240:
275:. Both these games have a rapidly increasing number of positions with each move. The total number of all possible positions, approximately 5×10 for chess and 10 (on a 19×19 board) for Go, is much too large to allow a brute force solution with current computing technology (compare the now solved, with great difficulty, Rubik's Cube at only about
62:
number of "configurations", with a relatively small, well-defined arsenal of "moves" that may be applicable to configurations and then lead to a new configuration. Solving the puzzle means to reach a designated "final configuration", a singular configuration, or one of a collection of configurations.
291:
Even this strategy is not possible with Go. Besides having hugely more positions to evaluate, no one so far has successfully constructed a set of simple rules for evaluating the strength of a Go position as has been done for chess, though neural networks trained through reinforcement learning can
247:
An algorithm to determine the minimum number of moves to solve Rubik's Cube was published in 1997 by
Richard Korf. While it had been known since 1995 that 20 was a lower bound on the number of moves for the solution in the worst case, Tom Rokicki proved in 2010 that no configuration requires more
98:
Instead of asking for a full solution, one can equivalently ask for a single move from an initial but not final configuration, where the move is the first of some optimal solution. An algorithm for the single-move version of the problem can be turned into an algorithm for the original problem by
292:
provide evaluations of a position that exceed human ability. Evaluation algorithms are prone to make elementary mistakes so even for a limited look ahead with the goal limited to finding the strongest interim position, a God's algorithm has not been possible for Go.
303:
proved this to be so by calculating a database of all positions with ten or fewer pieces, providing a God's algorithm for all end games of draughts which was used to prove that all perfectly played games of draughts end in a draw. However, draughts with only
283:
positions). Consequently, a brute force determination of God's algorithm for these games is not possible. While chess computers have been built that are capable of beating even the best human players, they do not calculate the game all the way to the end.
75:
the puzzle is solvable from that initial configuration, otherwise it signals the impossibility of a solution). A solution is optimal if the sequence of moves is as short as possible. The highest value of this, among all initial configurations, is known as
323:
The magnitude of the set of positions of a puzzle does not entirely determine whether a God's algorithm is possible. The already solved Tower of Hanoi puzzle can have an arbitrary number of pieces, and the number of positions increases exponentially as
103:
while applying each move reported to the present configuration, until a final one is reached; conversely, any algorithm for the original problem can be turned into an algorithm for the single-move version by truncating its output to its first move.
288:, for instance, searched only 11 moves ahead (counting a move by each player as two moves), reducing the search space to only 10. After this, it assessed each position for advantage according to rules derived from human play and experience.
239:
267:
Some well known games with a very limited set of simple well-defined rules and moves have nevertheless never had their God's algorithm for a winning strategy determined. Examples are the board games
71:
An algorithm can be considered to solve such a puzzle if it takes as input an arbitrary initial configuration and produces as output a sequence of moves leading to a final configuration (
461:), p. 207: "...the Pyraminx is much simpler than the Magic Cube... Nicholas Hammond has shown that God's Algorithm is at most 21 moves (including the four trivial vertex moves). "
221:
376:
349:
583:
996:
890:
Schaeffer, Jonathan; Burch, Neil; Björnsson, Yngvi; Kishimoto, Akihiro; Müller, Martin; Lake, Robert; Lu, Paul; Sutphen, Steve (14 September 2007).
37:. It refers to any algorithm which produces a solution having the fewest possible moves. The allusion to the deity is based on the notion that an
1700:
1694:
1612:
965:
234:
1374:
1369:
1364:
1359:
1065:
188:
puzzle, a God's algorithm is known for any given number of disks. The number of moves increases exponentially with the number of disks
95:
indexed by initial configurations would allow solutions to be found very quickly, but would require an extraordinary amount of memory.
1679:
137:
87:
Some writers, such as David Joyner, consider that for an algorithm to be properly referred to as "God's algorithm", it should also be
950:
850:
458:
1410:
1483:
84:
value. God's algorithm, then, for a given puzzle, is an algorithm that solves the puzzle and produces only optimal solutions.
883:
868:
823:
808:
437:
1034:
989:
1463:
145:
1731:
1721:
1508:
285:
91:, meaning that the algorithm does not require extraordinary amounts of memory or time. For example, using a giant
891:
1736:
1468:
982:
599:
1685:
814:
Davis, Darryl N.; Chalabi, T.; Berbank-Green, B., "Artificial-life, agents and Go", in
Mohammadian, Masoud,
168:
can be solved in 80 single-tile moves or 43 multi-tile moves in the worst case. For its generalization the
1726:
1663:
1538:
1498:
351:. Nevertheless, the solution algorithm is applicable to any size problem, with a running time scaling as
100:
511:
453:
by Ernö Rubik, Tamás Varga, Gerzson Kéri, György Marx, and Tamás
Vekerdy (1987, Oxford University Press,
299:(checkers) has long been suspected of being "played out" by its expert practitioners. In 2007 Schaeffer
1642:
1390:
1310:
63:
To solve the puzzle a sequence of moves is applied, starting from some arbitrary initial configuration.
1647:
1417:
405:
906:
399:
1300:
1085:
249:
1493:
1292:
1060:
1044:
960:, held in Berkeley, California, 10–16 August 1980, pp. 307–312, Birkhauser Boston Inc, 1983
561:
141:
1448:
1427:
1080:
1563:
1336:
1166:
1039:
961:
946:
930:
922:
879:
864:
846:
819:
804:
454:
433:
113:
34:
193:
1523:
1458:
1422:
1405:
1400:
1254:
1128:
1024:
914:
256:
354:
327:
1528:
1453:
1151:
185:
1075:
1005:
320:, in the database, is a much easier problem to solve –of the same order as Rubik's cube.
117:
23:
910:
430:
Jedburgh
Justice and Kentish Fire: The Origins of English in Ten Phrases and Expressions
1602:
1548:
1395:
1259:
1029:
839:
621:
387:
165:
121:
565:
1715:
1533:
1231:
1123:
1019:
566:"Finding a shortest solution for the N × N extension of the 15-puzzle is intractable"
532:
129:
27:
1553:
1473:
1241:
1203:
133:
92:
472:
1607:
1586:
1518:
1513:
1443:
1277:
1269:
1208:
1190:
1133:
1070:
674:
410:
392:
252:
1223:
1213:
59:
38:
926:
638:
Rokicki, Tomas; Kociemba, Herbert; Davidson, Morley; Dethridge, John (2010).
1637:
1543:
1503:
1171:
1161:
1118:
918:
591:
272:
148:, in which the configurations are the vertices, and the moves are the arcs.
125:
934:
958:
Proceedings of the Fourth
International Congress on Mathematical Education
956:
Singmaster, David, "The educational value of the
Hungarian 'Magic Cube'",
1558:
1488:
1478:
1318:
1249:
1198:
1100:
1095:
1090:
595:
572:. National Conference on Artificial Intelligence, 1986. pp. 168–172.
476:
296:
974:
173:
1328:
679:
55:
30:
176:, so it is not known whether there is a practical God's algorithm.
16:
Algorithm for solving a puzzle or game in the fewest possible moves
1282:
1156:
268:
238:
622:
Finding optimal solutions to Rubik's Cube using pattern databases
816:
New
Frontiers in Computational Intelligence and its Applications
978:
41:
being would know an optimal step from any given configuration.
22:
is a notion originating in discussions of ways to solve the
628:(AAAI-97), Providence, Rhode Island, Jul 1997, pp. 700–705.
756:
Moore & Mertens, chapter 1.3, "Playing chess with God"
639:
473:"Rubik's Cube quest for speedy solution comes to an end"
510:
A. Brüngger, A. Marzetta, K. Fukuda and J. Nievergelt,
259:
had "rashly conjectured" this number to be 20 in 1980.
172:-puzzle, the problem of finding an optimal solution is
531:
Norskog, Bruce; Davidson, Morley (December 8, 2010).
357:
330:
196:
1672:
1656:
1625:
1595:
1579:
1572:
1436:
1383:
1350:
1327:
1309:
1291:
1268:
1240:
1222:
1189:
1180:
1142:
1109:
1053:
1012:
584:"An optimal solution to the Towers of Hanoi Puzzle"
512:
The parallel search bench ZRAM and its applications
838:
370:
343:
255:on the length of optimal solutions. Mathematician
215:
112:Well-known puzzles fitting this description are
876:Catalysis, God's Algorithm, and the Green Demon
26:puzzle, but which can also be applied to other
990:
8:
626:Proc. Natl. Conf. on Artificial Intelligence
1576:
1186:
997:
983:
975:
778:Moore & Mertens, "Notes" to chapter 1
488:
486:
362:
356:
335:
329:
201:
195:
533:"The Fifteen Puzzle can be solved in 43
140:. These have in common that they can be
421:
859:Moore, Cristopher; Mertens, Stephan,
829:Fraser, Rober (ed); Hannah, W. (ed),
7:
1701:1982 World Rubik's Cube Championship
831:The Draught Players' Weekly Magazine
818:, pp. 125–139, IOS Press, 2000
1695:The Simple Solution to Rubik's Cube
878:, Amsterdam University Press, 2009
833:, vol. 2, Glasgow: J H Berry, 1885.
471:Jonathan Fildes (August 11, 2010).
845:. Johns Hopkins University Press.
588:Universidad Autónoma de Manizales
235:Optimal solutions for Rubik's Cube
138:missionaries and cannibals problem
14:
1066:Rubik's family cubes of all sizes
132:is also covered, as well as many
863:, Oxford University Press, 2011
1680:Rubik's Cube in popular culture
1:
582:Rueda, Carlos (August 2000).
516:Annals of Operations Research
248:than 20 moves. Thus, 20 is a
943:Notes on Rubik's Magic Cube
747:Fraser & Hannah, p. 197
1753:
1643:Thistlethwaite's algorithm
841:Adventures in Group Theory
312:positions and even fewer,
232:
861:The Nature of Computation
128:. The one-person game of
80:, or, more formally, the
1086:5×5×5 (Professor's Cube)
675:"Chess Position Ranking"
541:Domain of the Cube Forum
492:Singmaster, p. 311, 1980
451:Rubik's Cubic Compendium
243:A scrambled Rubik's Cube
1686:Rubik, the Amazing Cube
1081:4×4×4 (Rubik's Revenge)
919:10.1126/science.1144079
216:{\displaystyle 2^{n}-1}
1664:World Cube Association
1539:Anthony Michael Brooks
1499:Krishnam Raju Gadiraju
837:Joyner, David (2002).
372:
345:
244:
217:
142:modeled mathematically
101:invoking it repeatedly
54:The notion applies to
1657:Official organization
1311:Truncated icosahedron
373:
371:{\displaystyle 2^{n}}
346:
344:{\displaystyle 3^{n}}
242:
218:
1076:3×3×3 (Rubik's Cube)
892:"Checkers Is Solved"
640:"God's Number is 20"
432:, Hachette UK, 2014
428:Paul Anthony Jones,
400:Proofs from THE BOOK
355:
328:
228:
194:
1351:Virtual combination
1183:combination puzzles
1145:combination puzzles
1071:2×2×2 (Pocket Cube)
941:Singmaster, David,
911:2007Sci...317.1518S
905:(5844): 1518–1522.
570:Proceedings AAAI-86
295:On the other hand,
1732:Mathematical games
1648:Rubik's Cube group
1494:Prithveesh K. Bhat
1418:Rubik's Revolution
1293:Great dodecahedron
1045:Oskar van Deventer
874:Rothenberg, Gadi,
803:, MIT Press, 2004
726:Mohammadian, p. 11
620:Richard E. Korf, "
562:Manfred K. Warmuth
521:(1999), pp. 45–63.
406:Rubik's Cube group
368:
341:
245:
213:
152:Mechanical puzzles
114:mechanical puzzles
58:that can assume a
35:mathematical games
1722:Search algorithms
1709:
1708:
1621:
1620:
1346:
1345:
1110:Variations of the
1040:Panagiotis Verdes
966:978-0-8176-3082-9
663:Rothenberg, p. 11
1744:
1673:Related articles
1577:
1524:David Singmaster
1484:Shotaro Makisumi
1459:Jessica Fridrich
1437:Renowned solvers
1353:puzzles (>3D)
1301:Alexander's Star
1255:Pyraminx Crystal
1187:
1129:Nine-Colour Cube
1101:8×8×8 (V-Cube 8)
1096:7×7×7 (V-Cube 7)
1091:6×6×6 (V-Cube 6)
1013:Puzzle inventors
999:
992:
985:
976:
945:, Penguin, 1981
938:
896:
856:
844:
801:What is Thought?
788:
785:
779:
776:
770:
763:
757:
754:
748:
745:
739:
736:
730:
718:
712:
709:
703:
702:Singmaster, 1981
700:
694:
691:
685:
684:
670:
664:
661:
655:
654:
652:
650:
635:
629:
618:
612:
611:
609:
607:
598:. Archived from
579:
573:
558:
552:
551:
549:
547:
528:
522:
508:
502:
501:Joyner, page 149
499:
493:
490:
481:
480:
468:
462:
447:
441:
426:
377:
375:
374:
369:
367:
366:
350:
348:
347:
342:
340:
339:
319:
317:
311:
309:
282:
280:
257:David Singmaster
224:
222:
220:
219:
214:
206:
205:
1752:
1751:
1747:
1746:
1745:
1743:
1742:
1741:
1712:
1711:
1710:
1705:
1668:
1652:
1633:God's algorithm
1617:
1591:
1568:
1529:Ron van Bruchem
1454:Bob Burton, Jr.
1449:Édouard Chambon
1432:
1428:Rubik's Triamid
1379:
1352:
1342:
1323:
1305:
1287:
1264:
1236:
1218:
1182:
1176:
1152:Helicopter Cube
1144:
1138:
1111:
1105:
1049:
1008:
1003:
972:
894:
889:
853:
836:
799:Baum, Eric B.,
796:
791:
786:
782:
777:
773:
764:
760:
755:
751:
746:
742:
737:
733:
729:
719:
715:
710:
706:
701:
697:
692:
688:
672:
671:
667:
662:
658:
648:
646:
637:
636:
632:
619:
615:
605:
603:
581:
580:
576:
560:Daniel Ratner,
559:
555:
545:
543:
530:
529:
525:
509:
505:
500:
496:
491:
484:
470:
469:
465:
448:
444:
427:
423:
419:
384:
358:
353:
352:
331:
326:
325:
315:
313:
307:
305:
278:
276:
265:
237:
231:
197:
192:
191:
189:
186:Towers of Hanoi
182:
180:Towers of Hanoi
162:
154:
110:
69:
52:
47:
20:God's algorithm
17:
12:
11:
5:
1750:
1748:
1740:
1739:
1734:
1729:
1724:
1714:
1713:
1707:
1706:
1704:
1703:
1698:
1691:
1690:
1689:
1676:
1674:
1670:
1669:
1667:
1666:
1660:
1658:
1654:
1653:
1651:
1650:
1645:
1640:
1635:
1629:
1627:
1623:
1622:
1619:
1618:
1616:
1615:
1610:
1605:
1603:Layer by Layer
1599:
1597:
1593:
1592:
1590:
1589:
1583:
1581:
1574:
1570:
1569:
1567:
1566:
1561:
1556:
1551:
1549:Feliks Zemdegs
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1464:Chris Hardwick
1461:
1456:
1451:
1446:
1440:
1438:
1434:
1433:
1431:
1430:
1425:
1420:
1415:
1414:
1413:
1411:Master Edition
1403:
1398:
1393:
1387:
1385:
1381:
1380:
1378:
1377:
1375:Magic 120-cell
1372:
1367:
1362:
1356:
1354:
1348:
1347:
1344:
1343:
1341:
1340:
1337:Rubik's Domino
1333:
1331:
1325:
1324:
1322:
1321:
1315:
1313:
1307:
1306:
1304:
1303:
1297:
1295:
1289:
1288:
1286:
1285:
1280:
1274:
1272:
1266:
1265:
1263:
1262:
1260:Skewb Ultimate
1257:
1252:
1246:
1244:
1238:
1237:
1235:
1234:
1228:
1226:
1220:
1219:
1217:
1216:
1211:
1206:
1201:
1195:
1193:
1184:
1178:
1177:
1175:
1174:
1169:
1164:
1159:
1154:
1148:
1146:
1140:
1139:
1137:
1136:
1131:
1126:
1121:
1115:
1113:
1107:
1106:
1104:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1057:
1055:
1051:
1050:
1048:
1047:
1042:
1037:
1032:
1027:
1022:
1016:
1014:
1010:
1009:
1004:
1002:
1001:
994:
987:
979:
970:
969:
954:
939:
887:
872:
857:
851:
834:
827:
812:
795:
792:
790:
789:
780:
771:
758:
749:
740:
731:
728:
727:
724:
720:
713:
704:
695:
686:
665:
656:
630:
613:
574:
553:
523:
503:
494:
482:
463:
442:
420:
418:
415:
414:
413:
408:
403:
396:
390:
388:Oracle machine
383:
380:
365:
361:
338:
334:
264:
263:Unsolved games
261:
233:Main article:
230:
227:
212:
209:
204:
200:
181:
178:
166:Fifteen puzzle
161:
155:
153:
150:
146:directed graph
136:, such as the
122:Tower of Hanoi
109:
106:
68:
65:
51:
48:
46:
43:
15:
13:
10:
9:
6:
4:
3:
2:
1749:
1738:
1735:
1733:
1730:
1728:
1727:Logic puzzles
1725:
1723:
1720:
1719:
1717:
1702:
1699:
1697:
1696:
1692:
1688:
1687:
1683:
1682:
1681:
1678:
1677:
1675:
1671:
1665:
1662:
1661:
1659:
1655:
1649:
1646:
1644:
1641:
1639:
1636:
1634:
1631:
1630:
1628:
1624:
1614:
1611:
1609:
1606:
1604:
1601:
1600:
1598:
1594:
1588:
1585:
1584:
1582:
1578:
1575:
1571:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1534:Eric Limeback
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1441:
1439:
1435:
1429:
1426:
1424:
1423:Rubik's Snake
1421:
1419:
1416:
1412:
1409:
1408:
1407:
1406:Rubik's Magic
1404:
1402:
1401:Rubik's Clock
1399:
1397:
1394:
1392:
1389:
1388:
1386:
1382:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1357:
1355:
1349:
1338:
1335:
1334:
1332:
1330:
1326:
1320:
1317:
1316:
1314:
1312:
1308:
1302:
1299:
1298:
1296:
1294:
1290:
1284:
1281:
1279:
1276:
1275:
1273:
1271:
1267:
1261:
1258:
1256:
1253:
1251:
1248:
1247:
1245:
1243:
1239:
1233:
1232:Skewb Diamond
1230:
1229:
1227:
1225:
1221:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1196:
1194:
1192:
1188:
1185:
1179:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1150:
1149:
1147:
1141:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1116:
1114:
1108:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1058:
1056:
1054:Rubik's Cubes
1052:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1025:Larry Nichols
1023:
1021:
1018:
1017:
1015:
1011:
1007:
1000:
995:
993:
988:
986:
981:
980:
977:
973:
967:
963:
959:
955:
952:
951:0-907395-00-7
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
893:
888:
885:
881:
877:
873:
870:
866:
862:
858:
854:
852:0-8018-6947-1
848:
843:
842:
835:
832:
828:
825:
821:
817:
813:
810:
806:
802:
798:
797:
793:
784:
781:
775:
772:
768:
762:
759:
753:
750:
744:
741:
735:
732:
725:
722:
721:
717:
714:
708:
705:
699:
696:
690:
687:
682:
681:
676:
669:
666:
660:
657:
645:
641:
634:
631:
627:
623:
617:
614:
602:on 2004-06-05
601:
597:
593:
589:
585:
578:
575:
571:
567:
563:
557:
554:
542:
538:
536:
527:
524:
520:
517:
513:
507:
504:
498:
495:
489:
487:
483:
478:
474:
467:
464:
460:
459:0-19-853202-4
456:
452:
446:
443:
439:
435:
431:
425:
422:
416:
412:
409:
407:
404:
402:
401:
397:
394:
391:
389:
386:
385:
381:
379:
363:
359:
336:
332:
321:
302:
298:
293:
289:
287:
274:
270:
262:
260:
258:
254:
251:
241:
236:
226:
210:
207:
202:
198:
187:
179:
177:
175:
171:
167:
159:
156:
151:
149:
147:
143:
139:
135:
134:logic puzzles
131:
130:peg solitaire
127:
123:
119:
115:
107:
105:
102:
96:
94:
90:
85:
83:
79:
74:
66:
64:
61:
57:
49:
44:
42:
40:
36:
32:
29:
28:combinatorial
25:
21:
1737:Rubik's Cube
1693:
1684:
1632:
1580:Speedsolving
1554:Collin Burns
1509:Frank Morris
1474:Rowe Hessler
1391:Missing Link
1242:Dodecahedron
1204:Pyraminx Duo
1112:Rubik's Cube
1006:Rubik's Cube
971:
957:
942:
902:
898:
875:
860:
840:
830:
815:
800:
783:
774:
766:
761:
752:
743:
734:
723:Baum, p. 197
716:
711:Baum, p. 188
707:
698:
693:Baum, p. 199
689:
678:
673:John Tromp.
668:
659:
647:. Retrieved
643:
633:
625:
616:
604:. Retrieved
600:the original
587:
577:
569:
556:
544:. Retrieved
540:
534:
526:
518:
515:
506:
497:
466:
450:
445:
429:
424:
398:
395:(game of Go)
322:
300:
294:
290:
266:
246:
229:Rubik's Cube
183:
169:
163:
157:
118:Rubik's Cube
111:
97:
93:lookup table
88:
86:
81:
78:God's number
77:
72:
70:
53:
24:Rubik's Cube
19:
18:
1626:Mathematics
1608:CFOP method
1587:Speedcubing
1564:Mátyás Kuti
1519:Gilles Roux
1514:Lars Petrus
1444:Yu Nakajima
1396:Rubik's 360
1384:Derivatives
1370:MagicCube7D
1365:MagicCube5D
1360:MagicCube4D
1278:Impossiball
1270:Icosahedron
1209:Pyramorphix
1191:Tetrahedron
1143:Other cubic
1134:Sudoku Cube
1035:Tony Fisher
1030:Uwe Mèffert
738:Baum, p.197
411:Solved game
393:Divine move
253:upper bound
1716:Categories
1469:Kevin Hays
1224:Octahedron
1214:BrainTwist
1020:Ernő Rubik
884:9056295896
869:0191620807
824:9051994761
809:0262025485
794:References
765:Schaeffer
644:Cube20.org
438:1472116224
124:, and the
50:Definition
39:omniscient
1638:Superflip
1573:Solutions
1544:Mats Valk
1504:Tyson Mao
1181:Non-cubic
1172:Gear Cube
1162:Dino Cube
1124:Bump Cube
1119:Void Cube
927:0036-8075
769:, p. 1518
649:March 15,
606:March 15,
592:Manizales
546:March 15,
449:See e.g.
286:Deep Blue
208:−
126:15 puzzle
89:practical
1559:Max Park
1489:Toby Mao
1479:Leyan Lo
1319:Tuttminx
1250:Megaminx
1199:Pyraminx
1167:Square 1
1061:Overview
935:17641166
596:Colombia
564:(1986).
477:BBC News
382:See also
297:draughts
184:For the
160:-Puzzles
116:such as
108:Examples
67:Solution
1613:Optimal
1596:Methods
1339:(2x3x3)
907:Bibcode
899:Science
174:NP-hard
82:minimax
56:puzzles
31:puzzles
1329:Cuboid
964:
949:
933:
925:
882:
867:
849:
822:
807:
767:et al.
680:GitHub
457:
436:
301:et al.
120:, the
60:finite
1283:Dogic
1157:Skewb
895:(PDF)
787:Rueda
568:. in
535:moves
417:Notes
269:chess
250:sharp
144:as a
45:Scope
962:ISBN
947:ISBN
931:PMID
923:ISSN
880:ISBN
865:ISBN
847:ISBN
820:ISBN
805:ISBN
651:2022
608:2022
548:2022
455:ISBN
434:ISBN
271:and
164:The
33:and
915:doi
903:317
624:",
314:3.9
277:4.3
1718::
929:.
921:.
913:.
901:.
897:.
677:.
642:.
594:,
590:.
586:.
539:.
519:90
514:,
485:^
475:.
378:.
318:10
310:10
281:10
273:Go
225:.
73:if
998:e
991:t
984:v
968:.
953:.
937:.
917::
909::
886:.
871:.
855:.
826:.
811:.
683:.
653:.
610:.
550:.
537:"
479:.
440:.
364:n
360:2
337:n
333:3
316:×
308:×
306:5
279:×
223:)
211:1
203:n
199:2
190:(
170:n
158:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
