1751:
fixed points like (1 2 3 4 5 6 7 8 9 10 11 12 13)(14)(15). Under this permutation, a triple like 123 would map to 234, 345, ... (11, 12, 13), (12, 13, 1) and (13, 1, 2) before repeating. Denniston thus classified the 455 triples into 35 rows of 13 triples each, each row being the orbit of a given triple under the permutation. In order to construct a
Sylvester solution, no single-week Kirkman solution could use two triples from the same row, otherwise they would eventually collide when the permutation was applied to one of them. Solving Sylvester's problem is equivalent to finding one triple from each of the 35 rows such that the 35 triples together make a Kirkman solution. He then asked an
1130:
1456:. In words, is it possible for the girls to march every day for 13 weeks, such that every two girls march together exactly once each week and every three girls march together exactly once in the term of 13 weeks? This problem was much harder, and a computational solution would finally be provided in 1974 by
1861:
Solution 2: Day 1 Day 2 Day 3 Day 4 Week 1 ABC.DEF.GHI ADG.BEH.CFI AEI.BFG.CDH AFH.BDI.CEG Week 2 ABD.CEH.FGI ACF.BGH.DEI AEG.BCI.DFH AHI.BEF.CDG Week 3 ABE.CGH.DFI ACI.BFH.DEG ADH.BGI.CEF AFG.BCD.EHI Week 4 ABF.CGI.DEH
1853:
Solution 1: Day 1 Day 2 Day 3 Day 4 Week 1 ABC.DEF.GHI ADG.BEH.CFI AEI.BFG.CDH AFH.BDI.CEG Week 2 ABD.CEH.FGI ACF.BGH.DEI AEG.BCI.DFH AHI.BEF.CDG Week 3 ABE.CDI.FGH ACG.BDF.EHI ADH.BGI.CEF AFI.BCH.DEG Week 4 ABF.CEI.DGH
1631:
which listed for the first time all seven non-isomorphic solutions to the 15 schoolgirl problem, thus answering a long-standing question since the 1850s. The seven
Kirkman solutions correspond to four different Steiner systems when resolvability into parallel classes is removed as a constraint. Three
1360:
In 1850, Kirkman posed the 15 schoolgirl problem, which would become much more famous than the 1847 paper he had already written. Several solutions were received. Kirkman himself gave a solution that later would be found to be isomorphic to
Solution I above. Kirkman claimed it to be the only possible
1865:
Solution 2 has 54 automorphisms, generated by the permutations (A B D)(C H E)(F G I) and (A I F D E H)(B G). Applying the 9! = 362880 permutations of ABCDEFGHI, there are 362880/54 = 6720 different solutions all isomorphic to
Solution 2.
1750:
constructed it with a computer. Denniston's insight was to create a single-week
Kirkman solution in such a way that it could be permuted according to a specific permutation of cycle length 13 to create disjoint solutions for subsequent weeks; he chose a permutation with a single 13-cycle and two
1857:
Solution 1 has 42 automorphisms, generated by the permutations (A I D C F H)(B G) and (C F D H E I)(B G). Applying the 9! = 362880 permutations of ABCDEFGHI, there are 362880/42 = 8640 different solutions all isomorphic to
Solution 1.
954:
1552:
had invented it, and that his
Cambridge lectures had been the source of Kirkman's work. Kirkman quickly rebuffed his claims, stating that when he wrote his papers he had never been to Cambridge or heard of Sylvester's work. This priority dispute led to a falling out between Sylvester and
1570:
which cost
Kirkman recognition by the mathematical community in Europe), further contributing to his being sidelined by the mathematics establishment. His comprehensive 1847 paper in particular was forgotten, with many subsequent authors either crediting Steiner or Reiss, unaware of the
1962:
When two points as A and B of the line ABC are chosen, each of the five other lines through A is met by only one of the five other lines through B. The five points determined by the intersections of these pairs of lines, together with the two points A and B we designate a
1510:
to be 1 or 3 (mod 6) but left an open question as to when this would be realized, unaware that
Kirkman had already settled that question in 1847. As this paper was more widely read by the European mathematical establishment, triple systems later became known as
2270:
groups case where each pair of girls must be in the same group at some point, but we want to use as few days as possible. This can, for example, be used to schedule a rotating table plan, in which each pair of guests must at some point be at the same table.
1369:
though that geometry was not known at the time. However, in publishing his solutions to the schoolgirl problem, Kirkman neglected to refer readers to his own 1847 paper, and this omission would have serious consequences for invention and priority as seen
1862:
ACE.BDG.FHI ADI.BCH.EFG AGH.BEI.CDF Week 5 ABG.CDI.EFH ACH.BDF.EGI ADE.BHI.CFG AFI.BCE.DGH Week 6 ABH.CEI.DFG ACD.BFI.EGH AEF.BCG.DHI AGI.BDE.CFH Week 7 ABI.CDE.FGH ACG.BDH.EFI ADF.BEG.CHI AEH.BCF.DGI
1854:
ACD.BHI.EFG AEH.BCG.DFI AGI.BDE.CFH Week 5 ABG.CDE.FHI ACH.BEI.DFG ADI.BCF.EGH AEF.BDH.CGI Week 6 ABH.CDF.EGI ACI.BDG.EFH ADE.BFI.CGH AFG.BCE.DHI Week 7 ABI.CFG.DEH ACE.BFH.DGI ADF.BEG.CHI AGH.BCD.EFI
2226:
As this is a regrouping strategy where all groups are orthogonal, this process within the problem of organising a large group into smaller groups where no two people share the same group twice can be referred to as orthogonal regrouping.
1850:(J Combinatorial Theory, Vol 16 pp 273-285). There can indeed be 7 disjoint S(2,3,9) systems, and all such sets of 7 fall into two non-isomorphic categories of sizes 8640 and 6720, with 42 and 54 automorphisms respectively.
835:
1118:
1885:
In the 21st century, analogues of
Sylvester's problem have been visited by other authors under terms like "Disjoint Steiner systems" or "Disjoint Kirkman systems" or "LKTS" (Large Sets of Kirkman Triple Systems), for
1524:
answered the questions raised by Steiner, using both methodology and notation so similar to Kirkman's 1847 work (without acknowledging Kirkman), that subsequent authors such as Louise Cummings have called him out for
1663:
but the journal erroneously thought the problem had been solved already and rejected his paper in 1966, which was later found to be a serious mistake. His subsequent academic contributions were disrupted by the
1876:
and found that there could be up to 2 disjoint S(5,6,12) systems, up to 2 disjoint S(4,5,11) systems, and up to 5 disjoint S(3,4,10) systems. All such sets of 2 or 5 are respectively isomorphic to each other.
61:, that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks. Such Steiner systems that have a parallelism are also called
1538:
unified several disparate solutions presented thus far, and showed that there were three possible cyclic solution structures, one corresponding to Anstice's work, one based on Kirkman's solution, and one on
2215:
Many variations of the basic problem can be considered. Alan Hartman solves a problem of this type with the requirement that no trio walks in a row of four more than once using Steiner quadruple systems.
1263:
1576:
The schoolgirl puzzle's popularity itself was unaffected by Kirkman's academic conflicts, and in the late 19th and early 20th centuries the puzzle appeared in several recreational mathematics books by
1481:
and then cyclically shifting each subsequent day by one letter while leaving 0 unchanged (uppercase staying uppercase and lowercase staying lowercase). If the four triples without the 0 element (
1632:
of the Steiner systems have two possible ways of being separated into parallel classes, meaning two Kirkman solutions each, while the fourth has only one, giving seven Kirkman solutions overall.
994:
1766:
Day 1 ABJ CEM FKL HIN DGO Day 2 ACH DEI FGM JLN BKO Day 3 ADL BHM GIK CFN EJO Day 4 AEG BIL CJK DMN FHO Day 5 AFI BCD GHJ EKN LMO Day 6 AKM DFJ EHL BGN CIO Day 7 BEF CGL DHK IJM ANO
2106:
1169:
symbols, which may appear in any one of them shall be repeated in any other." Only two answers were received, one incorrect and the other correctly answering the question with
3768:
2148:
1036:
1740:
1454:
1422:
1844:
1343:= 1 or 3 (mod 6) (not necessarily resolvable ones, but triple systems in general). He also described resolvable triple systems in detail in that paper, particularly for
3103:
1685:
solved the Sylvester problem of constructing 13 disjoint Kirkman solutions and using them to cover all 455 triples on the 15 girls. His solution is discussed below.
40:
Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.
2190:
3576:
3399:
3377:
2268:
2248:
2064:
2044:
1592:. In his lifetime, Kirkman would complain about his serious mathematical work being eclipsed by the popularity of the schoolgirl problem. Kirkman died in 1895.
1506:
which reintroduced the concept of triple systems but did not mention resolvability into separate parallel classes. Steiner noted that it is necessary for
1982:
The schoolgirl problem consists in finding seven lines in the 5-space which do not intersect and such that any two lines always have a heptad in common.
1622:
which was the first paper to lay out all 80 solutions to the Steiner triple system of size 15. These included both resolvable and non-resolvable systems.
949:{\displaystyle 15!\times \left({\frac {1}{168}}+{\frac {1}{168}}+{\frac {1}{24}}+{\frac {1}{24}}+{\frac {1}{12}}+{\frac {1}{12}}+{\frac {1}{21}}\right)}
1846:
triples. This solution was known to Bays (1917) which was found again from a different direction by Earl Kramer and Dale Mesner in a 1974 paper titled
1286:
symbols, so that no pair of symbols shall be comprised more than once among them?". This is equivalent to repeating his 1844 question with the values
1042:
2223:
has gained interest that deals with 32 golfers who want to get to play with different people each day in groups of 4, over the course of 10 days.
2018:(1985), he noted that some solutions correspond to packings of PG(3,2), essentially as described by Conwell above, and he presented two of them.
1154:
32:
3455:
3300:
3190:
2956:
2775:
Araya, Makoto & Harada, Masaaki. (2010). Mutually Disjoint Steiner Systems S(5, 8, 24) and 5-(24, 12, 48) Designs. Electr. J. Comb.. 17.
2729:
3783:
3737:
1752:
3727:
3697:
3569:
1783:
As of 2021, it is not known whether there are other non-isomorphic solutions to Sylvester's problem, or how many solutions there are.
1347:= 9 and 15; resolvable triple systems are now known as Kirkman triple systems. He could not conclusively say for what other values of
3494:
3335:
2852:
1755:
computer to do exactly that search, which took him 7 hours to find this first-week solution, labeling the 15 girls with the letters
1172:
3773:
2292:, also generalizes Kirkman's schoolgirl problem. Kirkman's problem is the special case of the Oberwolfach problem in which the
2981:
2150:
days, with the requirement, again, that no pair of girls walk in the same row twice. The solution to this generalisation is a
1265:. As the question did not ask for anything more than the number of combinations, nothing was received about the conditions on
825:
From the number of automorphisms for each solution and the definition of an automorphism group, the total number of solutions
3830:
3825:
3747:
3562:
3394:
3372:
2205:
1650:
3763:
3778:
2289:
1995:
1138:
3071:
1801:
The corresponding Sylvester problem asks for 7 different S(2,3,9) systems of 12 triples each, together covering all
3346:
3033:
3835:
3536:
960:
124:
1137:
The problem has a long and storied history. This section is based on historical work done at different times by
3679:
3439:
3221:
2201:
1646:
1598:
1142:
80:
in 1922. The seven solutions are summarized in the table below, denoting the 15 girls with the letters A to O.
27:
1928:
together with the line through its diagonal points. Each point is on 7 lines, and there are 35 lines in all.
1323:
even though perfect balance would not be possible. He gave two different sequences of triple systems, one for
3712:
3112:
2844:
1972:
1917:
1747:
1659:
3722:
3157:
2344:
2069:
1991:
1932:
1925:
1773:
1694:
1545:
1466:
1376:
1150:
3319:
3799:
3667:
2796:
2220:
2151:
1512:
54:
2166:. It is this generalization of the problem that Kirkman discussed first, while the famous special case
1157:
at the time, asked the general question: "Determine the number of combinations that can be made out of
2746:
1924:
which has 15 points, 3 points to a line, 7 points and 7 lines in a plane. A plane can be considered a
1493:. The Pasch configuration would become important in isomorph rejection techniques in the 20th century.
1365:'s solution would be later found to be isomorphic to Solution II. Both solutions could be embedded in
1000:
3732:
3702:
3642:
3620:
3258:
2575:
2349:
2305:
2111:
1427:
1935:
in PG(5,2) with 63 points, 35 of which represent lines of PG(3,2). These 35 points form the surface
1700:
1382:
3742:
3707:
3687:
3585:
3315:
2836:
2275:
2193:
2011:
1869:
Thus there are 8640 + 6720 = 15360 solutions in total, falling into two non-isomorphic categories.
1743:
1682:
1669:
1665:
1457:
1804:
3717:
3637:
3274:
3137:
3003:
2962:
2813:
2317:
1332:
1316:
1307:) which comprehensively described and solved the problem of constructing triple systems of order
89:
2936:
2709:
1697:
in 1850 asked if 13 disjoint Kirkman systems of 35 triples each could be constructed to use all
3517:
2383:
1379:
asked if there could be 13 different solutions to the 15-schoolgirl problem that would use all
3514:
3490:
3484:
3451:
3416:
3331:
3296:
3292:
3186:
2952:
2848:
2725:
2603:
2380:
1976:
1615:
1611:
526:
73:
3075:
1872:
In addition to S(2,3,9), Kramer and Mesner examined other systems that could be derived from
3647:
3599:
3476:
3443:
3412:
3286:
3266:
3233:
3208:
3166:
3121:
2993:
2944:
2805:
2758:
2717:
2677:
2593:
2583:
2323:
1577:
1562:
over an unrelated matter (Cayley's choosing not to publish a series of papers by Kirkman on
120:
3133:
236:(A J M I B F C)(D H G N K E O).
132:(A M L K O C D)(B H N G I E J).
3541:
3532:
3464:
3129:
2935:
Banchero, Matías; Robledo, Franco; Romero, Pablo; Sartor, Pablo; Servetti, Camilo (2021).
2169:
1903:
1535:
731:(A B L C G D N)(E H K I O J F)
232:(A B I M F C J)(D N H K O L E)
128:(A K G E I L B)(C H M J N O D)
3426:
Ray-Chaudhuri, D.K.; Wilson, R.M. (1971), "Solution of Kirkman's schoolgirl problem, in
3262:
2579:
1490:
3630:
3480:
3390:
3368:
3179:
2598:
2563:
2279:
2253:
2233:
2049:
2029:
1891:
1873:
1585:
1890:> 15. Similar sets of disjoint Steiner systems have also been investigated for the
3819:
3327:
3311:
3246:
3148:
3007:
2966:
2682:
2665:
2334:
2285:
1940:
1605:
which discussed the early history of the field and corrected the historical omission.
1589:
1559:
1499:
1362:
332:(A H E)(B O K)(C F I)(D J L)(G N M)
23:
3278:
3238:
3213:
1597:
In 1918, Kirkman's serious mathematical work was brought back to wider attention by
1129:
3625:
3141:
2339:
2197:
1910:
1792:
1673:
1563:
1521:
1355:
triple systems exist; that problem would not be solved until the 1960s (see below).
434:(A L B O)(C I)(D K E N)(G J H M)
336:(A J B M)(D L E O)(F I)(G K H N)
2192:
was only proposed later. A complete solution to the general case was published by
2937:"Max-Diversity Orthogonal Regrouping of MBA Students Using a GRASP/VND Heuristic"
1668:
and rejected again. In 1968, the generalized theorem was proven independently by
2948:
2721:
1791:
The equivalent of the Kirkman problem for 9 schoolgirls results in S(2,3,9), an
3447:
2762:
1618:
on triple systems. This culminated in their famous and widely cited 1919 paper
3692:
3657:
3604:
3170:
2162:+ 3 elements occurs exactly once in each block of 3-element sets), known as a
1581:
1526:
69:
1113:{\displaystyle =2^{10}\times 3^{5}\times 5^{3}\times 7\times 11\times 13^{2}}
3522:
2696:
2388:
1798:
Day 1: 123 456 789 Day 2: 147 258 369 Day 3: 159 267 348 Day 4: 168 249 357
1567:
3548:
2607:
1967:
A heptad is determined by any two of its points. Each of the 28 points off
1769:
He stopped the search at that point, not looking to establish uniqueness.
2588:
1502:, completely unaware of Kirkman's 1847 paper, published his paper titled
1955:
3554:
2817:
1921:
1366:
135:
3125:
2998:
2943:. Lecture Notes in Computer Science. Vol. 12559. pp. 58–70.
3270:
3034:"The Mind-Bending Math Behind Spot It!, the Beloved Family Card Game"
2311:
735:(B G L)(C D N)(E F K)(H I O)
637:(A B C)(D L G)(E K H)(F J I)
537:(A B C)(D L G)(F J I)(E K H)
3152:
2809:
430:(A J M)(C F I)(D E K)(H O L)
2980:
Van Dam, Edwin R.; Haemers, Willem H.; Peek, Maurice B. M. (2003).
2431:
by Robin Wilson, Dept of Pure Mathematics, The Open University, UK
2329:
1913:
1637:
In the 1960s, it was proved that Kirkman triple systems exist for
1473:
solution, made by constructing the first day's five triples to be
1128:
2794:
Conwell, George M. (1910). "The 3-space PG(3,2) and its Group".
3558:
1299:
In 1847, at age 41, Thomas Kirkman published his paper titled
1282:
In 1846, Woolhouse asked: "How many triads can be made out of
1258:{\textstyle {\frac {n!}{q!(n-q)!}}\div {\frac {p!}{q!(p-q)!}}}
1165:
symbols in each; with this limitation, that no combination of
2308:
strategy for increasing interaction within classroom teaching
72:
solutions to the schoolgirl problem, as originally listed by
633:(A L)(B G)(E O)(H K)(F J)(I M)
533:(A L)(B G)(E O)(F J)(H K)(I M)
3344:
Hartman, Alan (1980), "Kirkman's trombone player problem",
1742:
triples on 15 girls. No solution was found until 1974 when
3428:
Combinatorics, University of California, Los Angeles, 1968
2564:"The Complete Enumeration of Triad Systems in 15 Elements"
2158:+ 3) with parallelism (that is, one in which each of the 6
16:
Combinatorics problem proposed by Thomas Penyngton Kirkman
3153:"On the triadic arrangements of seven and fifteen things"
3421:, vol. 2, Paris: Gauthier-Villars, pp. 183–188
1998:, and a partition of the lines into spreads is called a
2562:
Cole, F. N.; Cummings, Louise D.; White, H. S. (1917).
2230:
The Resolvable Coverings problem considers the general
1620:
Complete classification of triad systems on 15 elements
82:
2114:
1807:
1703:
1385:
1175:
3185:(2nd ed.), Boca Raton: Chapman & Hall/ CRC,
3078:
2367:
2256:
2236:
2172:
2072:
2052:
2032:
1430:
1045:
1003:
963:
838:
3361:
Collected Works of Lu Jiaxi on Combinatorial Designs
2212:) in 1965, but had not been published at that time.
1947:
there are 6 lines through it which do not intersect
3792:
3756:
3678:
3613:
3592:
2889:
2643:
3178:
3097:
2262:
2242:
2184:
2142:
2100:
2058:
2038:
1838:
1734:
1610:At about the same time, Cummings was working with
1485:) are taken and uppercase converted to lowercase (
1448:
1416:
1257:
1112:
1030:
988:
948:
3177:Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),
1795:isomorphic to the following triples on each day:
2402:
2400:
2568:Proceedings of the National Academy of Sciences
1971:lies in two heptads. There are 8 heptads. The
1960:
1331:= 9, 13, 25, etc. Using these propositions, he
3020:
2911:
2631:
2200:in 1968, though it had already been solved by
2010:. There are 56 spreads and 240 packings. When
1548:revisited the problem and tried to claim that
49:A solution to this problem is an example of a
3570:
3400:The Cambridge and Dublin Mathematical Journal
3378:The Cambridge and Dublin Mathematical Journal
3226:Bulletin of the American Mathematical Society
3201:Bulletin of the American Mathematical Society
2424:
2422:
2420:
2418:
2416:
2414:
2219:More recently a similar problem known as the
1931:The lines of PG(3,2) are identified by their
1824:
1811:
1720:
1707:
1558:In 1861-1862, Kirkman had a falling out with
1424:triples exactly once overall, observing that
1402:
1389:
8:
3385:, Macmillan, Barclay, and Macmillan: 191–204
2841:Finite Projective Spaces of Three Dimensions
2659:
2657:
2655:
2653:
2651:
2016:Finite Projective Spaces of Three Dimensions
3432:Proceedings of Symposia in Pure Mathematics
3407:, Macmillan, Barclay and Macmillan: 255–262
2866:
2708:Johnson, Tom; Jedrzejewski, Franck (2014).
2666:"Sylvester's problem of the 15 schoolgirls"
2296:graph consists of five disjoint triangles.
1529:. Kirkman himself expressed his bitterness.
1489:) they form what would later be called the
989:{\displaystyle =15!\times {\frac {13}{42}}}
3577:
3563:
3555:
3549:String (March, 2015) - Solution visualised
3489:(13th ed.), Dover, pp. 287–289,
3442:: American Mathematical Society: 187–203,
2710:"Kirkman's Ladies, A Combinatorial Design"
2506:
3537:"A design dilemma solved, minus designs."
3237:
3212:
3083:
3077:
2997:
2681:
2597:
2587:
2255:
2235:
2171:
2115:
2113:
2082:
2071:
2051:
2031:
1823:
1810:
1808:
1806:
1719:
1706:
1704:
1702:
1429:
1401:
1388:
1386:
1384:
1217:
1176:
1174:
1104:
1079:
1066:
1053:
1044:
1002:
976:
962:
931:
918:
905:
892:
879:
866:
853:
837:
3224:(1918), "An undervalued Kirkman paper",
3070:Bryant, Darryn; Danziger, Peter (2011),
2550:
2539:
2484:
2451:
2440:
1902:In 1910 the problem was addressed using
3363:, Huhhot: Inner Mongolia People's Press
3063:Mathematische Unterhaltungen und Spiele
2923:
2878:
2528:
2462:
2360:
1954:As there are seven days in a week, the
1327:= 7, 15, 19, 27, etc., and another for
1319:6). He also considered other values of
1304:
1277:when such a solution could be achieved.
3395:"Note on an unanswered prize question"
3199:Cole, F.N. (1922), "Kirkman parades",
2789:
2787:
2785:
2783:
2781:
2517:
2473:
2066:must be an odd multiple of 3 (that is
1958:is an important part of the solution:
1881:Larger systems and continuing research
1645:= 3 (mod 6). This was first proved by
3249:(1917), "Amusements in Mathematics",
2831:
2829:
2827:
2747:"A new result on Sylvester's problem"
2745:Zhou, Junling; Chang, Yanxun (2014).
2495:
2282:into edge-disjoint copies of a given
1994:of the points into lines is called a
7:
2406:
2101:{\displaystyle n\equiv 3{\pmod {6}}}
1916:with two elements is used with four
3486:Mathematical Recreations and Essays
3469:Mathematical Recreations and Essays
3324:Handbook of Combinatorics, Volume 2
2368:Graham, Grötschel & Lovász 1995
2090:
1848:Intersections Among Steiner Systems
1133:Original publication of the problem
829:isomorphic solutions is therefore:
3072:"On bipartite 2-factorizations of
2900:
2620:
2429:The Early History of Block Designs
2026:The problem can be generalized to
1815:
1711:
1657:) in 1965, and he submitted it to
1627:In 1922, Cole published his paper
1393:
14:
3181:Handbook of Combinatorial Designs
1776:composed a piece of music called
1772:The American minimalist composer
1361:solution but that was incorrect.
2986:Journal of Combinatorial Designs
2982:"Equitable resolvable coverings"
2143:{\textstyle {\frac {1}{2}}(n-1)}
1943:. For each of the 28 points off
1155:The Lady's and Gentleman's Diary
1031:{\displaystyle =404,756,352,000}
33:The Lady's and Gentleman's Diary
3239:10.1090/S0002-9904-1918-03086-3
3214:10.1090/S0002-9904-1922-03599-9
2890:Ray-Chaudhuri & Wilson 1971
2644:Ray-Chaudhuri & Wilson 1971
2083:
1894:in addition to triple systems.
1780:based on Denniston's solution.
1735:{\textstyle {15 \choose 3}=455}
1449:{\displaystyle 455=13\times 35}
1417:{\textstyle {15 \choose 3}=455}
3728:Cremona–Richmond configuration
3518:"Kirkman's schoolgirl problem"
3373:"On a Problem in Combinations"
3257:(2512), New York: Dover: 302,
2384:"Kirkman's Schoolgirl Problem"
2209:
2137:
2125:
2094:
2084:
2014:considered the problem in his
1654:
1335:that triple systems exist for
1246:
1234:
1205:
1193:
629:Tetrahedral group of order 12,
36:(pg.48). The problem states:
1:
1839:{\textstyle {9 \choose 3}=84}
1145:. The history is as follows:
3805:Kirkman's schoolgirl problem
3738:Grünbaum–Rigby configuration
3105:and the Oberwolfach problem"
2941:Variable Neighborhood Search
2683:10.1016/0012-365X(74)90004-1
1787:9 schoolgirls and extensions
1603:An Undervalued Kirkman Paper
1301:On a Problem in Combinations
809:
797:
785:
773:
761:
749:
737:
728:
711:
699:
687:
675:
663:
651:
639:
628:
611:
599:
587:
575:
563:
551:
539:
525:
508:
496:
484:
472:
460:
448:
436:
427:
410:
398:
386:
374:
362:
350:
338:
329:
312:
300:
288:
276:
264:
252:
240:
229:
212:
200:
188:
176:
164:
152:
140:
119:
68:There are exactly seven non-
20:Kirkman's schoolgirl problem
3698:Möbius–Kantor configuration
3291:, Dover Recreational Math,
3032:McRobbie, Linda Rodriguez.
2949:10.1007/978-3-030-69625-2_5
2722:10.1007/978-3-0348-0554-4_4
2108:), walking in triplets for
1975:PGL(3,2) is isomorphic the
3852:
3784:Bruck–Ryser–Chowla theorem
3021:Bryant & Danziger 2011
2912:Colbourn & Dinitz 2007
2763:10.1016/j.disc.2014.04.022
2664:Denniston, R.H.F. (1974).
2632:Colbourn & Dinitz 2007
3774:Szemerédi–Trotter theorem
3418:Récréations Mathématiques
3288:Amusements in Mathematics
3171:10.1080/14786445008646550
3764:Sylvester–Gallai theorem
3440:Providence, Rhode Island
1892:S(5,8,24) Steiner system
1599:Louise Duffield Cummings
1143:Louise Duffield Cummings
28:Thomas Penyngton Kirkman
3769:De Bruijn–Erdős theorem
3713:Desargues configuration
3113:Journal of Graph Theory
3098:{\displaystyle K_{n}-I}
2867:Ball & Coxeter 1987
2845:Oxford University Press
1973:projective linear group
1918:homogeneous coordinates
1748:University of Leicester
1689:
1660:Acta Mathematica Sinica
1504:Combinatorische Aufgabe
1475:0Gg, AbC, aDE, cef, BdF
230:Order 168, generated by
30:in 1850 as Query VI in
3448:10.1090/pspum/019/9959
3285:Dudeney, H.E. (1958),
3158:Philosophical Magazine
3099:
2697:Kirkman's Ladies audio
2345:Dijen K. Ray-Chaudhuri
2264:
2244:
2186:
2144:
2102:
2060:
2040:
1965:
1926:complete quadrilateral
1840:
1736:
1695:James Joseph Sylvester
1546:James Joseph Sylvester
1513:Steiner triple systems
1467:Robert Richard Anstice
1450:
1418:
1377:James Joseph Sylvester
1259:
1134:
1114:
1032:
990:
950:
729:Order 21, generated by
428:Order 24, generated by
330:Order 24, generated by
42:
3831:Mathematical problems
3800:Design of experiments
3100:
2797:Annals of Mathematics
2265:
2245:
2221:Social Golfer Problem
2187:
2164:Kirkman triple system
2152:Steiner triple system
2145:
2103:
2061:
2041:
1841:
1737:
1451:
1419:
1260:
1132:
1115:
1033:
991:
951:
55:Steiner triple system
51:Kirkman triple system
38:
3826:Combinatorial design
3733:Kummer configuration
3703:Pappus configuration
3586:Incidence structures
3076:
3038:Smithsonian Magazine
2751:Discrete Mathematics
2670:Discrete Mathematics
2589:10.1073/pnas.3.3.197
2350:Discrete mathematics
2306:Cooperative learning
2254:
2234:
2185:{\displaystyle n=15}
2170:
2112:
2070:
2050:
2030:
1805:
1701:
1428:
1383:
1173:
1043:
1001:
961:
836:
238:Related to PG(3,2).
3743:Klein configuration
3723:Schläfli double six
3708:Hesse configuration
3688:Complete quadrangle
3471:, London: Macmillan
3263:1917Natur.100..302.
3061:Ahrens, W. (1901),
2580:1917PNAS....3..197C
2330:Sports Competitions
2278:, of decomposing a
2276:Oberwolfach problem
2194:D. K. Ray-Chaudhuri
1986:Spreads and packing
1933:Plücker coordinates
1906:by George Conwell.
1690:Sylvester's problem
1670:D. K. Ray-Chaudhuri
1666:Cultural Revolution
1491:Pasch configuration
3718:Reye configuration
3515:Weisstein, Eric W.
3391:Kirkman, Thomas P.
3369:Kirkman, Thomas P.
3359:Lu, Jiaxi (1990),
3095:
3065:, Leipzig: Teubner
2869:, pp. 287−289
2837:Hirschfeld, J.W.P.
2716:. pp. 37–55.
2714:Looking at Numbers
2381:Weisstein, Eric W.
2318:Progressive dinner
2260:
2240:
2182:
2140:
2098:
2056:
2036:
1979:on the 8 heptads.
1836:
1732:
1601:in a paper titled
1487:abc, ade, cef, bdf
1483:AbC, aDE, cef, BdF
1477:on the 15 symbols
1446:
1414:
1255:
1135:
1110:
1028:
986:
946:
90:Automorphism group
3813:
3812:
3457:978-0-8218-1419-2
3330:: The MIT Press,
3316:Grötschel, Martin
3312:Graham, Ronald L.
3302:978-0-486-20473-4
3293:Mineola, New York
3192:978-1-58488-506-1
3126:10.1002/jgt.20538
2999:10.1002/jcd.10024
2958:978-3-030-69624-5
2731:978-3-0348-0553-7
2263:{\displaystyle g}
2243:{\displaystyle n}
2123:
2059:{\displaystyle n}
2039:{\displaystyle n}
1977:alternating group
1822:
1718:
1616:Henry Seely White
1612:Frank Nelson Cole
1400:
1253:
1212:
984:
939:
926:
913:
900:
887:
874:
861:
823:
822:
527:Tetrahedral group
74:Frank Nelson Cole
3843:
3836:Families of sets
3648:Projective plane
3600:Incidence matrix
3579:
3572:
3565:
3556:
3551:, Stack Exchange
3545:
3535:(June 9, 2015),
3533:Klarreich, Erica
3528:
3527:
3499:
3477:Ball, W.W. Rouse
3472:
3465:Rouse Ball, W.W.
3460:
3422:
3408:
3386:
3364:
3355:
3347:Ars Combinatoria
3340:
3305:
3281:
3271:10.1038/100302a0
3242:
3241:
3217:
3216:
3195:
3184:
3173:
3144:
3109:
3104:
3102:
3101:
3096:
3088:
3087:
3066:
3048:
3047:
3045:
3044:
3029:
3023:
3018:
3012:
3011:
3001:
2977:
2971:
2970:
2932:
2926:
2921:
2915:
2909:
2903:
2898:
2892:
2887:
2881:
2876:
2870:
2864:
2858:
2857:
2833:
2822:
2821:
2791:
2776:
2773:
2767:
2766:
2742:
2736:
2735:
2705:
2699:
2694:
2688:
2687:
2685:
2661:
2646:
2641:
2635:
2629:
2623:
2618:
2612:
2611:
2601:
2591:
2559:
2553:
2548:
2542:
2537:
2531:
2526:
2520:
2515:
2509:
2504:
2498:
2493:
2487:
2482:
2476:
2471:
2465:
2460:
2454:
2449:
2443:
2438:
2432:
2426:
2409:
2404:
2395:
2394:
2393:
2376:
2370:
2365:
2324:Speed Networking
2295:
2288:
2269:
2267:
2266:
2261:
2249:
2247:
2246:
2241:
2211:
2191:
2189:
2188:
2183:
2149:
2147:
2146:
2141:
2124:
2116:
2107:
2105:
2104:
2099:
2097:
2065:
2063:
2062:
2057:
2045:
2043:
2042:
2037:
1845:
1843:
1842:
1837:
1829:
1828:
1827:
1814:
1778:Kirkman's Ladies
1741:
1739:
1738:
1733:
1725:
1724:
1723:
1710:
1656:
1455:
1453:
1452:
1447:
1423:
1421:
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1405:
1392:
1264:
1262:
1261:
1256:
1254:
1252:
1226:
1218:
1213:
1211:
1185:
1177:
1153:, the editor of
1151:Wesley Woolhouse
1119:
1117:
1116:
1111:
1109:
1108:
1084:
1083:
1071:
1070:
1058:
1057:
1037:
1035:
1034:
1029:
995:
993:
992:
987:
985:
977:
955:
953:
952:
947:
945:
941:
940:
932:
927:
919:
914:
906:
901:
893:
888:
880:
875:
867:
862:
854:
83:
22:is a problem in
3851:
3850:
3846:
3845:
3844:
3842:
3841:
3840:
3816:
3815:
3814:
3809:
3788:
3752:
3674:
3609:
3605:Incidence graph
3588:
3583:
3542:Quanta Magazine
3531:
3513:
3512:
3509:
3504:
3497:
3481:Coxeter, H.S.M.
3475:
3463:
3458:
3425:
3411:
3389:
3367:
3358:
3343:
3338:
3310:
3303:
3284:
3245:
3220:
3198:
3193:
3176:
3147:
3107:
3079:
3074:
3073:
3069:
3060:
3056:
3051:
3042:
3040:
3031:
3030:
3026:
3019:
3015:
2979:
2978:
2974:
2959:
2934:
2933:
2929:
2922:
2918:
2910:
2906:
2899:
2895:
2888:
2884:
2877:
2873:
2865:
2861:
2855:
2835:
2834:
2825:
2810:10.2307/1967582
2793:
2792:
2779:
2774:
2770:
2744:
2743:
2739:
2732:
2707:
2706:
2702:
2695:
2691:
2663:
2662:
2649:
2642:
2638:
2630:
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2619:
2615:
2561:
2560:
2556:
2549:
2545:
2538:
2534:
2527:
2523:
2516:
2512:
2507:Rouse Ball 1892
2505:
2501:
2494:
2490:
2483:
2479:
2472:
2468:
2461:
2457:
2450:
2446:
2439:
2435:
2427:
2412:
2405:
2398:
2379:
2378:
2377:
2373:
2366:
2362:
2358:
2302:
2293:
2283:
2252:
2251:
2232:
2231:
2168:
2167:
2110:
2109:
2068:
2067:
2048:
2047:
2028:
2027:
2024:
2008:
2002:
1988:
1904:Galois geometry
1900:
1898:Galois geometry
1883:
1863:
1855:
1809:
1803:
1802:
1799:
1789:
1767:
1705:
1699:
1698:
1692:
1629:Kirkman Parades
1536:Benjamin Peirce
1479:0ABCDEFGabcdefg
1426:
1425:
1387:
1381:
1380:
1227:
1219:
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1127:
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958:
852:
848:
834:
833:
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813:
811:
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131:
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127:
86:Solution class
78:Kirkman Parades
47:
17:
12:
11:
5:
3849:
3847:
3839:
3838:
3833:
3828:
3818:
3817:
3811:
3810:
3808:
3807:
3802:
3796:
3794:
3790:
3789:
3787:
3786:
3781:
3779:Beck's theorem
3776:
3771:
3766:
3760:
3758:
3754:
3753:
3751:
3750:
3745:
3740:
3735:
3730:
3725:
3720:
3715:
3710:
3705:
3700:
3695:
3690:
3684:
3682:
3680:Configurations
3676:
3675:
3673:
3672:
3671:
3670:
3662:
3661:
3660:
3652:
3651:
3650:
3645:
3635:
3634:
3633:
3631:Steiner system
3628:
3617:
3615:
3611:
3610:
3608:
3607:
3602:
3596:
3594:
3593:Representation
3590:
3589:
3584:
3582:
3581:
3574:
3567:
3559:
3553:
3552:
3546:
3529:
3508:
3507:External links
3505:
3503:
3502:
3501:
3500:
3495:
3461:
3456:
3423:
3409:
3387:
3365:
3356:
3341:
3336:
3320:Lovász, László
3308:
3307:
3306:
3301:
3243:
3232:(7): 336–339,
3222:Cummings, L.D.
3218:
3207:(9): 435–437,
3196:
3191:
3174:
3165:(247): 50–53,
3145:
3094:
3091:
3086:
3082:
3067:
3057:
3055:
3052:
3050:
3049:
3024:
3013:
2992:(2): 113–123.
2972:
2957:
2927:
2916:
2904:
2893:
2882:
2871:
2859:
2853:
2823:
2777:
2768:
2737:
2730:
2700:
2689:
2676:(3): 229–233.
2647:
2636:
2624:
2613:
2574:(3): 197–199.
2554:
2543:
2532:
2521:
2510:
2499:
2488:
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2466:
2455:
2444:
2433:
2410:
2396:
2371:
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2352:
2347:
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2337:
2332:
2327:
2321:
2315:
2309:
2301:
2298:
2280:complete graph
2259:
2239:
2181:
2178:
2175:
2154:, an S(2, 3, 6
2139:
2136:
2133:
2130:
2127:
2122:
2119:
2096:
2093:
2089:
2086:
2081:
2078:
2075:
2055:
2035:
2023:
2022:Generalization
2020:
2006:
2000:
1990:In PG(3,2), a
1987:
1984:
1899:
1896:
1882:
1879:
1860:
1852:
1835:
1832:
1826:
1821:
1818:
1813:
1797:
1788:
1785:
1765:
1731:
1728:
1722:
1717:
1714:
1709:
1691:
1688:
1687:
1686:
1678:
1677:
1634:
1633:
1624:
1623:
1607:
1606:
1594:
1593:
1586:Wilhelm Ahrens
1573:
1572:
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1554:
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1531:
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1517:
1516:
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1494:
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1445:
1442:
1439:
1436:
1433:
1413:
1410:
1404:
1399:
1396:
1391:
1375:Also in 1850,
1372:
1371:
1357:
1356:
1296:
1295:
1279:
1278:
1251:
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1242:
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748:
736:
727:
723:
722:
710:
698:
686:
674:
662:
650:
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623:
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598:
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507:
495:
483:
471:
459:
447:
435:
426:
422:
421:
409:
397:
385:
373:
361:
349:
337:
328:
324:
323:
311:
299:
287:
275:
263:
251:
239:
228:
224:
223:
211:
199:
187:
175:
163:
151:
139:
118:
114:
113:
110:
107:
104:
101:
98:
95:
92:
87:
46:
43:
15:
13:
10:
9:
6:
4:
3:
2:
3848:
3837:
3834:
3832:
3829:
3827:
3824:
3823:
3821:
3806:
3803:
3801:
3798:
3797:
3795:
3791:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3761:
3759:
3755:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3685:
3683:
3681:
3677:
3669:
3666:
3665:
3663:
3659:
3656:
3655:
3654:Graph theory
3653:
3649:
3646:
3644:
3641:
3640:
3639:
3636:
3632:
3629:
3627:
3624:
3623:
3622:
3621:Combinatorics
3619:
3618:
3616:
3612:
3606:
3603:
3601:
3598:
3597:
3595:
3591:
3587:
3580:
3575:
3573:
3568:
3566:
3561:
3560:
3557:
3550:
3547:
3544:
3543:
3538:
3534:
3530:
3525:
3524:
3519:
3516:
3511:
3510:
3506:
3498:
3496:0-486-25357-0
3492:
3488:
3487:
3482:
3478:
3474:
3473:
3470:
3466:
3462:
3459:
3453:
3449:
3445:
3441:
3437:
3433:
3429:
3424:
3420:
3419:
3414:
3410:
3406:
3402:
3401:
3396:
3392:
3388:
3384:
3380:
3379:
3374:
3370:
3366:
3362:
3357:
3353:
3349:
3348:
3342:
3339:
3337:0-262-07171-1
3333:
3329:
3328:Cambridge, MA
3325:
3321:
3317:
3313:
3309:
3304:
3298:
3294:
3290:
3289:
3283:
3282:
3280:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3247:Dudeney, H.E.
3244:
3240:
3235:
3231:
3227:
3223:
3219:
3215:
3210:
3206:
3202:
3197:
3194:
3188:
3183:
3182:
3175:
3172:
3168:
3164:
3160:
3159:
3154:
3150:
3146:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3114:
3106:
3092:
3089:
3084:
3080:
3068:
3064:
3059:
3058:
3053:
3039:
3035:
3028:
3025:
3022:
3017:
3014:
3009:
3005:
3000:
2995:
2991:
2987:
2983:
2976:
2973:
2968:
2964:
2960:
2954:
2950:
2946:
2942:
2938:
2931:
2928:
2925:
2920:
2917:
2913:
2908:
2905:
2902:
2897:
2894:
2891:
2886:
2883:
2880:
2875:
2872:
2868:
2863:
2860:
2856:
2854:0-19-853536-8
2850:
2846:
2842:
2838:
2832:
2830:
2828:
2824:
2819:
2815:
2811:
2807:
2803:
2799:
2798:
2790:
2788:
2786:
2784:
2782:
2778:
2772:
2769:
2764:
2760:
2756:
2752:
2748:
2741:
2738:
2733:
2727:
2723:
2719:
2715:
2711:
2704:
2701:
2698:
2693:
2690:
2684:
2679:
2675:
2671:
2667:
2660:
2658:
2656:
2654:
2652:
2648:
2645:
2640:
2637:
2633:
2628:
2625:
2622:
2617:
2614:
2609:
2605:
2600:
2595:
2590:
2585:
2581:
2577:
2573:
2569:
2565:
2558:
2555:
2552:
2551:Cummings 1918
2547:
2544:
2541:
2540:Cummings 1918
2536:
2533:
2530:
2525:
2522:
2519:
2514:
2511:
2508:
2503:
2500:
2497:
2492:
2489:
2486:
2485:Cummings 1918
2481:
2478:
2475:
2470:
2467:
2464:
2459:
2456:
2453:
2452:Cummings 1918
2448:
2445:
2442:
2441:Cummings 1918
2437:
2434:
2430:
2425:
2423:
2421:
2419:
2417:
2415:
2411:
2408:
2403:
2401:
2397:
2391:
2390:
2385:
2382:
2375:
2372:
2369:
2364:
2361:
2355:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2335:Combinatorics
2333:
2331:
2328:
2325:
2322:
2320:party designs
2319:
2316:
2313:
2310:
2307:
2304:
2303:
2299:
2297:
2291:
2287:
2281:
2277:
2272:
2257:
2237:
2228:
2224:
2222:
2217:
2213:
2207:
2203:
2199:
2195:
2179:
2176:
2173:
2165:
2161:
2157:
2153:
2134:
2131:
2128:
2120:
2117:
2091:
2087:
2079:
2076:
2073:
2053:
2046:girls, where
2033:
2021:
2019:
2017:
2013:
2009:
2003:
1997:
1993:
1985:
1983:
1980:
1978:
1974:
1970:
1964:
1959:
1957:
1952:
1950:
1946:
1942:
1941:Klein quadric
1939:known as the
1938:
1934:
1929:
1927:
1923:
1919:
1915:
1912:
1907:
1905:
1897:
1895:
1893:
1889:
1880:
1878:
1875:
1870:
1867:
1859:
1851:
1849:
1833:
1830:
1819:
1816:
1796:
1794:
1786:
1784:
1781:
1779:
1775:
1770:
1764:
1762:
1758:
1754:
1749:
1745:
1744:RHF Denniston
1729:
1726:
1715:
1712:
1696:
1684:
1683:RHF Denniston
1680:
1679:
1675:
1671:
1667:
1662:
1661:
1652:
1648:
1644:
1640:
1636:
1635:
1630:
1626:
1625:
1621:
1617:
1613:
1609:
1608:
1604:
1600:
1596:
1595:
1591:
1590:Henry Dudeney
1587:
1583:
1579:
1578:Édouard Lucas
1575:
1574:
1569:
1565:
1561:
1560:Arthur Cayley
1557:
1556:
1551:
1547:
1543:
1542:
1537:
1533:
1532:
1528:
1523:
1519:
1518:
1514:
1509:
1505:
1501:
1500:Jakob Steiner
1497:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1468:
1464:
1463:
1459:
1458:RHF Denniston
1443:
1440:
1437:
1434:
1431:
1411:
1408:
1397:
1394:
1378:
1374:
1373:
1368:
1364:
1363:Arthur Cayley
1359:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1297:
1293:
1289:
1285:
1281:
1280:
1276:
1272:
1268:
1249:
1243:
1240:
1237:
1231:
1228:
1223:
1220:
1214:
1208:
1202:
1199:
1196:
1190:
1187:
1182:
1179:
1168:
1164:
1160:
1156:
1152:
1148:
1147:
1146:
1144:
1140:
1131:
1124:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1080:
1076:
1072:
1067:
1063:
1059:
1054:
1050:
1046:
1039:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
997:
981:
978:
973:
970:
967:
964:
957:
942:
936:
933:
928:
923:
920:
915:
910:
907:
902:
897:
894:
889:
884:
881:
876:
871:
868:
863:
858:
855:
849:
845:
842:
839:
832:
831:
830:
828:
726:Solution VII
725:
724:
625:
624:
528:
522:
521:
424:
423:
327:Solution III
326:
325:
226:
225:
137:
126:
122:
116:
115:
111:
108:
105:
102:
99:
96:
93:
91:
88:
85:
84:
81:
79:
75:
71:
66:
64:
60:
56:
53:, which is a
52:
44:
41:
37:
35:
34:
29:
25:
24:combinatorics
21:
3804:
3793:Applications
3626:Block design
3540:
3521:
3485:
3468:
3435:
3431:
3427:
3417:
3404:
3398:
3382:
3376:
3360:
3351:
3345:
3323:
3287:
3254:
3250:
3229:
3225:
3204:
3200:
3180:
3162:
3156:
3120:(1): 22–37,
3117:
3111:
3062:
3041:. Retrieved
3037:
3027:
3016:
2989:
2985:
2975:
2940:
2930:
2924:Hartman 1980
2919:
2907:
2896:
2885:
2879:Kirkman 1847
2874:
2862:
2840:
2804:(2): 60–76.
2801:
2795:
2771:
2754:
2750:
2740:
2713:
2703:
2692:
2673:
2669:
2639:
2627:
2616:
2571:
2567:
2557:
2546:
2535:
2529:Dudeney 1917
2524:
2513:
2502:
2491:
2480:
2469:
2463:Kirkman 1850
2458:
2447:
2436:
2428:
2387:
2374:
2363:
2273:
2229:
2225:
2218:
2214:
2198:R. M. Wilson
2163:
2159:
2155:
2025:
2015:
2005:
1999:
1989:
1981:
1968:
1966:
1961:
1953:
1948:
1944:
1936:
1930:
1911:Galois field
1908:
1901:
1887:
1884:
1871:
1868:
1864:
1856:
1847:
1800:
1793:affine plane
1790:
1782:
1777:
1771:
1768:
1760:
1756:
1753:Elliott 4130
1693:
1674:R. M. Wilson
1658:
1642:
1638:
1628:
1619:
1602:
1564:group theory
1549:
1522:Michel Reiss
1507:
1503:
1486:
1482:
1478:
1474:
1470:
1460:(see below).
1352:
1348:
1344:
1340:
1336:
1328:
1324:
1320:
1312:
1308:
1305:Kirkman 1847
1300:
1291:
1287:
1283:
1274:
1270:
1266:
1166:
1162:
1158:
1139:Robin Wilson
1136:
826:
824:
631:generated by
626:Solution VI
531:generated by
529:of order 12,
425:Solution IV
227:Solution II
125:generated by
77:
67:
62:
58:
50:
48:
39:
31:
26:proposed by
19:
18:
3664:Statistics
2518:Ahrens 1901
2474:Cayley 1850
2007:parallelism
1774:Tom Johnson
1469:provided a
523:Solution V
134:Related to
117:Solution I
59:parallelism
3820:Categories
3693:Fano plane
3658:Hypergraph
3149:Cayley, A.
3054:References
3043:2020-03-01
2496:Lucas 1883
2340:R M Wilson
2012:Hirschfeld
1582:Rouse Ball
1527:plagiarism
1353:resolvable
1339:values of
1315:= 1 or 3 (
70:isomorphic
63:resolvable
3643:Incidence
3523:MathWorld
3483:(1987) ,
3413:Lucas, É.
3295:: Dover,
3090:−
3008:120596961
2967:232314621
2757:: 15–19.
2407:Cole 1922
2389:MathWorld
2314:card game
2294:2-regular
2132:−
2077:≡
1992:partition
1963:"heptad".
1874:S(5,6,12)
1681:In 1974,
1568:polyhedra
1544:In 1861,
1539:Cayley's.
1534:In 1860,
1520:In 1859,
1498:In 1853,
1465:In 1852,
1441:×
1241:−
1215:÷
1200:−
1161:symbols,
1149:In 1844,
1098:×
1092:×
1086:×
1073:×
1060:×
974:×
846:×
827:including
57:having a
45:Solutions
3757:Theorems
3668:Blocking
3638:Geometry
3467:(1892),
3415:(1883),
3393:(1850),
3371:(1847),
3322:(1995),
3279:10245524
3151:(1850),
2839:(1985),
2608:16576216
2300:See also
2202:Lu Jiaxi
1920:to form
1647:Lu Jiaxi
1571:history.
1553:Kirkman.
1290:= 3 and
3354:: 19–26
3259:Bibcode
3142:7478839
3134:2833961
2914:, p. 13
2901:Lu 1990
2818:1967582
2634:, p. 13
2621:Lu 1990
2599:1091209
2576:Bibcode
2286:regular
2250:girls,
2206:Chinese
2001:packing
1922:PG(3,2)
1746:at the
1651:Chinese
1641:orders
1367:PG(3,2)
1141:and by
1125:History
136:PG(3,2)
3614:Fields
3493:
3454:
3334:
3299:
3277:
3251:Nature
3189:
3140:
3132:
3006:
2965:
2955:
2851:
2816:
2728:
2606:
2596:
2326:events
2312:Dobble
2208::
1996:spread
1956:heptad
1653::
1588:, and
1471:cyclic
1370:below.
1351:would
1333:proved
1311:where
112:Day 7
109:Day 6
106:Day 5
103:Day 4
100:Day 3
97:Day 2
94:Day 1
3275:S2CID
3138:S2CID
3108:(PDF)
3004:S2CID
2963:S2CID
2814:JSTOR
2356:Notes
2290:graph
1914:GF(2)
1273:, or
123:168,
121:Order
3748:Dual
3491:ISBN
3452:ISBN
3332:ISBN
3297:ISBN
3187:ISBN
2953:ISBN
2849:ISBN
2726:ISBN
2604:PMID
2274:The
2196:and
1909:The
1672:and
1614:and
1566:and
1294:= 2.
3444:doi
3436:XIX
3430:",
3267:doi
3255:100
3234:doi
3209:doi
3167:doi
3122:doi
2994:doi
2945:doi
2806:doi
2759:doi
2755:331
2718:doi
2678:doi
2594:PMC
2584:doi
2210:陆家羲
2088:mod
2004:or
1759:to
1730:455
1655:陆家羲
1639:all
1432:455
1412:455
1337:all
1317:mod
1026:000
1020:352
1014:756
1008:404
872:168
859:168
818:DIM
816:FGJ
814:CEK
812:BHO
810:ALN
806:EHN
804:DKO
802:CFI
800:BGL
798:AJM
794:IJN
792:EGO
790:CDL
788:BFM
786:AHK
782:ELM
780:DHJ
778:CGN
776:BIK
774:AFO
770:GKM
768:FHL
766:CJO
764:BDN
762:AEI
758:ILO
756:FKN
754:CHM
752:BEJ
750:ADG
746:MNO
744:JKL
742:GHI
740:DEF
738:ABC
733:and
720:GJM
718:DHK
716:CEO
714:BFI
712:ALN
708:EHL
706:DIM
704:CFJ
702:BGN
700:AKO
696:FLM
694:EGK
692:CDN
690:BHO
688:AIJ
684:EIN
682:DJO
680:CGL
678:BKM
676:AFH
672:HJN
670:FGO
668:CIK
666:BDL
664:AEM
660:ILO
658:FKN
656:CHM
654:BEJ
652:ADG
648:MNO
646:JKL
644:GHI
642:DEF
640:ABC
635:and
620:EGK
618:DIM
616:CFJ
614:BHO
612:ALN
608:GJM
606:EHL
604:CDN
602:BFI
600:AKO
596:FLM
594:DHK
592:CEO
590:BGN
588:AIJ
584:EIN
582:DJO
580:CGL
578:BKM
576:AFH
572:HJN
570:FGO
568:CIK
566:BDL
564:AEM
560:ILO
558:FKN
556:CHM
554:BEJ
552:ADG
548:MNO
546:JKL
544:GHI
542:DEF
540:ABC
535:and
517:EIK
515:DHM
513:CGO
511:BFJ
509:ALN
505:GKM
503:FHO
501:CEN
499:BDL
497:AIJ
493:ELM
491:DJO
489:CFI
487:BGN
485:AHK
481:EGJ
479:DIN
477:CHL
475:BKO
473:AFM
469:HJN
467:FGL
465:CDK
463:BIM
461:AEO
457:ILO
455:FKN
453:CJM
451:BEH
449:ADG
445:MNO
443:JKL
441:GHI
439:DEF
437:ABC
432:and
419:EGJ
417:DHM
415:CFI
413:BKO
411:ALN
407:GKM
405:FHO
403:CEN
401:BDL
399:AIJ
395:ELM
393:DIN
391:CGO
389:BFJ
387:AHK
383:EIK
381:DJO
379:CHL
377:BGN
375:AFM
371:HJN
369:FGL
367:CDK
365:BIM
363:AEO
359:ILO
357:FKN
355:CJM
353:BEH
351:ADG
347:MNO
345:JKL
343:GHI
341:DEF
339:ABC
334:and
321:EGJ
319:DHM
317:CFI
315:BKO
313:ALN
309:HJN
307:FGO
305:CEK
303:BDL
301:AIM
297:EIN
295:DJO
293:CGL
291:BFM
289:AHK
285:ELM
283:DIK
281:CHO
279:BGN
277:AFJ
273:GKM
271:FHL
269:CDN
267:BIJ
265:AEO
261:ILO
259:FKN
257:CJM
255:BEH
253:ADG
249:MNO
247:JKL
245:GHI
243:DEF
241:ABC
234:and
221:EGJ
219:DIK
217:CHO
215:BFM
213:ALN
209:ELM
207:DJO
205:CFI
203:BGN
201:AHK
197:EIN
195:DHM
193:CGL
191:BKO
189:AFJ
185:HJN
183:FGO
181:CEK
179:BDL
177:AIM
173:GKM
171:FHL
169:CDN
167:BIJ
165:AEO
161:ILO
159:FKN
157:CJM
155:BEH
153:ADG
149:MNO
147:JKL
145:GHI
143:DEF
141:ABC
130:and
76:in
3822::
3539:,
3520:.
3479:;
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