Probability Thread

IsThatA4x4 · Jul 16, 2022

zzoomer said:
Sounds like PBL but on 3x3. Should be the same number as sq1: 21^2 = 441

edit: you were pretty vague with your terminology ("oriented" relative to what?) and I may have misinterpreted what you are looking for.

This definitely matters whether it's PBL or PBL + separation but the number for PBL is actually higher.
The number you described is when both U and D layers have even parity, but both could also have odd parity (e.g. opp/adj PBL).
There are 2 different scenarios:
Even/even = 22^2 (because we have to include solved), -22 because both PLLs being the same would create duplicate cases = 462
Odd/odd = 22^2 again, but -22 again because of duplicates = 462

So we get a total of 462 x 2 = 924 PBL cases

924 is definitely more feasible to gen than 200 million, so I hope this is what @Imsoosm meant

ruffleduck · Jul 16, 2022

IsThatA4x4 said:
This definitely matters whether it's PBL or PBL + separation but the number for PBL is actually higher.
The number you described is when both U and D layers have even parity, but both could also have odd parity (e.g. opp/adj PBL).
There are 2 different scenarios:
Even/even = 22^2 (because we have to include solved), -22 because both PLLs being the same would create duplicate cases = 462
Odd/odd = 22^2 again, but -22 again because of duplicates = 462

So we get a total of 462 x 2 = 924 PBL cases

924 is definitely more feasible to gen than 200 million, so I hope this is what @Imsoosm meant

Thanks for the correction 22^2 instead of 21^2. I didn't multiply by 2 because I was taking into consideration Imsoosm's request to exclude z2 rotational symmetry

SYUTEO · Jul 28, 2022

Are the chances of a PLL skip higher if you learn full OLL?
I always wondered if that's true as I seem to get PLL skips more often the more OLLs I learn, or maybe I am just lucky.

OreKehStrah · Jul 28, 2022

SYUTEO said:
I always wondered if that's true as I seem to get PLL skips more often the more OLLs I learn, or maybe I am just lucky.

Ever so slightly yes. It increases the chances of you having 1LLL by using more OLLs.

DGCubes · Jul 28, 2022

No. Assuming you're not using different OLL algorithms to influence what PLL you get (e.g. choosing between R U' L' U R' U' L or R U R' U R U2 R' for the sune OLL), you have a 1/72 chance of skipping PLL regardless of whether you complete OLL with 1 or 2 algs.

EDIT (reply):

OreKehStrah said:
Ever so slightly yes. It increases the chances of you having 1LLL by using more OLLs.

While this is true, this does not increase the chances of a PLL skip, unless the alg you're doing is purposefully affecting PLL (like COLL or ZBLL algs).

ottozing · Aug 2, 2022

FMC related request:

For DR minus 4C4E, does anyone know the chances of getting a case that is solvable in x moves, with x being anywhere from 1-7? I wouldn't mind knowing the numbers for 4C2E and 3C2E as well but they're much less important, at least to me

I'm mostly curious because I just missed optimal by 1 on an attempt where I decided to look for more DR's instead of looking closely at the DR's I had, which I think was a mistake given the quality of the DR's (they were very good) and the quality of the scramble/EO's (they were pretty bad)
It'd be nice to know more or less how likely you are to get these move counts from DR triggers to know whether or not it's likely to be a waste of your time or not (especially considering the corners could end up being awful)

ottozing · Aug 8, 2022

ottozing said:
FMC related request:

For DR minus 4C4E, does anyone know the chances of getting a case that is solvable in x moves, with x being anywhere from 1-7? I wouldn't mind knowing the numbers for 4C2E and 3C2E as well but they're much less important, at least to me

I'm mostly curious because I just missed optimal by 1 on an attempt where I decided to look for more DR's instead of looking closely at the DR's I had, which I think was a mistake given the quality of the DR's (they were very good) and the quality of the scramble/EO's (they were pretty bad)
It'd be nice to know more or less how likely you are to get these move counts from DR triggers to know whether or not it's likely to be a waste of your time or not (especially considering the corners could end up being awful)

Bumping for visibility. Would really appreciate if anyone could help me with this

xyzzy · Aug 8, 2022

ottozing said:
FMC related request:

For DR minus 4C4E, does anyone know the chances of getting a case that is solvable in x moves, with x being anywhere from 1-7? I wouldn't mind knowing the numbers for 4C2E and 3C2E as well but they're much less important, at least to me

I'm mostly curious because I just missed optimal by 1 on an attempt where I decided to look for more DR's instead of looking closely at the DR's I had, which I think was a mistake given the quality of the DR's (they were very good) and the quality of the scramble/EO's (they were pretty bad)
It'd be nice to know more or less how likely you are to get these move counts from DR triggers to know whether or not it's likely to be a waste of your time or not (especially considering the corners could end up being awful)

Optimal move count to complete DR (U-D) from EO already being solved (F-B). F and B quarter turns are allowed in solving DR, i.e. EO can be temporarily broken.

# c, # e	0 move	1 move	2 moves	3 moves	4 moves	5 moves	6 moves	7 moves	8 moves	9 moves	10 moves	avg moves
0c0e	1	0	0	0	0	0	0	0	0	0	0	0.00
2c0e	0	0	0	0	0	24	16	16	0	0	0	5.86
3c0e	0	0	0	0	0	0	0	112	0	0	0	7.00
4c0e	0	0	0	0	0	0	64	220	128	8	0	7.19
5c0e	0	0	0	0	0	0	0	112	448	0	0	7.80
6c0e	0	0	0	0	0	0	0	0	136	480	0	8.78
7c0e	0	0	0	0	0	0	0	0	192	144	0	8.43
8c0e	0	0	0	0	0	0	2	2	28	36	18	8.77
0c1e	0	0	0	0	0	8	24	0	0	0	0	5.75
2c1e	0	0	0	0	0	48	376	1112	256	0	0	6.88
3c1e	0	0	0	16	120	552	1520	1328	48	0	0	6.16
4c1e	0	0	0	8	64	392	2120	5960	4712	184	0	7.15
5c1e	0	0	0	0	0	0	440	4344	11064	2072	0	7.82
6c1e	0	0	0	0	0	0	32	1664	11448	6552	16	8.25
7c1e	0	0	0	0	0	0	32	488	4144	5920	168	8.53
8c1e	0	0	0	0	0	0	0	96	776	1616	264	8.74
0c2e	0	0	0	0	2	16	62	80	8	0	0	6.45
2c2e	0	0	0	0	0	48	532	3540	4916	372	0	7.53
3c2e	0	0	0	0	0	184	1712	7272	9048	600	0	7.43
4c2e	0	2	16	84	476	2486	10520	28446	26718	1812	0	7.18
5c2e	0	0	0	0	64	704	6056	31392	48376	7488	0	7.59
6c2e	0	0	0	0	12	284	2832	19912	58196	21996	256	7.96
7c2e	0	0	0	0	16	144	1104	8200	31136	15696	152	8.09
8c2e	0	0	0	0	2	16	86	1222	6350	6440	332	8.39
0c3e	0	0	0	0	0	0	0	24	128	72	0	8.21
2c3e	0	0	0	0	0	0	16	1072	7024	4424	8	8.27
3c3e	0	0	0	0	16	144	1056	5576	14056	4208	32	7.84
4c3e	0	0	0	8	64	496	3776	21360	52480	15832	64	7.84
5c3e	0	0	0	0	64	768	6864	37088	66664	13984	8	7.69
6c3e	0	0	0	0	40	768	6904	37984	75024	17192	72	7.73
7c3e	0	0	0	0	0	240	2792	17920	43368	10936	8	7.82
8c3e	0	0	0	0	0	8	216	2936	11320	4752	32	8.07
0c4e	0	0	0	0	0	0	1	2	9	46	12	8.94
2c4e	0	0	0	0	0	0	0	0	648	2844	428	8.94
3c4e	0	0	0	0	0	0	32	368	2880	4336	224	8.56
4c4e	0	0	0	0	0	0	80	1582	12368	14944	426	8.48
5c4e	0	0	0	0	0	24	528	5976	22592	9984	96	8.08
6c4e	0	0	0	0	12	148	1240	9412	24336	7920	52	7.90
7c4e	0	0	0	16	104	624	3112	9648	9320	696	0	7.25
8c4e	0	0	1	2	9	88	772	2922	2082	144	0	7.23

Interesting that 0c1e and 2c0e are better on average than 3c1e, but they're also harder to set up to (so setting up to 0c1e or 2c0e isn't necessarily better overall).

The actual thing you asked for, 4c4e, is:
0.3% solvable in at most 6 moves
5.7% solvable in at most 7 moves
47.7% solvable in at most 8 moves
98.6% solvable in at most 9 moves
100% solvable in at most 10 moves

ottozing · Aug 9, 2022

xyzzy said:
Optimal move count to complete DR (U-D) from EO already being solved (F-B). F and B quarter turns are allowed in solving DR, i.e. EO can be temporarily broken.

# c, # e 0 move 1 move 2 moves 3 moves 4 moves 5 moves 6 moves 7 moves 8 moves 9 moves 10 moves avg moves
0c0e
1
0
0
0
0
0
0
0
0
0
0
0.00
2c0e
0
0
0
0
0
24
16
16
0
0
0
5.86
3c0e
0
0
0
0
0
0
0
112
0
0
0
7.00
4c0e
0
0
0
0
0
0
64
220
128
8
0
7.19
5c0e
0
0
0
0
0
0
0
112
448
0
0
7.80
6c0e
0
0
0
0
0
0
0
0
136
480
0
8.78
7c0e
0
0
0
0
0
0
0
0
192
144
0
8.43
8c0e
0
0
0
0
0
0
2
2
28
36
18
8.77
0c1e
0
0
0
0
0
8
24
0
0
0
0
5.75
2c1e
0
0
0
0
0
48
376
1112
256
0
0
6.88
3c1e
0
0
0
16
120
552
1520
1328
48
0
0
6.16
4c1e
0
0
0
8
64
392
2120
5960
4712
184
0
7.15
5c1e
0
0
0
0
0
0
440
4344
11064
2072
0
7.82
6c1e
0
0
0
0
0
0
32
1664
11448
6552
16
8.25
7c1e
0
0
0
0
0
0
32
488
4144
5920
168
8.53
8c1e
0
0
0
0
0
0
0
96
776
1616
264
8.74
0c2e
0
0
0
0
2
16
62
80
8
0
0
6.45
2c2e
0
0
0
0
0
48
532
3540
4916
372
0
7.53
3c2e
0
0
0
0
0
184
1712
7272
9048
600
0
7.43
4c2e
0
2
16
84
476
2486
10520
28446
26718
1812
0
7.18
5c2e
0
0
0
0
64
704
6056
31392
48376
7488
0
7.59
6c2e
0
0
0
0
12
284
2832
19912
58196
21996
256
7.96
7c2e
0
0
0
0
16
144
1104
8200
31136
15696
152
8.09
8c2e
0
0
0
0
2
16
86
1222
6350
6440
332
8.39
0c3e
0
0
0
0
0
0
0
24
128
72
0
8.21
2c3e
0
0
0
0
0
0
16
1072
7024
4424
8
8.27
3c3e
0
0
0
0
16
144
1056
5576
14056
4208
32
7.84
4c3e
0
0
0
8
64
496
3776
21360
52480
15832
64
7.84
5c3e
0
0
0
0
64
768
6864
37088
66664
13984
8
7.69
6c3e
0
0
0
0
40
768
6904
37984
75024
17192
72
7.73
7c3e
0
0
0
0
0
240
2792
17920
43368
10936
8
7.82
8c3e
0
0
0
0
0
8
216
2936
11320
4752
32
8.07
0c4e
0
0
0
0
0
0
1
2
9
46
12
8.94
2c4e
0
0
0
0
0
0
0
0
648
2844
428
8.94
3c4e
0
0
0
0
0
0
32
368
2880
4336
224
8.56
4c4e
0
0
0
0
0
0
80
1582
12368
14944
426
8.48
5c4e
0
0
0
0
0
24
528
5976
22592
9984
96
8.08
6c4e
0
0
0
0
12
148
1240
9412
24336
7920
52
7.90
7c4e
0
0
0
16
104
624
3112
9648
9320
696
0
7.25
8c4e
0
0
1
2
9
88
772
2922
2082
144
0
7.23

Interesting that 0c1e and 2c0e are better on average than 3c1e, but they're also harder to set up to (so setting up to 0c1e or 2c0e isn't necessarily better overall).

The actual thing you asked for, 4c4e, is:
0.3% solvable in at most 6 moves
5.7% solvable in at most 7 moves
47.7% solvable in at most 8 moves
98.6% solvable in at most 9 moves
100% solvable in at most 10 moves

Thanks a ton for doing this! I'll analyze this myself later today (btw, when I said 4C4E I was originally referring to what you've listed as 4C2E, but I should be able to get the info I need from this data table)

I already see some interesting things for me to look into outside the scope of normal DR triggers (4C4E/4C2E/3C2E) which might be worthwhile

xyzzy · Aug 9, 2022

ottozing said:
btw, when I said 4C4E I was originally referring to what you've listed as 4C2E

Ah, right, could've figured that out from context. I thought it was more common for XcYe to mean X corners twisted and Y E-slice edges out of place, rather than Y edges unsolved altogether.

Thank goodness I made the full table instead of just computing the results for one row, huh.

Timona · Aug 17, 2022

Idk if this is the right thread, but I want to know something. The average cross move count is 7 STM or maybe less. So I want to know the (Average or optimal, correct me if I'm wrong) movecount in STM for
- an xcross
- an xxcross
- EOLine
- EOCross
- EOxcross
- cross too, Cus I feel my estimations are wrong

For cross, xcross and xxcross, I'm regarding colour neutrality. For EOLine, EOCross and EOxcross, I'm ignoring colour neutrality and assuming a fixed orientation (Yellow top, Blue front )

Thom S. · Aug 17, 2022

I think to have heard that EOCross can always be solved in 10 moves.

Bruce MacKenzie · Aug 18, 2022

Thom S. said:
I think to have heard that EOCross can always be solved in 10 moves.

I did a cosets at depth calculation on the EO_Cross subgroup (ie the cube subgroup with the Up face edges solved). The deepest coset in the quarter turn metric is 9, the deepest coset in the face turn metric is 8 and the deepest coset in the slice turn metric is 7.

Depth	QTM	FTM	STM
0	1	1	1
1	10	15	21
2	73	158	275
3	500	1,394	2,815
4	3,078	9,809	21,136
5	15,528	46,381	88,454
6	57,180	97,254	76,592
7	91,654	34,966	786
8	21,849	102
9	207

xyzzy · Aug 18, 2022

Bruce MacKenzie said:
I did a cosets at depth calculation on the EO_Cross subgroup (ie the cube subgroup with the Up face edges solved).

EOCross is cross and edge orientation. So the intersection of (subgroup with U-face edges solved) and ⟨U, D, L, R, F2, B2⟩.

Bruce MacKenzie · Aug 18, 2022

xyzzy said:
EOCross is cross and edge orientation. So the intersection of (subgroup with U-face edges solved) and ⟨U, D, L, R, F2, B2⟩.

According to the WIKI database here EOCross is:

xyzzy · Aug 18, 2022

Bruce MacKenzie said:
According to the WIKI database here EOCross is:

View attachment 20449

The light-grey-coloured edges are meant to be ones that are oriented (wrt ⟨U, D, L, R, F2, B2⟩). Compare with, say, the pEOLine image, where the top four edges are left as dark grey.

(But also dang, these images are indeed confusing. There isn't even a description of what the edge colouring means.)

Bruce MacKenzie · Aug 18, 2022

Bruce MacKenzie said:
I did a cosets at depth calculation on the EO_Cross subgroup (ie the cube subgroup with the Up face edges solved). The deepest coset in the quarter turn metric is 9, the deepest coset in the face turn metric is 8 and the deepest coset in the slice turn metric is 7.

Depth
QTM
FTM
STM
0
1
1
1
1
10
15
21
2
73
158
275
3
500
1,394
2,815
4
3,078
9,809
21,136
5
15,528
46,381
88,454
6
57,180
97,254
76,592
7
91,654
34,966
786
8
21,849
102

9
207

As xyzzy pointed out the above calculation is for the CROSS subgroup cosets not the EO_CROSS subgroup cosets. I added the orientation of the other eight edge cubies and redid the analysis:

Depth	QTM	FTM	STM
0	1	1	1
1	10	15	21
2	85	178	313
3	744	1,982	4,329
4	6,194	21,041	55,730
5	49,009	204,732	643,385
6	360,175	1,645,039	5,444,086
7	2,277,947	8,477,633	16,008,710
8	9,529,222	12,917,628	2,173,545
9	11,407,300	1,061,851	120
10	699,495	140
11	58

I dumped out a representative member for ten of the depth 10 FTM cosets and solved them with a partial cube solver configured to solve to the EO_Cross subgroup and in each case got an optimal 10 move FTM solution so I am fairly confident in my numbers.

Generator: EO_CROSS FTM Antipodes	EO_CROSS Optimal FTM Solution
F' L' B' L' D' L2 R U F' U2 F' R U R' U2 R2 U2 F' U F	R F B D' U2 L D R B U
D F L' D R D2 L' U' B2 L2 B2 U L B L2 U' L' U2 B L B	R F' U' B D' L' F U' R B'
R U B F U2 L2 D B L U2 L' U' B' L' U2 B2 L B2 L2 B L'	R U B' U2 F2 R' D F' L2 B'
B' L2 F L D L B U B' R2 U R' U R B2 R' B' R2 B2 U2 R'	R U' L2 F' U2 L' B' D' R2 F
F U D2 B U R D2 L2 F U' F L' F U L' F' U L U2 L2	R U R L D' F L' B U2 L
R U F U2 L2 D U L' U2 B' L2 U B2 L' B2 L' B L U2 B L2 U L2	R U B U2 F2 L D F R2 B
U' D2 B L F L D L R' U F' R U F2 U2 R F2 R' F' U' F'	R U' L B' L D F' R U B
R D U2 B D2 L2 U D2 L' F L' F L' F2 L' D L2 D2 F' L' D	R U B D' L' F' U B' F2 L2
B F U' R' U' D F2 D2 B' R' B' U B2 U2 B R' U2 B2 R2 B' R' U2	R U B D' B' L2 B L U F
U L' F' L F2 U' L' U' F' U2 L U' F' B' R2 L2 D' U' R' D L'	R F' U B R D F U L' B2

Imsoosm · Aug 19, 2022

What's the average movecount just for EO? I'm starting to learn EO for FMC/ZZ so I want some statistics on how it would affect my FMC solves.

ruffleduck · Aug 20, 2022

Imsoosm said:
What's the average movecount just for EO? I'm starting to learn EO for FMC/ZZ so I want some statistics on how it would affect my FMC solves.

It depends on neutrality. IIRC, fixed (1 axis) averages low 4 HTM and fully neutral (3 axes) averages low 3

vidcapper · Aug 21, 2022

This may be the wrong place or been posted before, but...

Rubik's Cube solution unlocked by memorising 3915 final move sequences | New Scientist

For the first time, a speedcuber has demonstrated a solution to the Rubik’s cube that combines the two final steps of the puzzle’s solution into one

www.newscientist.com

Wow, i can't even remember all 16 algs for 4LLL!

Also...

Probability Thread

Veteran

Veteran

Regular

Veteran

Veteran

Platinum Member

Platinum Member

Super Moderator

Platinum Member

Super Moderator

Veteran

Veteran

Regular

Super Moderator

Regular

Super Moderator

Regular

Veteran

Veteran

Veteran

Similar threads