# Largest coset solved yet of actual Rubik's positions

Using Tom Rokicki's coset solver as well as his optimal solver I managed to do a full analysis of the corner and edge permutations of all the 3x3x3 cube positions that have the orientation in the solved state.

Here is the distribution table:
```0              1
1              4
2             10
3             36
4            123
5            368
6           1336
7           4928
8          16839
9          63920
10        257888
11       1019992
12       4317941
13      20240924
14     102343680
15     568081384
16    3458261494
17   22676234692
18  153062896516
19  935739718492
20 3412399473195
21 3865485362804
22 1259308558965
23    3845400456
24            12
9656672256000
```

The above table was generated after combining 495 Koceimba's coset. After symmetry reduction only 81 needed to be solved. Here is the distribution of each one of the 81 solved coset (the first column is the number of symmetries the coset has):

```                Depth     0     1     2     3     4     5     6     7     8     9              10              11              12              13              14              15              16              17              18              19              20              21              22              23              24           Total
8                U2U2     1     4    10    36   123   368  1320  4800 15495 54016          194334          656752         2222295         7814000        26402962        89183776       297590924       929624528      2573889614      5506671444      6551983325      3219955376       301913989          249300               8     19508428800
4            B3L2R2F3     0     0     0     0     0     0     8    32   116   432            1636            5928           24980          103608          458574         2231438        11992546        70176756       429202988      2347804894      7324331676      7329897766      1988201876         3993546               0     19508428800
1          B3L2R2F3D1     0     0     0     0     0     0     0     8    32   144             580            2521           10770           54980          282364         1595057         9632476        61871585       400895021      2280000673      7299016431      7406219631      2044376726         4469801               0     19508428800
1          B3L2U2R2F3     0     0     0     0     0     0     0     0    39   268            1194            5655           29891          153371          790913         4220673        23087957       128624179       696355920      3137483639      7660052358      6481589639      1373896128         2136976               0     19508428800
2        B3L2R2F3D3U3     0     0     0     0     0     0     0     0    24    96             376            1780            9910           46940          263159         1537484         9487305        61344651       400335887      2279988642      7299668060      7406557125      2044449679         4737682               0     19508428800
2      B3D2U2F3L2F2U2     0     0     0     0     0     0     0     0    16    64             358            1822           10085           51036          277079         1585519         9675538        62371249       405412817      2299581914      7310590193      7385966873      2028248314         4655923               0     19508428800
2      B3D2U2F3D2F2R2     0     0     0     0     0     0     0     0    16    96             468            2448           12500           61824          315788         1732452        10195612        64202339       411968096      2319901098      7331467781      7364089601      2000254139         4224542               0     19508428800
2      B3L2R2F3L2D1U3     0     0     0     0     0     0     0     0     8    32             204            1060            6538           34196          210094         1299008         8469914        56754114       380508316      2220016630      7275138986      7471623108      2089880340         4486252               0     19508428800
4    B3L2R2F3L2D1F2U3     0     0     0     0     0     0     0     0     8    32             184             952            6028           32486          202324         1259604         8212344        55423750       374093830      2201585450      7275658280      7491292560      2096041402         4619566               0     19508428800
2          B3L2R2F3L2     0     0     0     0     0     0     0     0     8    32             254            1352            7314           39866          246746         1499316         9586116        63047182       411683476      2326993792      7346032920      7358682024      1986657566         3950836               0     19508428800
2          B3L2R2F3D2     0     0     0     0     0     0     0     0     8    40             286            1328            7462           42738          260584         1558036         9898190        64655806       418416476      2343807762      7335855332      7339697362      1989776062         4451328               0     19508428800
2        B3L2R2F3D1U3     0     0     0     0     0     0     0     0     8    32             272            1226            6948           37310          228754         1364398         8789142        58267172       386209402      2229968192      7254490206      7459381432      2104489668         5194638               0     19508428800
4    B3L2R2F3D1R2F2U3     0     0     0     0     0     0     0     0     8    32             176             808            5586           31568          198556         1237880         8111604        54878130       370190264      2167596834      7174439542      7523969328      2201268664         6499820               0     19508428800
1        F3D3B2F2U1F3     0     0     0     0     0     0     0     0     6   108             817            4437           24017          132038          721691         3951028        22043373       124463711       680872669      3101606391      7656604639      6521806479      1393947188         2250208               0     19508428800
1        F3D3B2F2U1F1     0     0     0     0     0     0     0     0     6    89             710            4139           23054          127101          701929         3875368        21742881       123193262       675380882      3080144685      7634610583      6544395049      1421754355         2474707               0     19508428800
2      B3L2R2F3L2D3U3     0     0     0     0     0     0     0     0     4    16             136             808            5480           31778          206402         1302599         8619806        58174769       389497507      2260206183      7319017404      7430364094      2036867661         4134153               0     19508428800
2      B3D3F3L2B1U1F1     0     0     0     0     0     0     0     0     4    16             120             670            5124           30720          198990         1264090         8358162        56623614       380858450      2222027952      7265201204      7469252142      2099592346         5015196               0     19508428800
1      B3D1B1U2F3D3F3     0     0     0     0     0     0     0     0     2    16              82             515            3670           25721          175670         1186870         8087055        55904772       378932610      2212929632      7239041376      7478720300      2127973935         5446574               0     19508428800
1      B3D1B1U2F3D3F1     0     0     0     0     0     0     0     0     2    14              98             737            4793           30381          196572         1287567         8573798        58259325       390225482      2260262402      7311992244      7430212196      2043221411         4161778               0     19508428800
1        B3L2U2R2F3R2     0     0     0     0     0     0     0     0     0    57             575            3645           21437          126648          716786         4007849        22623825       128280797       698283864      3143945309      7656672090      6475529503      1375895823         2320592               0     19508428800
2      B3D2U2F3D3U3R2     0     0     0     0     0     0     0     0     0    32             284            1502            7679           42625          243539         1425293         8902917        58576017       387846497      2246916313      7294852973      7442756788      2062360511         4495830               0     19508428800
1        B3D2U2F3D3U2     0     0     0     0     0     0     0     0     0    32             171            1059            5848           39347          236103         1482590         9496669        62840191       409743245      2311940323      7306678363      7372953748      2028054001         4957110               0     19508428800
1        B3L2R2F3L2D1     0     0     0     0     0     0     0     0     0    30             169            1040            5546           34994          211339         1356073         8802313        59207036       393918272      2271226526      7317041121      7418270769      2034235640         4117932               0     19508428800
1      B3L2R2F3L2D1F2     0     0     0     0     0     0     0     0     0    28             140             920            5179           33762          206551         1313792         8548824        57663074       386063383      2243870055      7293554486      7446735584      2065835837         4597185               0     19508428800
1      B3L2R2F3D1R2F2     0     0     0     0     0     0     0     0     0    28             128             895            5277           34321          206817         1314629         8564074        57702986       385310568      2230897069      7247410666      7458841939      2112716870         5422533               0     19508428800
1        B3L2R2F3D1R2     0     0     0     0     0     0     0     0     0    28             150            1021            5546           34489          209205         1349293         8770469        59089321       393715358      2271774342      7318843166      7417821443      2032670506         4144463               0     19508428800
1        B3L2R2F3D1B2     0     0     0     0     0     0     0     0     0    22             195            1084            6032           35717          216144         1349830         8774637        58804101       390759279      2254949485      7290985503      7434538809      2063472610         4535352               0     19508428800
1        B3L2R2F3U3L2     0     0     0     0     0     0     0     0     0    16             134             800            4961           31474          199066         1289760         8578423        58245766       389823873      2257680418      7309101022      7432779048      2046506921         4187118               0     19508428800
2        B3L2R2F3U2F2     0     0     0     0     0     0     0     0     0     4             124             962            6002           37964          238070         1489040         9549068        62987854       411297030      2320810802      7312896936      7364053738      2020227170         4834036               0     19508428800
2        B3D2U2F3R2U2     0     0     0     0     0     0     0     0     0     4             220            1154            6367           36738          232593         1429213         9298756        61886575       407376460      2319516127      7350665556      7367413499      1986634448         3931090               0     19508428800
1      B3L2R2F3U3L2F2     0     0     0     0     0     0     0     0     0     2              44             455            3471           24511          169119         1156284         7956354        55370437       377516121      2219004578      7280361076      7473936844      2088208215         4721289               0     19508428800
1      B3L2R2U2R2U2F1     0     0     0     0     0     0     0     0     0     0             172             738            4180           21448          125057          731978         4781962        34107607       252224418      1703368534      6797799117      8007870820      2699279494         8113275               0     19508428800
4        B3D2U2F3D2U2     0     0     0     0     0     0     0     0     0     0             136             924            6300           39674          260236         1635022        10572950        68956214       440624724      2414389322      7365243560      7264443514      1937506494         4749730               0     19508428800
2        B3L2R2F3R2F2     0     0     0     0     0     0     0     0     0     0             124             762            5418           33190          216052         1371846         9030662        60498710       400504826      2288301936      7302841986      7399267630      2041615332         4740326               0     19508428800
2      B3L2R2F3R2U2F2     0     0     0     0     0     0     0     0     0     0              68             638            4796           31912          206811         1312318         8614081        58411756       391029571      2259818964      7282010338      7429505632      2072348735         5133180               0     19508428800
1  F3L2R2D2R2U2F3D3U3     0     0     0     0     0     0     0     0     0     0              64             264            1779            9678           70407          483074         3705312        29158566       230478351      1629917416      6745388211      8086137133      2774570276         8508269               0     19508428800
1  B3L2R2U2R2U2F1D3U3     0     0     0     0     0     0     0     0     0     0              64             268            1830            9850           69895          478229         3671562        28972801       229260801      1621066302      6714966276      8094316430      2806243972         9370520               0     19508428800
1      B3L2R2F3U3L2R2     0     0     0     0     0     0     0     0     0     0              63             689            4794           32339          211193         1368533         9024499        60702261       400977141      2283759836      7288207279      7403241965      2055789431         5108777               0     19508428800
2      B3L2R2F3U2F2D3     0     0     0     0     0     0     0     0     0     0              44             528            4085           27680          191370         1260088         8395622        57281396       385645152      2240501369      7270527562      7449892726      2089450564         5250613               1     19508428800
2    B3L2R2F3D1U1R2F2     0     0     0     0     0     0     0     0     0     0              44             500            4030           28092          184040         1201342         8012460        54895872       373086496      2196450538      7237904504      7496133360      2135022826         5504696               0     19508428800
1    B3L2R1U3L2U1R1F3     0     0     0     0     0     0     0     0     0     0              22             146             940            7505           59123          454767         3590425        28842355       228975153      1619563839      6721386026      8096580978      2800202711         8764810               0     19508428800
8        F3L2R2D2U2F3     0     0     0     0     0     0     0     0     0     0              18              72             622            3504           30674          248648         2094914        18101936       156781036      1250930688      6178660788      8470184760      3416646348        14744792               0     19508428800
1      F3D3U2B2F2U3B3     0     0     0     0     0     0     0     0     0     0              18             144            1093            8425           64757          491184         3807036        30036124       235049089      1641889815      6739653530      8073523019      2775638877         8265689               0     19508428800
1      F3D3U2B2F2U3B1     0     0     0     0     0     0     0     0     0     0              14             148            1142            8627           66021          498638         3834492        30103738       234981695      1642446750      6755617525      8073477701      2759713511         7678798               0     19508428800
1    B3D3B2F2D2U3F3U2     0     0     0     0     0     0     0     0     0     0              14             114             894            7475           60670          471842         3723885        29756055       234706963      1647132577      6764978506      8068848912      2750743468         7997425               0     19508428800
1    B3D3B2F2D2U3F1U2     0     0     0     0     0     0     0     0     0     0              14              94             776            7237           59476          466587         3711873        29798906       235017245      1646650296      6746777338      8068693867      2768647678         8597413               0     19508428800
1    F3D3B2F2D2U3B3U3     0     0     0     0     0     0     0     0     0     0              12             114             858            7299           57165          452879         3605299        29039676       230545665      1629361019      6745522937      8087241821      2774482464         8111592               0     19508428800
1    B3D1U2B3R2F1U1B3     0     0     0     0     0     0     0     0     0     0              12              68             676            6158           53838          431204         3492917        28399232       227126877      1612147161      6720032042      8104718692      2803508038         8511885               0     19508428800
1  B3D1U2F3U2B1D3U2F3     0     0     0     0     0     0     0     0     0     0              10              84             765            6764           55775          441973         3551709        28768319       228995182      1618435641      6703504417      8097177818      2818106542         9383801               0     19508428800
1  F3D3B2F2D2U3B3L2U3     0     0     0     0     0     0     0     0     0     0               8              72             708            6630           56300          447673         3607763        29221936       232028859      1634982319      6741701484      8080953335      2776819278         8602435               0     19508428800
1    F3D3B2F2D2U3B3D2     0     0     0     0     0     0     0     0     0     0               8              92             766            7057           57998          461827         3685752        29650761       234225853      1642999014      6741768360      8072422936      2774475663         8672713               0     19508428800
1    F3D3B2F2D2U3B1U3     0     0     0     0     0     0     0     0     0     0               8              98             836            6999           56873          450698         3586232        28943050       230255668      1629145715      6746472888      8087625108      2773841895         8042732               0     19508428800
1  B3D3U2L2R2U3B3R2U3     0     0     0     0     0     0     0     0     0     0               8              70             748            7161           57526          453086         3629807        29289898       232264208      1636155450      6743481467      8079748012      2774780636         8560723               0     19508428800
1  B3D3L2R2D2U3B1L2U3     0     0     0     0     0     0     0     0     0     0               8              56             618            5684           50658          419710         3447954        28193393       225870105      1601431296      6669346707      8114058449      2855498350        10105812               0     19508428800
1    B3L2R2U2R2U2F1U3     0     0     0     0     0     0     0     0     0     0               4             320            1628           10992           71431          524208         3878174        30199051       235700866      1645586385      6733837194      8068716337      2780725103         9177107               0     19508428800
1    F3L2R2D2R2U2F3U3     0     0     0     0     0     0     0     0     0     0               2             310            1642           11014           71603          527480         3893258        30271332       236064489      1649624271      6756453373      8065361455      2757730033         8418538               0     19508428800
1    F3D3U2B2F2U3B3U3     0     0     0     0     0     0     0     0     0     0               2              68             712            6185           52619          425737         3453080        28126960       225047877      1598994533      6677500986      8117172791      2848159124         9488126               0     19508428800
1    F3D3U2B2F2U3B1U3     0     0     0     0     0     0     0     0     0     0               2              94             878            6763           55629          442561         3500948        28225933       225342457      1604138421      6719159350      8113376845      2806155136         8023783               0     19508428800
1    F3D3B2F2D2U3B1D2     0     0     0     0     0     0     0     0     0     0               2              74             801            6932           57724          464881         3696497        29662606       234322936      1645261860      6761574636      8070818438      2754561804         7999609               0     19508428800
1    B3L2R2U2R2U2F1D3     0     0     0     0     0     0     0     0     0     0               0             482            2230           13201           79441          550666         3922000        29990553       233192657      1634749655      6737588232      8080356817      2779429840         8553026               0     19508428800
1  F3L2U2R2D2U2F3D3U2     0     0     0     0     0     0     0     0     0     0               0             152             938            7681           55429          454560         3571564        28923325       230713375      1631813601      6732545010      8083888326      2787328084         9126755               0     19508428800
1    F3L2U2R2D2U2F3U2     0     0     0     0     0     0     0     0     0     0               0              66            1378            8937           71167          520725         4032341        31520949       245056660      1691248516      6792711464      8022246717      2712341390         8668490               0     19508428800
1  F3D3U2B2F2U3B3U1R2     0     0     0     0     0     0     0     0     0     0               0              50             518            5330           48778          408225         3347049        27601680       222950615      1596899657      6698937239      8120362981      2828930201         8936477               0     19508428800
1  F3D3F3U2B1D3U2F3D1     0     0     0     0     0     0     0     0     0     0               0              48             528            5158           46419          398393         3334222        27698990       224069390      1600410406      6690727433      8116206517      2836036408         9494888               0     19508428800
1    F3D3U2B2F2U3B1R2     0     0     0     0     0     0     0     0     0     0               0              44             558            6393           55209          456613         3694073        29974012       236906721      1655898584      6760895869      8059310080      2752661970         8568674               0     19508428800
1    B3D3L2R2D2U3B1L2     0     0     0     0     0     0     0     0     0     0               0              42             576            6180           55238          454739         3689957        29918297       236607194      1651445506      6737289172      8063112427      2776572263         9277209               0     19508428800
4      F3L2R2D2U2F3U3     0     0     0     0     0     0     0     0     0     0               0              36             190            2464           22664          214454         1878806        16690348       148921734      1211741330      6121453044      8510512736      3481937962        15053032               0     19508428800
1  F3L2R2D2U2F3U3L2U3     0     0     0     0     0     0     0     0     0     0               0              10             166            1801           18573          184311         1761448        16316551       147259087      1202325714      6098304199      8520320351      3506870927        15065662               0     19508428800
2    F3L2R2D2U2F3U3R2     0     0     0     0     0     0     0     0     0     0               0               8             108            1540           17106          173866         1686974        15957678       145097062      1193135930      6092072750      8530116168      3515340400        14829210               0     19508428800
2  F3L2R2D2U2F3D3L2F2     0     0     0     0     0     0     0     0     0     0               0               8             104            1616           18050          182194         1743933        16276269       147070470      1203683957      6116815431      8520182559      3488566412        13887797               0     19508428800
4  F3L2R2D2U2F3D3B2F2     0     0     0     0     0     0     0     0     0     0               0               8             140            1696           17176          165308         1599096        15168320       139273768      1157430860      6009463010      8564725450      3603861210        16722758               0     19508428800
2  B3U3L2R2B2F2D2U3B3     0     0     0     0     0     0     0     0     0     0               0               8             186            1635           18830          189483         1785964        16452899       147925079      1204382991      6108377561      8519024172      3496106780        14163212               0     19508428800
1  B3L2R2F3D1B1L2R2F3     0     0     0     0     0     0     0     0     0     0               0               6             145            1911           19715          192439         1809091        16678672       149643532      1216124979      6125583193      8506534882      3477158724        14681511               0     19508428800
2    F3L2R2D2U2F3D3F2     0     0     0     0     0     0     0     0     0     0               0               4              94            1588           16956          173164         1680504        15860146       144489494      1190271898      6085834868      8532945060      3522192484        14962540               0     19508428800
8    F3L2R2D2U2F3D3U3     0     0     0     0     0     0     0     0     0     0               0               0             304            3608           32388          250368         2065536        17940760       156112622      1249436492      6197733266      8472780140      3398270284        13803032               0     19508428800
2  F3L2R2D2U2F3U3L2U2     0     0     0     0     0     0     0     0     0     0               0               0              96            1420           15694          164748         1629096        15503694       142030852      1175320556      6048410076      8547361036      3562128586        15862946               0     19508428800
2  F3L2R2D2U2F3D3B2U2     0     0     0     0     0     0     0     0     0     0               0               0              96            1294           16574          172886         1667894        15809564       144286486      1187544954      6072600044      8535340038      3535643306        15345664               0     19508428800
2  B3U3L2R2U2B2F2U3B3     0     0     0     0     0     0     0     0     0     0               0               0              80            1368           17610          179646         1733692        16106724       145360780      1188745012      6067246670      8534073490      3539855568        15108160               0     19508428800
1F3L2R2D2U2F3D3R2U3R2     0     0     0     0     0     0     0     0     0     0               0               0              76            1394           17446          177598         1728357        16216149       147308799      1206971083      6121336660      8516372002      3483823062        14476174               0     19508428800
2F3L2R2D2U2F3D3R2F2U3     0     0     0     0     0     0     0     0     0     0               0               0              76            1515           17760          179364         1710493        15960384       144768337      1188040702      6076191391      8535236166      3531526343        14796269               0     19508428800
1  F3L2R2D2U2F3D3B2U1     0     0     0     0     0     0     0     0     0     0               0               0              76            1247           16077          166520         1657312        15714056       144118344      1188819026      6084871089      8534492123      3523551502        15021428               0     19508428800
Total     1     4    10    36   123   368  1336  4928 16839 63920          257888         1019992         4317941        20240924       102343680       568081384      3458261494     22676234692    153062896516    935739718492   3412399473195   3865485362804   1259308558965      3845400456              12   9656672256000
```

An interesting result of my current calculation is the discovery of a new position at distance 24, its generating sequence is:

F- B+ R- U+ B+ U- D- R+ U- B- R+ U- F- R+ B+ U+ F- D- D- U+ B- F- U- U-

This combined with the 8 positions at distance 24 found by Silviu in the trivial coset (U2U2 in the above table) is a proof that positions with the solved orientation have a diameter of 24 and that they are 12 antipodes of them.

I would like to thank Tom Rokicki for sharing his amazing coset solver as well as his optimal solver without which the above calculation is simply impossible.

The next step is to solve the flipped cosset that contains Reid's position.

## Comment viewing options

### QTM analysis

I should have specified that the above analysis is for QTM.