# Slice turn metric, anyone?

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 4368975951874This 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

## Comment viewing options

### Slice turn vs Axial Turn

### SPTM (ATM) exact count

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

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 000000001736596179Since 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 R' R2 M' M M2 L' L L2 U U' U2 E' E E2 D' D D2 F F' F2 S S' S2 B' B 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 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

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

How did you get these numbers?

### 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

### SQTM

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

### 18*s

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

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

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

### 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.

## Axial solutions

Here's the Superflip optimally in 14 SPTM twists:

D2(F2B1)(L3R3)(D1U1)F2D1(L3R1)(D3U1)R1(F3B3)(L2R2)D3B2(L3R1)

Here are the only six distance 16's after 781,183 symmetric state solutions. (The rest were distance 15 or less.)

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