Probabilities for corner cycles

blah · Jul 13, 2010

Code:

8         5040    5040       1/8
6+2       3360    8400      1/12
5+3       2688   11088      1/15
4+4       1260   12348      1/32
4+2+2     1260   13608      1/32
3+3+2     1120   14728      1/36
2+2+2+2    105   14833     1/384
7         5760   20593       1/7
5+2       4032   24625      1/10
4+3       3360   27985      1/12
3+2+2     1680   29665      1/24
6         3360   33025      1/12
4+2       2520   35545      1/16
3+3       1120   36665      1/36
2+2+2      420   37085      1/96
5         1344   38429      1/30
3+2       1120   39549      1/36
4          420   39969      1/96
2+2        210   40179     1/192
3          112   40291     1/360
2           28   40319    1/1440
0            1   40320   1/40320

Another view:

Code:

7         5760       1/7   
8         5040       1/8   P
5+2       4032      1/10   P
4+3       3360      1/12   P
6         3360      1/12   P
6+2       3360      1/12
5+3       2688      1/15
4+2       2520      1/16
3+2+2     1680      1/24
5         1344      1/30
4+2+2     1260      1/32   P
4+4       1260      1/32
3+3+2     1120      1/36   P
3+2       1120      1/36   P
3+3       1120      1/36
2+2+2      420      1/96   P
4          420      1/96   P
2+2        210     1/192
3          112     1/360
2+2+2+2    105     1/384
2           28    1/1440   P
0            1   1/40320

By the 20% rule, anything below a pure 5-cycle is considered lucky. So even 4+4, 4+2+2, and 2+2+2+2 are considered lucky?

Edges will be coming soon, they're a little ugly.

JeffDelucia · Jul 13, 2010

So a corners skip BLD is 1/40320 chance?

Baian Liu · Jul 13, 2010

JeffDelucia said:
So a corners skip BLD is 1/40320 chance?

Corner permutation, I believe.

Kirjava · Jul 13, 2010

blah said:
By the 20% rule, anything below a pure 5-cycle is considered lucky. So even 4+4, 4+2+2, and 2+2+2+2 are considered lucky?

Another reason why the term 'lucky' needs to be phased out.

blah · Jul 13, 2010

8! = 40320.

Sakarie · Jul 13, 2010

Yeah, this (very interesting) statistics would see corners in place but unoriented as solved. For a total skip, you have to multiply by 3^7.

blah · Jul 14, 2010

Didn't want to create a new thread, so here's a (much more interesting) page for corner orientations.

Is it a coincidence that there are exactly 100 cases? Anyone wanna explain?

Now, to combine orientation and permutation...

Escher · Jul 14, 2010

blah said:
Didn't want to create a new thread, so here's a (much more interesting) page for corner orientations.

Looks like it's time for me to generate good algorithms for them all

amostay2004 · Jul 14, 2010

100 cases doesn't seem too bad at all for 1 look CO

I think 3OP corners could potentially be as fast as BH/freestyle, or at least come very close

blah · Jul 14, 2010

LOL noobs. 5OP is the cool thing now. Only 7 more algs to learn

blah · Jul 14, 2010

Oh and commutators will always be better than nOP because braindead commies < braindead nOP. Move count.

amostay2004 · Jul 14, 2010

blah said:
LOL noobs. 5OP is the cool thing now. Only 7 more algs to learn

Does that mean Orientation + 5-cycle corners?

blah · Jul 14, 2010

Yes. To be able to do *any* 5-cycle (with corners oriented) within 2 setup moves, you only need seven 5-cycle algs, excluding mirrors and inverses. But I haven't really given it much thought yet - maybe someone can find even fewer algs

Oh and 2 algs max to solve all corners.

Only 4+4 requires both algs to be 5-cycles, for the remaining cases you need a maximum of one 5-cycle together with a 3-cycle/double 2-swap.

blah · Jul 14, 2010

And there's a much easier approach than learning all 100 cases

I need to give it some thought first before I publish it.

blah · Jul 14, 2010

Updated with new tables. As per intuition, your best bet is to learn 1-look algs for all the cases with 5 or 6 corners twisted. (You probably already know 1-look algs for everything up to 4 corners twisted anyway

)

Note some counterintuitive things like how case 8 is more common than cases 1, 2, and 3 combined!

Here's the link.

Lucas Garron · Jul 14, 2010

Code:

PermutationStructures[8, "EvenOnly" -> False]

(7)            5760   5760    1/7
(8)            5040   10800   1/8
(5)(2)         4032   14832   1/10
(6)            3360   18192   1/12
(4)(3)         3360   21552   1/12
(6)(2)         3360   24912   1/12
(5)(3)         2688   27600   1/15
(4)(2)         2520   30120   1/16
(3)(2)(2)      1680   31800   1/24
(5)            1344   33144   1/30
(4)(2)(2)      1260   34404   1/32
(4)(4)         1260   35664   1/32
(3)(2)         1120   36784   1/36
(3)(3)         1120   37904   1/36
(3)(3)(2)      1120   39024   1/36
(4)            420    39444   1/96
(2)(2)(2)      420    39864   1/96
(2)(2)         210    40074   1/192
(3)            112    40186   1/360
(2)(2)(2)(2)   105    40291   1/384
(2)            28     40319   1/1440
()             1      40320   1/40320

Code:

PermutationStructures[8, "EvenOnly" -> True]

(7)            5760   5760    1/7
(6)(2)         3360   9120    1/12
(5)(3)         2688   11808   1/15
(4)(2)         2520   14328   1/16
(3)(2)(2)      1680   16008   1/24
(5)            1344   17352   1/30
(4)(4)         1260   18612   1/32
(3)(3)         1120   19732   1/36
(2)(2)         210    19942   1/192
(3)            112    20054   1/360
(2)(2)(2)(2)   105    20159   1/384
()             1      20160   1/40320

Code:

PermutationStructures[12, "EvenOnly" -> False]

(11)                 43545600   43545600    1/11
(12)                 39916800   83462400    1/12
(9)(2)               26611200   110073600   1/18
(10)                 23950080   134023680   1/20
(10)(2)              23950080   157973760   1/20
(8)(3)               19958400   177932160   1/24
(9)(3)               17740800   195672960   1/27
(7)(4)               17107200   212780160   1/28
(6)(5)               15966720   228746880   1/30
(8)(2)               14968800   243715680   1/32
(8)(4)               14968800   258684480   1/32
(7)(5)               13685760   272370240   1/35
(6)(3)(2)            13305600   285675840   1/36
(5)(4)(2)            11975040   297650880   1/40
(7)(3)               11404800   309055680   1/42
(7)(3)(2)            11404800   320460480   1/42
(6)(4)               9979200    330439680   1/48
(6)(4)(2)            9979200    340418880   1/48
(9)                  8870400    349289280   1/54
(7)(2)(2)            8553600    357842880   1/56
(5)(3)(2)            7983360    365826240   1/60
(5)(4)(3)            7983360    373809600   1/60
(8)(2)(2)            7484400    381294000   1/64
(6)(6)               6652800    387946800   1/72
(7)(2)               5702400    393649200   1/84
(5)(3)(3)            5322240    398971440   1/90
(6)(2)(2)            4989600    403961040   1/96
(4)(3)(2)(2)         4989600    408950640   1/96
(4)(4)(3)            4989600    413940240   1/96
(5)(5)               4790016    418730256   1/100
(5)(5)(2)            4790016    423520272   1/100
(6)(3)               4435200    427955472   1/108
(6)(3)(3)            4435200    432390672   1/108
(5)(4)               3991680    436382352   1/120
(5)(3)(2)(2)         3991680    440374032   1/120
(4)(4)(2)            3742200    444116232   1/128
(4)(3)(2)            3326400    447442632   1/144
(4)(3)(3)            3326400    450769032   1/144
(4)(3)(3)(2)         3326400    454095432   1/144
(8)                  2494800    456590232   1/192
(5)(2)(2)            1995840    458586072   1/240
(5)(2)(2)(2)         1995840    460581912   1/240
(4)(4)(2)(2)         1871100    462453012   1/256
(6)(2)               1663200    464116212   1/288
(3)(3)(2)(2)         1663200    465779412   1/288
(6)(2)(2)(2)         1663200    467442612   1/288
(3)(3)(3)(2)         1478400    468921012   1/324
(5)(3)               1330560    470251572   1/360
(4)(2)(2)(2)         1247400    471498972   1/384
(4)(4)(4)            1247400    472746372   1/384
(4)(2)(2)            623700     473370072   1/768
(4)(4)               623700     473993772   1/768
(7)                  570240     474564012   1/840
(3)(3)(2)            554400     475118412   1/864
(3)(2)(2)(2)         554400     475672812   1/864
(3)(3)(2)(2)(2)      554400     476227212   1/864
(3)(3)(3)            492800     476720012   1/972
(3)(2)(2)(2)(2)      415800     477135812   1/1152
(5)(2)               399168     477534980   1/1200
(4)(3)               332640     477867620   1/1440
(4)(2)(2)(2)(2)      311850     478179470   1/1536
(3)(3)(3)(3)         246400     478425870   1/1944
(3)(2)(2)            166320     478592190   1/2880
(6)                  110880     478703070   1/4320
(4)(2)               83160      478786230   1/5760
(2)(2)(2)(2)(2)      62370      478848600   1/7680
(2)(2)(2)(2)         51975      478900575   1/9216
(3)(3)               36960      478937535   1/12960
(5)                  19008      478956543   1/25200
(3)(2)               15840      478972383   1/30240
(2)(2)(2)            13860      478986243   1/34560
(2)(2)(2)(2)(2)(2)   10395      478996638   1/46080
(4)                  2970       478999608   1/161280
(2)(2)               1485       479001093   1/322560
(3)                  440        479001533   1/1088640
(2)                  66         479001599   1/7257600
()                   1          479001600   1/479001600

Code:

PermutationStructures[12, "EvenOnly" -> True]

(11)                 43545600   43545600    1/11
(10)(2)              23950080   67495680    1/20
(9)(3)               17740800   85236480    1/27
(8)(2)               14968800   100205280   1/32
(8)(4)               14968800   115174080   1/32
(7)(5)               13685760   128859840   1/35
(6)(3)(2)            13305600   142165440   1/36
(5)(4)(2)            11975040   154140480   1/40
(7)(3)               11404800   165545280   1/42
(6)(4)               9979200    175524480   1/48
(9)                  8870400    184394880   1/54
(7)(2)(2)            8553600    192948480   1/56
(6)(6)               6652800    199601280   1/72
(5)(3)(3)            5322240    204923520   1/90
(4)(4)(3)            4989600    209913120   1/96
(5)(5)               4790016    214703136   1/100
(5)(3)(2)(2)         3991680    218694816   1/120
(4)(3)(2)            3326400    222021216   1/144
(4)(3)(3)(2)         3326400    225347616   1/144
(5)(2)(2)            1995840    227343456   1/240
(4)(4)(2)(2)         1871100    229214556   1/256
(6)(2)               1663200    230877756   1/288
(3)(3)(2)(2)         1663200    232540956   1/288
(6)(2)(2)(2)         1663200    234204156   1/288
(5)(3)               1330560    235534716   1/360
(4)(2)(2)(2)         1247400    236782116   1/384
(4)(4)               623700     237405816   1/768
(7)                  570240     237976056   1/840
(3)(3)(3)            492800     238468856   1/972
(3)(2)(2)(2)(2)      415800     238884656   1/1152
(3)(3)(3)(3)         246400     239131056   1/1944
(3)(2)(2)            166320     239297376   1/2880
(4)(2)               83160      239380536   1/5760
(2)(2)(2)(2)         51975      239432511   1/9216
(3)(3)               36960      239469471   1/12960
(5)                  19008      239488479   1/25200
(2)(2)(2)(2)(2)(2)   10395      239498874   1/46080
(2)(2)               1485       239500359   1/322560
(3)                  440        239500799   1/1088640
()                   1          239500800   1/479001600

Code:

<< Combinatorica`
Options[PermutationStructures] := {"EvenOnly" -> False};
PermutationStructures[n_, OptionsPattern[]] := Block[{cycles, data},
  cycles = {#, (n!/(Times @@ #)/(Times @@ ((Length /@ (# // 
                Gather))!)))} & /@ Partitions[n];
  If[OptionValue["EvenOnly"], 
   cycles = Select[cycles, EvenQ[Length[Select[#[[1]], EvenQ]]] &]];
  data = SortBy[cycles, Last] // Reverse;
  {
     "(" <> StringJoin[Riffle[ToString /@ DeleteCases[#, 1], ")("]] <>
         ")" & /@ data[[All, 1]],
     data[[All, 2]],
     Accumulate[data[[All, 2]]],
     ToString[#, InputForm] & /@ (data[[All, 2]]/n!)
     } // Transpose // TableForm
  ]

Lucas Garron · Jul 14, 2010

Hmm.

Code:

Export["PermutationStructures.gif", 
 Table[ListLogPlot[(PermutationStructures[i])[[All, 2]], 
   PlotLabel -> i, Ticks -> None], {i, 1, 40, 1}]]

Ranzha · Jul 14, 2010

blah said:
Updated with new tables. As per intuition, your best bet is to learn 1-look algs for all the cases with 5 or 6 corners twisted. (You probably already know 1-look algs for everything up to 4 corners twisted anyway )

Note some counterintuitive things like how case 8 is more common than cases 1, 2, and 3 combined!

Here's the link.

I'm quite sure you already know why, but this is for the others who may not know.
This is because it'd be less likely to have one layer's corner orientation solved (1/81) than to have three corners in a layer oriented (1/27). Three times less likely.
Then, if you generate all of the cases (including mirrors/inverses), you'd get this:

Notation system:

A corner that needs to be twisted anticlockwise is labelled as "2".
A corner that needs to be twisted clockwise is labelled as "4".
A corner that is oriented is labelled as "0".
This corresponds to the number of times (R' D' R D) must be applied to orient the corner if it were in the top layer.

Say you had the bowtie/L OCLL.
Like this:
0 2
4 0
This'd be read as 0204, clockwise from the UBL corner around the U layer.
For the other cases, it'd be U/D.

0 2
4 0
0 0
0 0

The third and fourth rows are the D layer, starting with DFL, then DFR, DBL, and DBR.
Read as written, from left to right, top to bottom.
02040000.
Yeah.

Do remember that for corner orientation to be solvable, the values of all of the labels should add up to a multiple of six.
Thus, 2 + 4 = 6, and 6 is divisible by six. Therefore, Bowtie/L OCLL is solvable.

With one layer's CO solved:
Bottom is 0000, so it's left out of the typed string.
27 cases including mirrors and inverses, which form into 8 groups:
Solved (O), Headlights (U), Chameleon (T), Antisune (A), Bowtie (L), Sune (S), Pi (P), and Double Sune (H).

0000 O
0024 U
0042 T
0204 T
0222 A
0240 L
0402 U
0420 L
0444 S
2004 L
2022 A
2040 U
2202 A
2220 A
2244 P
2400 T
2424 P
2442 H
4002 L
4020 T
4044 S
4200 U
4224 H
4242 P
4404 S
4422 P
4440 S

Then, cases for one corner in bottom layer disoriented.
54 cases if the disoriented corner in the bottom layer is fixed in the DFR position.
Thus, the first four digits of the notation are the U layer, and then the D layer covers the last digit.

If the disoriented corner is a "2".

00042
00222
00402
02022
02202
02442
04002
04242
04422
20022
20202
20442
22002
22242
22422
24042
24222
24402
40002
40242
40422
42042
42222
42402
44022
44202
44442

If the disoriented corner is a "4".

00024
00204
00444
02004
02244
02424
04044
04224
04404
20004
20244
20424
22044
22224
22404
24024
24204
24444
40044
40224
40404
42024
42204
42444
44004
44244
44424

Do realize that there are 54 × 4 = 216 cases for this is the D-layer misoriented corner is not in a fixed position, but that's for another day.

So yeah, 81 total cases.

Probabilities for corner cycles

blah

brah

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