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| '''Kirkman's schoolgirl problem''' is a problem in [[combinatorics]] proposed by Rev. [[Thomas Penyngton Kirkman]] in 1850 as Query VI in ''[[The Lady's and Gentleman's Diary]]'' (pg.48). The problem states:
| | The author is known as Wilber Pegues. Ohio is exactly where her house is. Distributing production is exactly where her primary income arrives from. The preferred hobby for him and his kids is style and he'll be beginning something else alongside with it.<br><br>Feel free to surf to my blog - [https://www.machlitim.org.il/subdomain/megila/end/node/12300 tarot readings] |
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| <blockquote>
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| 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.<ref name="graham1995">{{harv|Graham|Grötschel|Lovász|1995}}</ref>
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| </blockquote>
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| == Solution ==
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| If the girls are numbered from 01 to 15, the following arrangement is one solution:<ref name="essays"/>
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| {| class="wikitable"
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| |-
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| ! Sun.
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| ! Mon.
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| ! Tues.
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| ! Wed.
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| ! Thurs.
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| ! Fri.
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| ! Sat.
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| |-
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| | 01, 06, 11
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| | 01, 02, 05
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| | 02, 03, 06
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| | 05, 06, 09
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| | 03, 05, 11
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| | 05, 07, 13
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| | 11, 13, 04
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| |-
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| | 02, 07, 12
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| | 03, 04, 07
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| | 04, 05, 08
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| | 07, 08, 11
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| | 04, 06, 12
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| | 06, 08, 14
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| | 12, 14, 05
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| |-
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| | 03, 08, 13
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| | 08, 09, 12
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| | 09, 10, 13
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| | 12, 13, 01
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| | 07, 09, 15
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| | 09, 11, 02
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| | 15, 02, 08
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| |-
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| | 04, 09, 14
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| | 10, 11, 14
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| | 11, 12, 15
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| | 14, 15, 03
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| | 08, 10, 01
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| | 10, 12, 03
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| | 01, 03, 09
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| |-
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| | 05, 10, 15
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| | 13, 15, 06
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| | 14, 01, 07
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| | 02, 04, 10
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| | 13, 14, 02
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| | 15, 01, 04
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| | 06, 07, 10
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| |}
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| A solution to this problem is an example of a ''Kirkman triple system'',<ref name="mathworld">{{MathWorld |title= Kirkman's Schoolgirl Problem |urlname= KirkmansSchoolgirlProblem}}</ref> which is a [[Steiner triple system]] having a ''parallelism'', that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks.
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| There are seven non-[[isomorphic]] solutions to the schoolgirl problem.<ref>{{harv|Cole|1922}}</ref> Two of these are ''packings'' of the finite [[projective space]] PG(3,2).<ref>{{harv|Hirschfeld|1985|loc=pg.75}}</ref> A packing of a projective space is a [[Partition (mathematics)|partition]] of the lines of the space into ''spreads'', and a spread is a partition of the points of the space into lines. These "packing" solutions can be visualized as relations between a tetrahedron and its vertices, edges, and faces.<ref>{{harvnb|Falcone|Pavone|2011}}</ref>
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| ==History==
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| The first solution was published by [[Arthur Cayley]].<ref>{{harvnb|Cayley|1850}}</ref> This was shortly followed by Kirkman's own solution<ref>{{harvnb|Kirkman|1850}}</ref> which was given as a special case of his considerations on combinatorial arrangements published three years prior.<ref name="kirkman1847"/> [[J. J. Sylvester]] also investigated the problem and ended up declaring that Kirkman stole the idea from him. The puzzle appeared in several recreational mathematics books at the turn of the century by Lucas,<ref>{{harvnb|Lucas|1883}}</ref> Rouse Ball,<ref>{{harvnb|Rouse Ball|1892}}</ref> Ahrens,<ref>{{harvnb|Ahrens|1901}}</ref> and Dudeney.<ref>{{harvnb|Dudeney|1917}}</ref> | |
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| Kirkman often complained about the fact that his substantial paper {{harv|Kirkman|1847}} was totally eclipsed by the popular interest in the schoolgirl problem.<ref>{{harvnb|Cummings|1918}}</ref>
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| == Generalization ==
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| The problem can be generalized to <math>n</math> girls, where <math>n</math> must be an odd multiple of 3 (that is <math>n \equiv 3 \pmod{6}</math>), walking in triplets for <math>\frac{1}{2}(n-1)</math> days, with the requirement, again, that no pair of girls walk in the same row twice. The solution to this generalisation is a [[Steiner triple system]], an S(2, 3, 6''t'' + 3) with parallelism (that is, one in which each of the 6''t'' + 3 elements occurs exactly once in each block of 3-element sets), known as a ''Kirkman triple system''.<ref name="essays">{{harv|Ball|Coxeter|1974}}</ref> It is this generalization of the problem that Kirkman discussed first, while the famous special case <math>n=15</math> was only proposed later.<ref name="kirkman1847">{{harvnb|Kirkman|1847}}</ref> A complete solution to the general case was published by [[D. K. Ray-Chaudhuri]] and [[R. M. Wilson]] in 1968,<ref>{{harvnb|Ray-Chaudhuri|Wilson|1971}}</ref> though it had already been solved by Lu Jiaxi in 1965,<ref>{{harvnb|Jiaxi|1990}}</ref> but had not been published at that time.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=p. 13}}</ref>
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| 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<ref name="hartman1980">{{harv|Hartman|1980}}</ref> using Steiner quadruple systems.
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| More recently a similar problem known as the Social Golfer Problem has gained interest that deals with 20 golfers who want to get to play with different people each day in groups of 4.
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| As this is a regrouping strategy where all groups are orthogonal, this process within the problem of organising a large group into a small groups where no two people share the same group twice can be referred to as orthogonal regrouping. However, this term is currently not commonly used and evidence suggests that there isn't a common name for the process.
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| == Other applications ==
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| * [[Progressive dinner]] party designs
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| * [[Speed Networking]] events
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| * [[Cooperative learning]] strategy for increasing interaction within classroom teaching
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| * Sports Competitions
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| == Notes ==
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| <references/>
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| ==References==
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| *{{citation|last=Ahrens|first=W.|title=Mathematische Unterhaltungen und Spiele|year=1901|publisher=Teubner|location=Leipzig}}
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| *{{citation|last= Ball |first= W.W. Rouse |authorlink= W. W. Rouse Ball |coauthors= [[H.S.M. Coxeter]] |title=Mathematical Recreations & Essays |year= 1974 |publisher= University of Toronto Press |location= Toronto and Buffalo |isbn= 0-8020-1844-0}}
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| *{{citation|last=Cayley|first=A.|authorlink=Arthur Cayley|title=On the triadic arrangements of seven and fifteen things|journal=Phil. Mag.|volume=37|year=1850|pages=50–53}}
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| *{{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=1-58488-506-8|edition=2nd Edition}}
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| *{{citation|last=Cole|first=F.W.|title=Kirkman parades|journal=Bulletin of the American Mathematical Society|volume=28|year=1922|pages=435–437|doi=10.1090/S0002-9904-1922-03599-9}}
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| *{{citation|last=Cummings|first=L.D.|title=An undervalued Kirkman paper|journal=Bulletin of the American Mathematical Society|volume=24|year=1918|pages=336–339|doi=10.1090/S0002-9904-1918-03086-3}}
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| *{{citation|last=Dudeney|first=H.E.|title=Amusements in Mathematics|publisher=Dover|place=New York|year=1917}}
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| *{{citation|last1=Falcone|first1=Giovanni|last2=Pavone|first2=Marco|title=Kirkman's Tetrahedron and the Fifteen Schoolgirl Problem|journal=American Mathematical Monthly|volume=118|year=2011|pages=887–900|doi=10.4169/amer.math.monthly.118.10.887}}
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| *{{citation|last= Graham |first= Ronald L. |authorlink= Ronald Graham |coauthors= [[Martin Grötschel]], [[László Lovász]] |title= Handbook of Combinatorics, Volume 2 |year= 1995 |publisher= The MIT Press |location= [[Cambridge, MA]] |isbn= 0-262-07171-1 }}
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| *{{citation|last= Hartman |first= Alan |title=Kirkman's trombone player problem|journal=[[Ars Combinatoria (journal)|Ars Combinatoria]]|volume=10 |pages=19–26|year=1980}}
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| *{{citation|last=Hirschfeld|first=J.W.P.|title=Finite Projective Spaces of Three Dimensions|publisher=Oxford University Press|location=Oxford|year=1985|isbn=0-19-853536-8}}
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| *{{citation|last=Jiaxi|first=Lu|title=Collected Works of Lu Jiaxi on Combinatorial Designs|publisher=Inner Mongolia People's Press|location=Huhhot|year=1990}}
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| *{{citation|last= Kirkman |first=Thomas P. |authorlink= Thomas Kirkman |title= On a Problem in Combinations |journal= [[The Cambridge and Dublin Mathematical Journal]] |volume= II |pages= 191–204 | publisher = Macmillan, Barclay, and Macmillan |year= 1847}}
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| *{{citation|last=Kirkman|first=Thomas P.|authorlink=Thomas Kirkman|title=Note on an unanswered prize question|journal=[[The Cambridge and Dublin Mathematical Journal]]| volume=5|pages=255–262|publisher= Macmillan, Barclay and Macmillan|year=1850}}
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| *{{citation|last=Lucas|first=É.|title=Récréations Mathématiques|volume=2|publisher=Gauthier-Villars|location=Paris|year=1883}}
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| *{{citation|last1=Ray-Chaudhuri|first1=D.K.|last2=Wilson|first2=R.M.|title=Solution of Kirkman's schoolgirl problem, in ''Combinatorics, University of California, Los Angeles, 1968''|journal=Proc. Sympos. Pure Math.|volume=XIX|publisher=American Mathematical Society|place=Providence, R.I.|year=1971|pages=187–203}}
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| *{{citation|last=Rouse Ball|first=W.W.|authorlink=W.W. Rouse Ball|title=Mathematical Recreations and Essays|publisher=Macmillan|place=London|year=1892}}
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| [[Category:Design theory]]
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| [[Category:Set families]]
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| [[Category:Mathematical problems]]
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The author is known as Wilber Pegues. Ohio is exactly where her house is. Distributing production is exactly where her primary income arrives from. The preferred hobby for him and his kids is style and he'll be beginning something else alongside with it.
Feel free to surf to my blog - tarot readings