80:
775:
1293:
711:
1351:
115:
1166:
20:
1139:
1173:
1159:
1152:
1145:
1187:
1180:
2856:
2611:(the size of the search space) is over 3.3×10 (Löbbing and Wegener, 1995). We would not want to try to solve this problem using brute force, but by using human insight and ingenuity we can solve the knight's tour without much difficulty. We see that the cardinality of a combinatorial optimization problem is not necessarily indicative of its difficulty.
1781:
1308:
onward moves. When calculating the number of onward moves for each candidate square, we do not count moves that revisit any square already visited. It is possible to have two or more choices for which the number of onward moves is equal; there are various methods for breaking such ties, including one
706:
A tour reported in the fifth book of
Bhagavantabaskaraby by Bhat Nilakantha, a cyclopedic work in Sanskrit on ritual, law and politics, written either about 1600 or about 1700 describes three knight's tours. The tours are not only reentrant but also symmetrical, and the verses are based on the same
1075:
knight's tours, and a much greater number of sequences of knight moves of the same length. It is well beyond the capacity of modern computers (or networks of computers) to perform operations on such a large set. However, the size of this number is not indicative of the difficulty of the problem,
1545:
47:
such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed (or re-entrant); otherwise, it is open.
1286:
A graphical representation of
Warnsdorf's Rule. Each square contains an integer giving the number of moves that the knight could make from that square. In this case, the rule tells us to move to the square with the smallest integer in it, namely 2.
1370:, and each neuron is initialized randomly to be either "active" or "inactive" (output of 1 or 0), with 1 implying that the neuron is part of the solution. Each neuron also has a state function (described below) which is initialized to 0.
1533:
1373:
When the network is allowed to run, each neuron can change its state and output based on the states and outputs of its neighbors (those exactly one knight's move away) according to the following transition rules:
1776:{\displaystyle V_{t+1}(N_{i,j})=\left\{{\begin{array}{ll}1&{\mbox{if}}\,\,U_{t+1}(N_{i,j})>3\\0&{\mbox{if}}\,\,U_{t+1}(N_{i,j})<0\\V_{t}(N_{i,j})&{\mbox{otherwise}},\end{array}}\right.}
28:
374:
For example, the first line can be read from left to right or by moving from the first square to the second line, third syllable (2.3) and then to 1.5 to 2.7 to 4.8 to 3.6 to 4.4 to 3.2.
148:
or 'arrangement in the steps of a horse'. The same verse in four lines of eight syllables each can be read from left to right or by following the path of the knight on tour. Since the
707:
tour, starting from different squares. Nilakantha's work is an extraordinary achievement being a fully symmetric closed tour, predating the work of Euler (1759) by at least 60 years.
86:
showing all possible paths for a knight's tour on a standard 8 × 8 chessboard. The numbers on each node indicate the number of possible moves that can be made from that position.
766:
saw Anand making 13 consecutive knight moves (albeit using both knights); online commentators jested that Anand was trying to solve the knight's tour problem during the game.
2754:
2012:
1930:
1848:
1382:
1086:
By dividing the board into smaller pieces, constructing tours on each piece, and patching the pieces together, one can construct tours on most rectangular boards in
2494:
2061:
1338:
A computer program that finds a knight's tour for any starting position using
Warnsdorf's rule was written by Gordon Horsington and published in 1984 in the book
2035:
1970:
1950:
1888:
1868:
1806:
2339:
2063:. When the network converges, either the network encodes a knight's tour or a series of two or more independent circuits within the same board.
1312:
This rule may also more generally be applied to any graph. In graph-theoretic terms, each move is made to the adjacent vertex with the least
152:
used for
Sanskrit are syllabic, each syllable can be thought of as representing a square on a chessboard. Rudrata's example is as follows:
2454:
Löbbing, Martin; Wegener, Ingo (1996). "The number of knight's tours equals 33,439,123,484,294—counting with binary decision diagrams".
2017:
Although divergent cases are possible, the network should eventually converge, which occurs when no neuron changes its state from time
1304:
for finding a single knight's tour. The knight is moved so that it always proceeds to the square from which the knight will have the
957:
136:(5.15), a Sanskrit work on Poetics, the pattern of a knight's tour on a half-board has been presented as an elaborate poetic figure (
122:, a chess-playing machine hoax. This particular solution is closed (circular), and can thus be completed from any point on the board.
722:, 260) also forming a knight's tour – no fully magic tours exist on an 8x8 board (although they do exist on larger boards)
843:
proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight's tour. For any
2938:
2762:
2538:
730:. The first procedure for completing the knight's tour was Warnsdorf's rule, first described in 1823 by H. C. von Warnsdorf.
2745:
2943:
923:
closed tours is half this number, since every tour can be traced in reverse. There are 9,862 undirected closed tours on a
2792:
2649:
2406:
2506:
2860:
2823:
2197:
2087:
2715:
2593:
2160:
390:
2291:
909:
closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are
2370:
2948:
1335:
was first described in "Des Rösselsprungs einfachste und allgemeinste Lösung" by H. C. von
Warnsdorf in 1823.
1324:
in general, on many graphs that occur in practice this heuristic is able to successfully locate a solution in
2490:
2146:
755:
2874:
1080:
1035:
There are several ways to find a knight's tour on a given board with a computer. Some of these methods are
2528:
67:
students. Variations of the knight's tour problem involve chessboards of different sizes than the usual
2072:
56:
2933:
1363:
919:
906:
409:
meter) where the second verse can be derived from the first verse by performing a Knight's tour on a
99:
2683:
79:
1317:
914:
91:
684:
The 20th verse that can be obtained by performing Knight's tour on the above verse is as follows:
2928:
2890:
Philip, Anish (2013). "A Generalized Pseudo-Knight?s Tour
Algorithm for Encryption of an Image".
2152:
747:
2189:
2178:
1975:
1893:
1811:
2870:
sequence A001230 (Number of undirected closed knight's tours on a 2n X 2n chessboard)
2753:. DATA ANALYTICS 2018: The Seventh International Conference on Data Analytics. Athens, greece:
2678:
2600:
The knight's tour problem is a classic combinatorial optimization problem. ... The cardinality
910:
126:
The earliest known reference to the knight's tour problem dates back to the 9th century AD. In
2583:
1366:
implementation. The network is set up such that every legal knight's move is represented by a
1313:
1087:
149:
2475:
2324:
2138:
2266:
1292:
774:
102:. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in
8:
2077:
2040:
1076:
which can be solved "by using human insight and ingenuity ... without much difficulty."
2907:
2696:
2362:
2092:
2082:
2020:
1955:
1935:
1873:
1853:
1791:
1052:
726:
After
Nilakantha, one of the first mathematicians to investigate the knight's tour was
413:
board, starting from the top-left corner. The transliterated 19th verse is as follows:
2743:
2644:
2819:
2758:
2589:
2555:
2534:
2234:
2217:
2193:
2156:
1367:
759:
710:
83:
2911:
2700:
2366:
2899:
2796:
2688:
2639:
2463:
2358:
2354:
2229:
2118:
715:
406:
64:
60:
2747:
A Predictive Data
Analytic for the Hardness of Hamiltonian Cycle Problem Instances
2669:
Pohl, Ira (July 1967). "A method for finding
Hamilton paths and Knight's tours".
2558:
2471:
2112:
1528:{\displaystyle U_{t+1}(N_{i,j})=U_{t}(N_{i,j})+2-\sum _{N\in G(N_{i,j})}V_{t}(N)}
1296:
A very large (130 × 130) square open knight's tour created using
Warnsdorf's Rule
763:
378:
119:
2505:. Department of Computer Science, Australian National University. Archived from
98:. The problem of finding a closed knight's tour is similarly an instance of the
2838:
Y. Takefuji, K. C. Lee. "Neural network computing for knight's tour problems."
2624:
2391:
727:
719:
394:
382:
40:
2903:
2250:
1595:
2922:
2480:
See attached comment by Brendan McKay, Feb 18, 1997, for the corrected count.
2185:
2142:
742:
737:
group of writers used it, among many others. The most notable example is the
385:, during the 14th century, in his 1,008-verse magnum opus praising the deity
699:
It is believed that Desika composed all 1,008 verses (including the special
2744:
Van Horn, Gijs; Olij, Richard; Sleegers, Joeri; Van den Berg, Daan (2018).
2123:
95:
2800:
2692:
1325:
1321:
1090:– that is, in a time proportional to the number of squares on the board.
1055:
for a knight's tour is impractical on all but the smallest boards. On an
103:
27:
19:
1350:
2437:
2423:
386:
44:
2879:
2884:
2563:
2325:"MathWorld News: There Are No Magic Knight's Tours on the Chessboard"
1332:
1301:
1040:
1036:
114:
402:
2467:
2218:"Solution of the Knight's Hamiltonian Path Problem on Chessboards"
2866:
2716:"A Warnsdorff-Rule Algorithm for Knight's Tours on Square Boards"
1362:
The knight's tour problem also lends itself to being solved by a
127:
2252:
Kavyalankara of Rudrata (Sanskrit text, with Hindi translation);
2855:
2723:
2216:
Conrad, A.; Hindrichs, T.; Morsy, H. & Wegener, I. (1994).
734:
90:
The knight's tour problem is an instance of the more general
2869:
2215:
1770:
952:
2791:. ACM Southeast Regional Conference. New York, New York:
2267:"Indian Institute of Information Technology, Bangalore"
2625:"An Efficient Algorithm for the Knight's Tour Problem"
1757:
1666:
1604:
741:
knight's tour which sets the order of the chapters in
31:
An animation of an open knight's tour on a 5 × 5 board
2553:
2340:"Which Rectangular Chessboards Have a Knight's Tour?"
2043:
2023:
1978:
1958:
1938:
1896:
1876:
1856:
1814:
1794:
1548:
1385:
63:
to find a knight's tour is a common problem given to
2533:. Society for Industrial & Applied Mathematics.
2117:(MS thesis). San José State University. p. 3.
703:mentioned above) in a single night as a challenge.
2789:Finding Re-entrant Knight's Tours on N-by-M Boards
2389:
2177:
2055:
2029:
2006:
1964:
1944:
1924:
1882:
1862:
1842:
1800:
1775:
1527:
1309:devised by Pohl and another by Squirrel and Cull.
2292:"Bridge-India: Paduka Sahasram by Vedanta Desika"
2255:. Delhitraversal: Parimal Sanskrit Series No. 30.
71:, as well as irregular (non-rectangular) boards.
2920:
2880:Introduction to Knight's tours by George Jelliss
2713:
2337:
1099:
1079:
865:one or more of these three conditions are met:
2885:Knight's tours complete notes by George Jelliss
2786:
2782:
2780:
2737:
2530:Branching Programs and Binary Decision Diagrams
2248:
1030:
804:one or more of these three conditions are met:
2453:
143:
137:
131:
2489:
1850:is the state of the neuron connecting square
2777:
2289:
2211:
2209:
2175:
1328:. The knight's tour is such a special case.
905:board, there are exactly 26,534,728,821,064
800:, a closed knight's tour is always possible
2816:Century/Acorn User Book of Computer Puzzles
2588:, John Wiley & Sons, pp. 449–450,
2577:
2575:
2526:
2385:
2383:
2137:
1345:
1340:Century/Acorn User Book of Computer Puzzles
938:Number of directed tours (open and closed)
2682:
2643:
2233:
2206:
2122:
1673:
1672:
1611:
1610:
1046:
778:A radially symmetric closed knight's tour
2875:H. C. von Warnsdorf 1823 in Google Books
2622:
2572:
2380:
1349:
1291:
773:
709:
113:
78:
26:
18:
16:Mathematical problem set on a chessboard
2664:
2662:
2449:
2447:
2014:is the set of neighbors of the neuron.
1808:represents discrete intervals of time,
1165:
59:of finding a knight's tour. Creating a
2921:
2889:
405:verses containing 32 letters each (in
2813:
2581:
2554:
2176:Deitel, H. M.; Deitel, P. J. (2003).
2110:
1135:
861:, a knight's tour is always possible
23:An open knight's tour of a chessboard
2714:Squirrel, Douglas; Cull, P. (1996).
2668:
2659:
2585:Evolutionary Optimization Algorithms
2444:
2456:Electronic Journal of Combinatorics
1093:
13:
896:
14:
2960:
2848:
2787:Alwan, Karla; Waters, K. (1992).
2438:"Knight's Tours on 4 by N Boards"
2424:"Knight's Tours on 3 by N Boards"
2390:Cull, P.; De Curtins, J. (1978).
2180:Java How To Program Fifth Edition
2114:Knight's Tours and Zeta Functions
1932:is the output of the neuron from
718:(its diagonals do not sum to its
2854:
2655:from the original on 2022-10-09.
2412:from the original on 2022-10-09.
1358:board solved by a neural network
1185:
1178:
1172:
1171:
1164:
1158:
1157:
1151:
1150:
1144:
1143:
1137:
2832:
2807:
2707:
2616:
2547:
2520:
2483:
2430:
2416:
2331:
2088:Longest uncrossed knight's path
1186:
1059:board, for instance, there are
401:) has composed two consecutive
118:The knight's tour as solved by
2359:10.1080/0025570X.1991.11977627
2317:
2308:
2283:
2259:
2242:
2169:
2131:
2104:
2001:
1982:
1919:
1900:
1837:
1818:
1751:
1732:
1709:
1690:
1647:
1628:
1584:
1565:
1522:
1516:
1501:
1482:
1456:
1437:
1421:
1402:
1179:
701:Chaturanga Turanga Padabandham
696:sA dhyA thA pa ka rA sa rA ||
693:dhu ran ha sAm sa nna thA dhA
1:
2645:10.1016/S0166-218X(96)00010-8
2148:The Oxford Companion to Chess
1081:Divide-and-conquer algorithms
756:World Chess Championship 2010
687:sThi thA sa ma ya rA ja thpA
2939:Hamiltonian paths and cycles
2632:Discrete Applied Mathematics
2503:Technical Report TR-CS-97-03
2314:A History of Chess by Murray
2235:10.1016/0166-218X(92)00170-Q
2222:Discrete Applied Mathematics
2111:Brown, Alfred James (2017).
1031:Finding tours with computers
782:Schwenk proved that for any
769:
690:ga tha rA mA dha kE ga vi |
39:is a sequence of moves of a
7:
2944:Mathematical chess problems
2290:Bridge-india (2011-08-05).
2066:
144:
138:
132:
10:
2965:
2818:. Century Communications.
2814:Dally, Simon, ed. (1984).
2145:(1996) . "knight's tour".
2073:Abu Bakr bin Yahya al-Suli
2007:{\displaystyle G(N_{i,j})}
1925:{\displaystyle V(N_{i,j})}
1843:{\displaystyle U(N_{i,j})}
1354:Closed knight's tour on a
109:
2904:10.1109/MPOT.2012.2219651
2671:Communications of the ACM
2392:"Knight's Tour Revisited"
2338:Allen J. Schwenk (1991).
733:In the 20th century, the
100:Hamiltonian cycle problem
74:
2098:
1346:Neural network solutions
1318:Hamiltonian path problem
92:Hamiltonian path problem
2462:(1). Research Paper 5.
2153:Oxford University Press
1024:19,591,828,170,979,904
2623:Parberry, Ian (1997).
2495:"Knight's Tours on an
2124:10.31979/etd.e7ra-46ny
2057:
2031:
2008:
1966:
1946:
1926:
1884:
1864:
1844:
1802:
1777:
1529:
1359:
1300:Warnsdorf's rule is a
1297:
1047:Brute-force algorithms
779:
754:The sixth game of the
723:
123:
87:
32:
24:
2949:Mathematical problems
2842:, 4(5):249–254, 1992.
2801:10.1145/503720.503806
2693:10.1145/363427.363463
2249:Satyadev, Chaudhary.
2058:
2032:
2009:
1967:
1947:
1927:
1885:
1865:
1845:
1803:
1778:
1530:
1353:
1295:
777:
713:
389:'s divine sandals of
381:poet and philosopher
150:Indic writing systems
117:
82:
53:knight's tour problem
30:
22:
2863:at Wikimedia Commons
2795:. pp. 377–382.
2527:Wegener, I. (2000).
2347:Mathematics Magazine
2041:
2021:
1976:
1956:
1936:
1894:
1874:
1854:
1812:
1792:
1546:
1383:
748:Life a User's Manual
57:mathematical problem
2582:Simon, Dan (2013),
2399:Fibonacci Quarterly
2078:Eight queens puzzle
2056:{\displaystyle t+1}
1039:, while others are
2757:. pp. 91–96.
2556:Weisstein, Eric W.
2093:Self-avoiding walk
2083:George Koltanowski
2053:
2027:
2004:
1962:
1942:
1922:
1880:
1860:
1840:
1798:
1773:
1768:
1761:
1670:
1608:
1525:
1505:
1360:
1298:
1053:brute-force search
780:
724:
124:
88:
33:
25:
2859:Media related to
2764:978-1-61208-681-1
2540:978-0-89871-458-6
2030:{\displaystyle t}
1965:{\displaystyle j}
1945:{\displaystyle i}
1883:{\displaystyle j}
1863:{\displaystyle i}
1801:{\displaystyle t}
1760:
1669:
1607:
1468:
1284:
1283:
1028:
1027:
917:). The number of
760:Viswanathan Anand
682:
681:
677:
669:
661:
653:
645:
637:
629:
621:
611:
603:
595:
587:
579:
571:
563:
555:
545:
537:
529:
521:
513:
505:
497:
489:
479:
471:
463:
455:
447:
439:
431:
423:
372:
371:
261:
260:
2956:
2934:Graph algorithms
2915:
2868:
2858:
2843:
2836:
2830:
2829:
2811:
2805:
2804:
2784:
2775:
2774:
2772:
2771:
2752:
2741:
2735:
2734:
2732:
2731:
2720:
2711:
2705:
2704:
2686:
2666:
2657:
2656:
2654:
2647:
2629:
2620:
2614:
2613:
2579:
2570:
2569:
2568:
2551:
2545:
2544:
2524:
2518:
2517:
2515:
2514:
2498:
2487:
2481:
2479:
2451:
2442:
2441:
2434:
2428:
2427:
2420:
2414:
2413:
2411:
2396:
2387:
2378:
2377:
2375:
2369:. Archived from
2344:
2335:
2329:
2328:
2321:
2315:
2312:
2306:
2305:
2303:
2302:
2287:
2281:
2280:
2278:
2277:
2263:
2257:
2256:
2246:
2240:
2239:
2237:
2213:
2204:
2203:
2184:(5th ed.).
2183:
2173:
2167:
2166:
2151:(2nd ed.).
2135:
2129:
2128:
2126:
2108:
2062:
2060:
2059:
2054:
2036:
2034:
2033:
2028:
2013:
2011:
2010:
2005:
2000:
1999:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1931:
1929:
1928:
1923:
1918:
1917:
1889:
1887:
1886:
1881:
1869:
1867:
1866:
1861:
1849:
1847:
1846:
1841:
1836:
1835:
1807:
1805:
1804:
1799:
1782:
1780:
1779:
1774:
1772:
1769:
1762:
1758:
1750:
1749:
1731:
1730:
1708:
1707:
1689:
1688:
1671:
1667:
1646:
1645:
1627:
1626:
1609:
1605:
1583:
1582:
1564:
1563:
1534:
1532:
1531:
1526:
1515:
1514:
1504:
1500:
1499:
1455:
1454:
1436:
1435:
1420:
1419:
1401:
1400:
1357:
1189:
1188:
1182:
1181:
1175:
1174:
1168:
1167:
1161:
1160:
1154:
1153:
1147:
1146:
1141:
1140:
1100:
1094:Warnsdorf's rule
1074:
1073:
1070:
1067:
1064:
1058:
1016:165,575,218,320
955:
930:
929:
926:
904:
852:
791:
740:
716:semimagic square
675:
667:
659:
651:
643:
635:
627:
619:
609:
601:
593:
585:
577:
569:
561:
553:
543:
535:
527:
519:
511:
503:
495:
487:
477:
469:
461:
453:
445:
437:
429:
421:
416:
415:
412:
397:(in chapter 30:
266:
265:
263:transliterated:
155:
154:
147:
145:turagapadabandha
141:
135:
70:
65:computer science
2964:
2963:
2959:
2958:
2957:
2955:
2954:
2953:
2919:
2918:
2892:IEEE Potentials
2851:
2846:
2837:
2833:
2826:
2812:
2808:
2785:
2778:
2769:
2767:
2765:
2750:
2742:
2738:
2729:
2727:
2718:
2712:
2708:
2684:10.1.1.412.8410
2667:
2660:
2652:
2627:
2621:
2617:
2605:
2596:
2580:
2573:
2552:
2548:
2541:
2525:
2521:
2512:
2510:
2496:
2488:
2484:
2452:
2445:
2436:
2435:
2431:
2422:
2421:
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2155:. p. 204.
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2019:
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1316:. Although the
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902:
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897:Number of tours
844:
783:
772:
764:Veselin Topalov
738:
410:
399:Chitra Paddhati
395:Paduka Sahasram
112:
77:
68:
17:
12:
11:
5:
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2929:Chess problems
2917:
2916:
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2861:Knight's Tours
2850:
2849:External links
2847:
2845:
2844:
2840:Neurocomputing
2831:
2825:978-0712605410
2824:
2806:
2776:
2763:
2736:
2706:
2677:(7): 446–449.
2658:
2638:(3): 251–260.
2615:
2603:
2594:
2571:
2559:"Knight Graph"
2546:
2539:
2519:
2482:
2443:
2429:
2415:
2379:
2376:on 2019-05-26.
2353:(5): 325–332.
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2228:(2): 125–134.
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2199:978-0131016217
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2143:Whyld, Kenneth
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1364:neural network
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936:
898:
895:
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893:
883:
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832:
822:
816:
771:
768:
728:Leonhard Euler
720:magic constant
680:
679:
671:
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631:
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383:Vedanta Desika
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139:citra-alaṅkāra
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84:Knight's graph
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2595:9781118659502
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2560:
2557:
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2542:
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2531:
2523:
2509:on 2013-09-28
2508:
2504:
2500:
2492:
2491:Brendan McKay
2486:
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2469:
2468:10.37236/1229
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2186:Prentice Hall
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2162:0-19-280049-3
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2139:Hooper, David
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2015:
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877:
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871:
868:
867:
866:
864:
860:
856:
851:
847:
842:
838:
831:= 4, 6, or 8.
830:
826:
823:
820:
817:
814:
810:
807:
806:
805:
803:
799:
795:
790:
786:
776:
767:
765:
761:
757:
752:
750:
749:
744:
743:Georges Perec
736:
731:
729:
721:
717:
712:
708:
704:
702:
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694:
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688:
685:
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426:
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418:
417:
414:
408:
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396:
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388:
384:
380:
379:Sri Vaishnava
375:
367:
364:
361:
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186:
183:
182:
178:
175:
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169:
166:
163:
160:
157:
156:
153:
151:
146:
142:) called the
140:
134:
129:
121:
116:
107:
105:
101:
97:
93:
85:
81:
72:
66:
62:
58:
54:
49:
46:
42:
38:
37:knight's tour
29:
21:
2898:(6): 10–16.
2895:
2891:
2839:
2834:
2815:
2809:
2788:
2768:. Retrieved
2746:
2739:
2728:. Retrieved
2722:
2709:
2674:
2670:
2635:
2631:
2618:
2608:
2601:
2599:
2584:
2562:
2549:
2529:
2522:
2511:. Retrieved
2507:the original
2502:
2485:
2459:
2455:
2432:
2418:
2402:
2398:
2371:the original
2350:
2346:
2333:
2319:
2310:
2299:. Retrieved
2296:Bridge-India
2295:
2285:
2274:. Retrieved
2270:
2261:
2251:
2244:
2225:
2221:
2179:
2171:
2147:
2133:
2113:
2106:
2016:
1787:
1372:
1361:
1339:
1337:
1330:
1311:
1305:
1299:
1085:
1050:
1034:
945:
941:
933:
918:
900:
889:
885:
882:= 3, 5, or 6
879:
875:
869:
862:
858:
854:
849:
845:
840:
836:
834:
828:
824:
821:= 1, 2, or 4
818:
815:are both odd
812:
808:
801:
797:
793:
788:
784:
781:
753:
746:
732:
725:
705:
700:
698:
695:
692:
689:
686:
683:
674:
666:
658:
650:
642:
634:
626:
618:
608:
600:
592:
584:
576:
568:
560:
552:
542:
534:
526:
518:
510:
502:
494:
486:
476:
468:
460:
452:
444:
436:
428:
420:
398:
376:
373:
262:
133:Kavyalankara
125:
96:graph theory
89:
52:
50:
36:
34:
2499:Chessboard"
2405:: 276–285.
2188:. pp.
1326:linear time
1088:linear time
915:reflections
853:board with
839:and Conrad
792:board with
104:linear time
2923:Categories
2770:2018-11-27
2730:2011-08-21
2513:2013-09-22
2301:2019-10-16
2276:2019-10-11
1870:to square
1041:heuristics
1037:algorithms
1008:6,637,920
950:(sequence
920:undirected
387:Ranganatha
45:chessboard
2679:CiteSeerX
2564:MathWorld
1759:otherwise
1477:∈
1470:∑
1466:−
1333:heuristic
1302:heuristic
911:rotations
770:Existence
745:'s novel
407:Anushtubh
391:Srirangam
2912:39213422
2701:14100648
2650:Archived
2493:(1997).
2407:Archived
2367:28726833
2067:See also
907:directed
888:= 4 and
878:= 3 and
872:= 1 or 2
827:= 3 and
758:between
403:Sanskrit
120:the Turk
2476:1368332
2190:326–328
1356:24 × 24
1322:NP-hard
956:in the
953:A165134
927:board.
739:10 × 10
128:Rudrata
110:History
61:program
55:is the
2910:
2822:
2761:
2724:GitHub
2699:
2681:
2592:
2537:
2474:
2365:
2196:
2159:
1972:, and
1788:where
1368:neuron
1314:degree
1306:fewest
1000:1,728
940:on an
901:On an
863:unless
841:et al.
837:et al.
802:unless
735:Oulipo
75:Theory
41:knight
2908:S2CID
2751:(PDF)
2719:(PDF)
2697:S2CID
2653:(PDF)
2628:(PDF)
2497:8 × 8
2410:(PDF)
2395:(PDF)
2374:(PDF)
2363:S2CID
2343:(PDF)
2099:Notes
1057:8 × 8
948:board
925:6 × 6
903:8 × 8
835:Cull
559:thpA
475:dhyA
419:sThi
411:4 × 8
69:8 × 8
43:on a
2867:OEIS
2820:ISBN
2759:ISBN
2590:ISBN
2535:ISBN
2194:ISBN
2157:ISBN
1713:<
1651:>
1331:The
958:OEIS
913:and
892:= 4.
811:and
762:and
668:(22)
665:nna
660:(13)
657:dha
652:(28)
636:(32)
628:(15)
620:(18)
617:ran
610:(12)
602:(25)
586:(21)
578:(14)
570:(17)
567:dhu
554:(31)
544:(23)
541:thA
528:(27)
525:thA
520:(10)
517:tha
512:(29)
501:thA
496:(19)
488:(16)
478:(26)
470:(11)
462:(24)
459:dhA
446:(20)
443:sAm
430:(30)
377:The
51:The
2900:doi
2797:doi
2793:ACM
2755:XPS
2689:doi
2640:doi
2607:of
2464:doi
2355:doi
2230:doi
2119:doi
2037:to
1952:to
1320:is
1072:532
1069:410
1066:364
1063:267
676:(5)
673:ya
649:pa
644:(7)
641:ja
633:rA
625:ga
607:mA
599:sA
594:(6)
591:rA
583:sa
575:kE
562:(8)
551:sa
536:(4)
533:ma
509:ka
504:(2)
493:ha
485:vi
467:rA
454:(3)
451:sa
438:(9)
435:ga
427:rA
422:(1)
368:lī
342:nā
316:lī
290:lī
257:ली
231:ना
205:ली
179:ली
130:'s
94:in
2925::
2906:.
2896:32
2894:.
2779:^
2721:.
2695:.
2687:.
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