' b  [inside 4-TFS] ---> b' R  U2 R'  @ 0000000010279 nodes with 0000005671903 4-TFS probes
  Solution [243] =  L2 U' F' f' R2 B' u2 d2 D  R' b' [inside 4-TFS] ---> b  R  U2 R'  @ 0000000010280 nodes with 0000005671904 4-TFS probes
  Solution [244] =  L2 U' F' f' R2 B' u2 d2 D  R' b2 [inside 4-TFS] ---> b2 R  U2 R'  @ 0000000010281 nodes with 0000005671905 4-TFS probes
  Solution [245] =  L2 U' F' f' R2 B' u2 d2 D  R' B  [inside 4-TFS] ---> B' R  U2 R'  @ 0000000010282 nodes with 0000005671906 4-TFS probes
  Solution [246] =  L2 U' F' f' R2 B' u2 d2 D  R' B' [inside 4-TFS] ---> B  R  U2 R'  @ 0000000010283 nodes with 0000005671907 4-TFS probes
  Solution [247] =  L2 U' F' f' R2 B' u2 d2 D  R' B2 [inside 4-TFS] ---> B2 R  U2 R'  @ 0000000010284 nodes with 0000005671908 4-TFS probes
 ?Solution [248] =  L2 U' b' B' U2 u2 d2 f  b' B' U' [inside 3-TFS] ---> D' U  R'  @ 0000000014777 nodes with 0000000223923 3-TFS probes          tive!
  Solution [249] =  L2 U' b' B' U2 u2 d2 f  b' B' U2 [inside 3-TFS] ---> D' U2 R'  @ 0000000014778 nodes with 0000000223924 3-TFS probes
  Solution [250] =  L2 U' b' B' U2 u2 d2 f  b' B' u  [inside 3-TFS] ---> D' u' R'  @ 0000000014779 nodes with 0000000223925 3-TFS probes
  Solution [251] =  L2 U' b' B' U2 u2 d2 f  b' B' u' [inside 3-TFS] ---> D' u  R'  @ 0000000014780 nodes with 0000000223926 3-TFS probes
  Solution [252] =  L2 U' b' B' U2 u2 d2 f  b' B' u2 [inside 3-TFS] ---> D' u2 R'  @ 0000000014781 nodes with 0000000223927 3-TFS probes
  Solution [253] =  L2 U' b' B' U2 u2 d2 f  b' B' d  [inside 3-TFS] ---> D' d' R'  @ 0000000014782 nodes with 0000000223928 3-TFS probes
  Solution [254] =  L2 U' b' B' U2 u2 d2 f  b' B' d' [inside 3-TFS] ---> D' d  R'  @ 0000000014783 nodes with 0000000223929 3-TFS probes
  Solution [255] =  L2 U' b' B' U2 u2 d2 f  b' B' d2 [inside 3-TFS] ---> D' d2 R'  @ 0000000014784 nodes with 0000000223930 3-TFS probes
  Completed searching 0000000000024909  nodes through DEPTHS 11-16:     LEAF NODES: 0000000009239511 cut BRANCHES: 0000029679219459 pruned
  TFS Database probes 0000000105173517: 1-TFS probes 0000000000000253: 2-TFS probes 0000000000007842: 3-TFS probes 0000000000224040
  TFS Database misses 0000000121724548: 4-TFS probes 0000000005675374: 5-TFS probes 0000000099266008: 6-TFS probes 0000000000000000


  ****

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

  Search completed.

In summary, I would consider it a bug that the move generator is using rules that assume the orientation of the cube doesn't matter, but the program still considers the orientation of the cube to matter for determining whether or not the cube is solved. So in addition to not finding the real optimal solution, it has to search extra deep to find the optimal "original cube orientation" solution. In fact, it might be a little bit of luck involved in this case that it even finds a solution that cancels down to what I'm presuming to be the optimal "original cube orientation" solution.

I was wondering if anyone has looked for corner swap solutions. In addition to the traditional case:

F2 U' F2 D R2 B2 U B2 D' R2 U r2 U2 r2 U2 u2 r2 u2

there are essentially four other cases, it appears to me:

R2 D' R F2 U' F2 D R2 B2 U B2 D' R2 U r2 U2 r2 U2 u2 r2 u2 R' D R2

L2 U L2 F2 U' F2 D R2 B2 U B2 D' R2 U r2 U2 r2 U2 u2 r2 u2 L2 U' L2

R' F2 U' F2 D R2 B2 U B2 D' R2 U r2 U2 r2 U2 u2 r2 u2 R

R2 F2 U' F2 D R2 B2 U B2 D' R2 U r2 U2 r2 U2 u2 r2 u2 R2

These can take good advantage of corners pruning, but the solutions still seem to be a bit deep for my SSTM program.
 
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I know this kind of thing has been discussed before somewhere, but I don't think it ever got as far as trying to assign specific numbers to each move. The problem is that not only does it depend on the person, but it also depends on the cube itself. It is harder to perform slice turns quickly on some cubes, while others might not have as fast outer layers. And depending on tensions, performing a lot of D moves (say, the common <R,U,D> E-perm) may or may not be easy.
It is possibly too subjective. Perhaps the best idea would be to just cluster all of the solves together, stratified by turn metric, and let the cubers decide for themselves.

I tried the following scramble on OO4x4x4 version 1.2.3 and 1.2.8:

L l r' F f b' R r l' F2 f2 b2 L l U u d' R2 r2 l2
Thanks Bruce, this is the kind of feedback I need! Specific scrambles, nice summary of what you think is going on, etc.

OO_4x4x4 presently has a "preferred orientation" but, theoretically, it still should find any solution, even one that might be optimal at another orientation. I have not examined that subset of code in a while, it is clearly time to revisit.

This has me a little concerned:

Code:
Completed searching 0000000000000000  nodes through DEPTHS 06-11:     LEAF NODES: 0000000000006318 cut BRANCHES: 0000000000000677 pruned
Depth 6 filtered out every node! Yet, somehow, only 6318 were leaf nodes (moves pruned after move generation but before sending them onward to the search) and only 677 branches (moves not yet generated because there was no possible way to solve those 677 cube positions in the remaining depth of search). Definitely a move generator bug.

I am in the process now of flattening out a sparsely-populated matrix into a collapsed linear array of moves with no dependency on axis of rotation or move group that will still avoid cycling positions. It's a little more complex than it sounds.

Thanks for an excellent bug report. Stay tuned!

I know this kind of thing has been discussed before somewhere, but I don't think it ever got as far as trying to assign specific numbers to each move. The problem is that not only does it depend on the person, but it also depends on the cube itself. It is harder to perform slice turns quickly on some cubes, while others might not have as fast outer layers. And depending on tensions, performing a lot of D moves (say, the common <R,U,D> E-perm) may or may not be easy.
It is possibly too subjective. Perhaps the best idea would be to just cluster all of the solves together, stratified by turn metric, and let the cubers decide for themselves.

I tried the following scramble on OO4x4x4 version 1.2.3 and 1.2.8:

L l r' F f b' R r l' F2 f2 b2 L l U u d' R2 r2 l2
Thanks Bruce, this is the kind of feedback I need! Specific scrambles, nice summary of what you think is going on, etc.

OO_4x4x4 presently has a "preferred orientation" but, theoretically, it still should find any solution, even one that might be optimal at another orientation. I have not examined that subset of code in a while, it is clearly time to revisit.

This has me a little concerned:

Code:
Completed searching 0000000000000000  nodes through DEPTHS 06-11:     LEAF NODES: 0000000000006318 cut BRANCHES: 0000000000000677 pruned
Depth 6 filtered out every node! Yet, somehow, only 6318 were leaf nodes (moves pruned after move generation but before sending them onward to the search) and only 677 branches (moves not yet generated because there was no possible way to solve those 677 cube positions in the remaining depth of search). Definitely a move generator bug.

I am in the process now of flattening out a sparsely-populated matrix into a collapsed linear array of moves with no dependency on axis of rotation or move group that will still avoid cycling positions. It's a little more complex than it sounds.

Thanks for an excellent bug report. Stay tuned!

UPDATE: I spent the last day re-coding the move generator and verifying the uniqueness of positions in the TFS databases. It turns out there were duplicate entries in the TFS as a result of the move generator still producing moves that had functional equivalents earlier in the process, despite the fact that it used rules for axis rotation/move grouping + my parallel rotation pruner.

At depths 1 and 2, all positions were unique. At depth 3, there are only 28812 unique positions, not 28836 as with the "rotation rules" version of the move generator. At depth 4, there are 806694 unique positions, not 810648 with my parallel leaf node pruner or 810891 using only the Jakube move generator idea.

I obtained these numbers by using a full-width non-descriminater, generating all 36 possible moves at each depth, computing the "true 4x4x4 index" using that huge formula that enumerates all of the possible positions, and storing that 4x4x4 true index in an ever-growing linked list. For each move generated, I computed the true index, scanned the list, and if there was no match I appended the linked list with the new entry. If an index was already generated and a "cube match" was found, I printed out the turn sequence and the index that matched. Since I also stored every unique cube and its turn sequence and index, I can now check to see that the collision data is correct.

It looks like move generation is a little more complex than I thought, so I'll be implementing a new version of it from scratch. This will not be a rules-based version, it will be a baseline native generator with no duplication and no orientation preferences. The trick is, how to accomplish all of this and still make it fast.
 
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