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Largest coset solved yet of actual Rubik's positions
Submitted by mdlazreg on Sun, 02/28/2010 - 23:32.
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:
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):
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.
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.
