Slice turn metric, anyone?

Anyone got a reasonable upper or lower bound for slice turn metric? (Clearly, the 20 result gives us a trivial upper bound of 20.) Lower bounds? Here are position count results; these are the easy ones. I'm hoping this table will magically extend itself. First table is exact count at that level:
dist mod M+inv       mod M     positions
 0           1           1             1
 1           4           4            27
 2          13          19           501
 3         150         236          9175
 4        1920        3642        164900
 5       31341       61457       2912447
 6      533133     1062321      50839041
 7     9180455    18336013     879457304
 8   157399245   314710818   15103187300
 9  2684527137  5368523290  257676327458
10 45511823396 91021483374 4368975951874
This table is cumulative counts.
dist mod M+inv       mod M     positions
 0           1           1             1
 1           5           5            28
 2          18          24           529
 3         168         260          9704
 4        2088        3902        174604
 5       33429       65359       3087051
 6      566562     1127680      53926092
 7     9747017    19463693     933383396
 8   167146262   334174511   16036570696
 9  2851673399  5702697801  273712898154
10 48363496795 96724181175 4642688850028

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Axial solutions

I ended up calling this the simultaneously-possible turn metric (in 3D it's the same as the axial turn metric or robot turn metric with Bruce's numbers).

Here's the Superflip optimally in 14 SPTM twists:

Here are the only six distance 16's after 781,183 symmetric state solutions. (The rest were distance 15 or less.)
U1F1U2F2L2U2R3B2U3L3F3R2F2U1R1D3F2B1U1R1 (Superflip + 2 opposing corners disoriented)
U1F1U1F2D2F2R3D1L3F2D2F2B3L2U2F3D1R3U1B3 (Superflip + 2 adjacent corners disoriented)
U1F1U1R2U1D1R3F1U3R1L1U3B1R3U1D1R2U1B1D3 (Superflip + 2 same-face [but not adjacent] corners disoriented)
U1B1U1B2U1L3U2B3F1L3R1B3D3L2D2L1F2L2B1D3 (Superflip + 3 non-adjacent corners disoriented—all 3 adjacent to a common, oriented corner)

I have numbers to tack onto Bruce's, but I'm about to attain distance 9, so I'll wait for that.

Slice turn vs Axial Turn

Just to be clear: Andrew is talking about the Axial Turn metric or the SPTM (simultaneous-possible-turn metric) for the 3x3 here (which you clearly see since at distance 1 there are 45 distinct positions). The slice-turn metric only has 27 (moves, and positions at distance-1).

SPTM (ATM) exact count

Exact count at each distance through 9 for SPTM (or ATM).
d  mod M + inv        mod M      positions
0            1            1              1
1            6            6             45
2           27           45           1347
3          578          994          39631
4        12912        25382        1152290
5       348938       692436       32717804
6      9602778     19189952      917301226
7    266133337    532097030    25514695958
8   7348986607  14697402495   705324642686
9 201985646743 403965402851 19389615510216

My counts for 3x3x3 with 27 nodes at depth 1

I get different counts. This was done using my 5x5x5 move generator with only the 3x3x3 subset enabled. The numbers are cumulative. No "depth 0" result, since that does not evoke my move generator.
  Depth 01: Nodes 000000000000000027
  Depth 02: Nodes 000000000000000567
  Depth 03: Nodes 000000000000011259
  Depth 04: Nodes 000000000000223155
  Depth 05: Nodes 000000000004422195
  Depth 06: Nodes 000000000087633171
  Depth 07: Nodes 000000001736596179
Since I have a different count for depth 2 and beyond, here is the list of my legal moves for depths 1 and 2. I'm hoping I have the M, E, and S rotations defined properly. If I don't, it should be apparent by perusing the list below.
R  M  
R  M2 
R  L  
R  L2 
R  U  
R  U' 
R  U2 
R  E' 
R  E  
R  E2 
R  D' 
R  D  
R  D2 
R  F  
R  F' 
R  F2 
R  S  
R  S' 
R  S2 
R  B' 
R  B  
R  B2 
R' M' 
R' M2 
R' L' 
R' L2 
R' U  
R' U' 
R' U2 
R' E' 
R' E  
R' E2 
R' D' 
R' D  
R' D2 
R' F  
R' F' 
R' F2 
R' S  
R' S' 
R' S2 
R' B' 
R' B  
R' B2 
R2 M' 
R2 M  
R2 L' 
R2 L  
R2 U  
R2 U' 
R2 U2 
R2 E' 
R2 E  
R2 E2 
R2 D' 
R2 D  
R2 D2 
R2 F  
R2 F' 
R2 F2 
R2 S  
R2 S' 
R2 S2 
R2 B' 
R2 B  
R2 B2 
M' L  
M' L2 
M' U  
M' U' 
M' U2 
M' E' 
M' E  
M' E2 
M' D' 
M' D  
M' D2 
M' F  
M' F' 
M' F2 
M' S  
M' S' 
M' S2 
M' B' 
M' B  
M' B2 
M  L' 
M  L2 
M  U  
M  U' 
M  U2 
M  E' 
M  E  
M  E2 
M  D' 
M  D  
M  D2 
M  F  
M  F' 
M  F2 
M  S  
M  S' 
M  S2 
M  B' 
M  B  
M  B2 
M2 L' 
M2 L  
M2 U  
M2 U' 
M2 U2 
M2 E' 
M2 E  
M2 E2 
M2 D' 
M2 D  
M2 D2 
M2 F  
M2 F' 
M2 F2 
M2 S  
M2 S' 
M2 S2 
M2 B' 
M2 B  
M2 B2 
L' U  
L' U' 
L' U2 
L' E' 
L' E  
L' E2 
L' D' 
L' D  
L' D2 
L' F  
L' F' 
L' F2 
L' S  
L' S' 
L' S2 
L' B' 
L' B  
L' B2 
L  U  
L  U' 
L  U2 
L  E' 
L  E  
L  E2 
L  D' 
L  D  
L  D2 
L  F  
L  F' 
L  F2 
L  S  
L  S' 
L  S2 
L  B' 
L  B  
L  B2 
L2 U  
L2 U' 
L2 U2 
L2 E' 
L2 E  
L2 E2 
L2 D' 
L2 D  
L2 D2 
L2 F  
L2 F' 
L2 F2 
L2 S  
L2 S' 
L2 S2 
L2 B' 
L2 B  
L2 B2 
U  R  
U  R' 
U  R2 
U  M' 
U  M  
U  M2 
U  L' 
U  L  
U  L2 
U  E  
U  E2 
U  D  
U  D2 
U  F  
U  F' 
U  F2 
U  S  
U  S' 
U  S2 
U  B' 
U  B  
U  B2 
U' R  
U' R' 
U' R2 
U' M' 
U' M  
U' M2 
U' L' 
U' L  
U' L2 
U' E' 
U' E2 
U' D' 
U' D2 
U' F  
U' F' 
U' F2 
U' S  
U' S' 
U' S2 
U' B' 
U' B  
U' B2 
U2 R  
U2 R' 
U2 R2 
U2 M' 
U2 M  
U2 M2 
U2 L' 
U2 L  
U2 L2 
U2 E' 
U2 E  
U2 D' 
U2 D  
U2 F  
U2 F' 
U2 F2 
U2 S  
U2 S' 
U2 S2 
U2 B' 
U2 B  
U2 B2 
E' R  
E' R' 
E' R2 
E' M' 
E' M  
E' M2 
E' L' 
E' L  
E' L2 
E' D  
E' D2 
E' F  
E' F' 
E' F2 
E' S  
E' S' 
E' S2 
E' B' 
E' B  
E' B2 
E  R  
E  R' 
E  R2 
E  M' 
E  M  
E  M2 
E  L' 
E  L  
E  L2 
E  D' 
E  D2 
E  F  
E  F' 
E  F2 
E  S  
E  S' 
E  S2 
E  B' 
E  B  
E  B2 
E2 R  
E2 R' 
E2 R2 
E2 M' 
E2 M  
E2 M2 
E2 L' 
E2 L  
E2 L2 
E2 D' 
E2 D  
E2 F  
E2 F' 
E2 F2 
E2 S  
E2 S' 
E2 S2 
E2 B' 
E2 B  
E2 B2 
D' R  
D' R' 
D' R2 
D' M' 
D' M  
D' M2 
D' L' 
D' L  
D' L2 
D' F  
D' F' 
D' F2 
D' S  
D' S' 
D' S2 
D' B' 
D' B  
D' B2 
D  R  
D  R' 
D  R2 
D  M' 
D  M  
D  M2 
D  L' 
D  L  
D  L2 
D  F  
D  F' 
D  F2 
D  S  
D  S' 
D  S2 
D  B' 
D  B  
D  B2 
D2 R  
D2 R' 
D2 R2 
D2 M' 
D2 M  
D2 M2 
D2 L' 
D2 L  
D2 L2 
D2 F  
D2 F' 
D2 F2 
D2 S  
D2 S' 
D2 S2 
D2 B' 
D2 B  
D2 B2 
F  R  
F  R' 
F  R2 
F  M' 
F  M  
F  M2 
F  L' 
F  L  
F  L2 
F  U  
F  U' 
F  U2 
F  E' 
F  E  
F  E2 
F  D' 
F  D  
F  D2 
F  S' 
F  S2 
F  B  
F  B2 
F' R  
F' R' 
F' R2 
F' M' 
F' M  
F' M2 
F' L' 
F' L  
F' L2 
F' U  
F' U' 
F' U2 
F' E' 
F' E  
F' E2 
F' D' 
F' D  
F' D2 
F' S  
F' S2 
F' B' 
F' B2 
F2 R  
F2 R' 
F2 R2 
F2 M' 
F2 M  
F2 M2 
F2 L' 
F2 L  
F2 L2 
F2 U  
F2 U' 
F2 U2 
F2 E' 
F2 E  
F2 E2 
F2 D' 
F2 D  
F2 D2 
F2 S  
F2 S' 
F2 B' 
F2 B  
S  R  
S  R' 
S  R2 
S  M' 
S  M  
S  M2 
S  L' 
S  L  
S  L2 
S  U  
S  U' 
S  U2 
S  E' 
S  E  
S  E2 
S  D' 
S  D  
S  D2 
S  B  
S  B2 
S' R  
S' R' 
S' R2 
S' M' 
S' M  
S' M2 
S' L' 
S' L  
S' L2 
S' U  
S' U' 
S' U2 
S' E' 
S' E  
S' E2 
S' D' 
S' D  
S' D2 
S' B' 
S' B2 
S2 R  
S2 R' 
S2 R2 
S2 M' 
S2 M  
S2 M2 
S2 L' 
S2 L  
S2 L2 
S2 U  
S2 U' 
S2 U2 
S2 E' 
S2 E  
S2 E2 
S2 D' 
S2 D  
S2 D2 
S2 B' 
S2 B  
B' R  
B' R' 
B' R2 
B' M' 
B' M  
B' M2 
B' L' 
B' L  
B' L2 
B' U  
B' U' 
B' U2 
B' E' 
B' E  
B' E2 
B' D' 
B' D  
B' D2 
B  R  
B  R' 
B  R2 
B  M' 
B  M  
B  M2 
B  L' 
B  L  
B  L2 
B  U  
B  U' 
B  U2 
B  E' 
B  E  
B  E2 
B  D' 
B  D  
B  D2 
B2 R  
B2 R' 
B2 R2 
B2 M' 
B2 M  
B2 M2 
B2 L' 
B2 L  
B2 L2 
B2 U  
B2 U' 
B2 U2 
B2 E' 
B2 E  
B2 E2 
B2 D' 
B2 D  
B2 D2 

Well, clearly your list has p

Well, clearly your list has positions that are not distinct - that is, they differ only in the orientation of the cube. For example R M and R2 L' produce the same 3x3x3 position except in a different orientation. Of course, Tom treated these as the same position.

The reason I showed the list

The reason I showed the list was that so someone could point out what I was missing. I had not considered your example, thank you. Now I need to apply a rule to my move generator that prevents such a pair of moves from being played.

At present, I disallow:

1. 3 consecutive moves made on the same axis of rotation
2. 2 consecutive moves made on the same axis if both moves are in the same direction or distance

I need a third rule of suppression.

So let me add 11

 d  mod M + inv      mod M     positions
-- ------------ ------------- --------------
 9   2684527137    5368523290   257676327458
10  45511823396   91021483374  4368975951874
11 766963392901 1533913901305 73627651087473

Other metrics

And SQTM? And void cube?

How did you get these numbers?

SQTM? I prefer QSTM.

SQTM? I prefer QSTM.

Axes metric?

What about axes metric, i.e. the number of changes of axis plus one?

some axial metric numbers

I've seen a number of variations as to what to call this metric (axial metric, axis turn metric, etc.). Anyway, the results from my hash table program are given below.

Rubik's cube - axial metric

dist  positions     pos mod M   positions     positions
          mod M  (cumulative)              (cumulative)
 0            1             1           1             1
 1            6             7          45            46
 2           45            52        1347          1393
 3          994          1046       39631         41024 
 4        25382         26428     1152290       1193314
 5       692436        718864    32717804      33911118
 6     19189952      19908816   917301226     951212344


I think SQTM is a pretty artificial metric; it lacks the bipartite nature of the QTM, and also the natural intuitiveness of HTM. So unless I really lack for something to do, I probably won't investigate it. (If there's a good reason to compare four metrics on the same group, then maybe it will be worthwhile.)

Void is certainly interesting; we've already shown it's 20f* though. (This is through a combination of an earlier result on this board that shows it's at least 20, plus the 20 bound for the cube-with-centers that also shows the void cube can be solved in 20.)

I got these numbers using the same coset ideas (indeed, almost the same program) as I used for the earlier QTM and HTM count results; you'll find an explanation of those ideas under the earlier postings.

some SQTM numbers

Modifying an old hash-table-based program of mine, I've duplicated the results for the number of positions and mod M positions up to distance 7 (STM).

I also note that my program counted the number of positions (mod M) that are of order 1260 in the <U,D,L,R,F,B> group. There are 10 at STM distance 5, 396 at distance 6, and 10628 at distance 7.

I could make a simple change to the program to generate the SQTM numbers as well, out to distance 8.

Rubik's Cube: SQTM

dist  pos mod M   pos mod M    positions    positions   order-1260
                (cumulative)              (cumulative)   (mod M)
 0            1           1            1            1        0
 1            2           3           18           19        0
 2            9          12          243          262        0
 3           76          88         3240         3502        0
 4          925        1013        42815        46317        0
 5        11739       12752       558930       605247        6
 6       151114      163866      7234281      7839528       76
 7      1938907     2102773     92990550    100830078     1762
 8     24755348    26858121   1187984046   1288814124    24762


Thanks, Bruce!

The deepest position I've found so far (just running some random ones) is 17s*. Does anyone know of an 18s*? (I'm using the new Cube Explorer which supports slice turn metric.)

Distance 18s*'s

I've now found millions of distance 18s* positions.

So now the search is for 19s* positions.


I wanted to proudly announce that I found 84 of them…
I'm solving all cubes having S6 symmetry or higher. There are almost 4,000 of them so this could be done on my computer in one weekend. This was my only reason for choosing this set instead of another symmetry set but I have the impression that these cubes are pretty hard: so far I have 0.8% of 18s* and 37% of 17s* and this rates will probably raise because I solved easy-looking cubes first.

S6 cubes

Sorry, in the previous post, I meant 8% of 18s* not 0.8%!

I was optimistic, in fact it took more than 3 days to complete this calculation.

here are the results:

Number of Cubes: 3896
Cubes solved optimally: 3896

0s*: 1
3s*: 1
4s*: 1
5s*: 1
7s*: 1
8s*: 6
10s*: 8
11s*: 4
12s*: 22
13s*: 62
14s*: 162
15s*: 458
16s*: 1010
17s*: 1798
18s*: 361
Cubes with antisymmetry: 1372


Very nice.

I've completed 63 full cosets in the STM, focusing on those that have the greatest distances, and all of them have distance 18s*; that is, no distance 19s* positions, but millions of distance 18s* positions.

These 63 cosets represent about 1.2 trillion positions (not including conjugations with M).

I'd say the chances of an 19s* existing are looking mighty slim. Yet, 18s*'s are much more numerous than 20f*'s.