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Kirkman's task about schoolgirls

Original task publication

Kirkman's schoolgirl task is a combinatorial problem proposed by Thomas Penington Kirkman in 1850 as Question VI in The Lady's and Gentleman's Diary (magazine of entertaining mathematics published between 1841 and 1871). The task says:

Fifteen young girls at school walk three in a row for seven days (every day), they need to be distributed for each walk so that no two girls walk in the same row ( Graham, Grötschel, Lovász 1995 ).

Content

Solution

If the girls are numbered from 0 to 14, the following distribution will be one of the solutions [1] :

Resurrected
sony		Pone
little whiteie		TuesdayWednesdayThursdayFridaySaturday
0, 5, 100, 1, 41, 2, 54, 5, 82, 4, 104, 6, 1210, 12, 3
1, 6, 112, 3, 63, 4, 76, 7, 103, 5, 115, 7, 1311, 13, 4
2, 7, 127, 8, 118, 9, 1211, 12, 06, 8, 148, 10, 114, 1, 7
3, 8, 139, 10, 1310, 11, 1413, 14, 27, 9, 09, 11, 20, 2, 8
4, 9, 1412, 14, 513, 0, 61, 3, 912, 13, 114, 0, 35, 6, 9

Solving this problem is an example of Kirkman's system of triples [2] ; This means that it is a Steiner system of triples , which has parallelism , that is, it has a partition of blocks of a system of triples into parallel classes, which are a partition of points into disjoint blocks.

There are seven non - isomorphic solutions to the schoolgirl problem [3] . Two of them can be visualized as the relationship between the tetrahedron and its vertices, edges and edges [4] . An approach using three-dimensional projective geometry over is given below.

Decision XOR triples

If the girls are renumbered with binary numbers from 0001 to 1111, the following distribution is a solution, such that for any three girls forming a group, the bitwise XOR of two numbers gives the third:

Resurrected
sony		Pone
little whiteie		TuesdayWednesdayThursdayFridaySaturday
0001, 0010, 00110001, 0100, 01010001, 0110, 01110001, 1000, 10010001, 1010, 10110001, 1100, 11010001, 1110, 1111
0100, 1000, 11000010, 1000, 10100010, 1001, 10110010, 1100, 11100010, 1101, 11110010, 0100, 01100010, 0101, 0111
0101, 1010, 11110011, 1101, 11100011, 1100, 11110011, 0101, 01100011, 0100, 01110011, 1001, 10100011, 1000, 1011
0110, 1011, 11010110, 1001, 11110100, 1010, 11100100, 1011, 11110101, 1001, 11000101, 1011, 11100100, 1001, 1101
0111, 1001, 11100111, 1011, 11000101, 1000, 11010111, 1010, 11010110, 1000, 11100111, 1000, 11110110, 1010, 1100

This solution has a geometric interpretation associated with the Galois geometry and PG (3,2) . Take a tetrahedron and renumber its vertices as 0001, 0010, 0100, and 1000. We number the six centers of the edges as the XOR of the edges of the edges. Assign the four centers of the faces of the label equal to the XOR of the three vertices, and the center of the body give the label 1111. Then 35 triples and the XOR solution corresponds exactly to the 35 direct PG (3,2).

History

The first decision was published by Arthur Cayley [5] . It was quickly followed by the decision of Kirkman himself [6] , which was given as a special case of his combinatorial placement, published three years earlier [7] . D. D. Sylvester also investigated the problem and concluded that Kirkman had stolen the idea from him. The puzzle appeared in several entertaining math books at the turn of the century by Lukas [8] , Rose Ball [9] , Aarents [10] and Dyudeni [11] .

Kirkman often explained that his large article ( Kirkman 1847 ) was completely caused by a huge interest in the problem [12] .

Galois geometry

In 1910, the task was considered by George Conwell with the help of Galois geometry [13] .

The Galois field with two elements was used with four homogeneous coordinates to form a PG (3,2) with 15 points, 3 points on a straight line, 7 points and 7 lines on a plane. The plane can be considered a complete quadrilateral along with straight lines through its diagonal points. Each point lies on 7 lines and in total there are 35 lines.

The straight spaces PG (3,2) are defined by their pluker coordinates in PG (5,2) with 63 points, 35 of which represent straight lines in PG (3,2). These 35 points form the surface S , known as the . For each of the 28 points not lying on S , there are 6 lines through this point that do not intersect S [14] .

As the number of days in a week, the seven plays an important role in the decision:

 If two points A and B on a straight line ABC are selected, each of the five other straight lines through A intersects only one of the five straight lines passing through B. Five points that result from the intersection of these pairs, together with two points A and B, are called "seven" ( Conwell 1910 , 68). 

A seven is defined by its two points. Each of the 28 points outside S lies on two sevens. There are 8 sevens. PGL (3,2) is isomorphic to the alternating group on 8 sevens [15] .

The problem of schoolgirls consists of finding seven non-intersecting lines in a 5-dimensional space, such that any two lines always have a common seven [16] .

Generalization

The task can be generalized ton {\ displaystyle n}   female students wheren {\ displaystyle n}   must be a number equal to the product of an odd number by 3 (i.e.,n≡3(mod6) {\ displaystyle n \ equiv 3 {\ pmod {6}}}   strolling in threesone2(n-one) {\ displaystyle {\ frac {1} {2}} (n-1)}   days with the condition, again, that no pair of girls walks twice in the same row [17] . The solution to this generalization is the Steiner S (2, 3, 6 t + 3) system of triples with parallelism (that is, a system in which every 6 t + 3 elements are exactly once in each block of 3-element sets), known as the Kirkman system Template: Rouse Ball, Coxeter . This is a generalization of the problem, which Kirkman originally discussed, and the famous special casen=15 {\ displaystyle n = 15}   he discussed later [7] . The full solution of the general case was published by DK Rei-Chadhuri and R. M. Wilson in 1968 [18] , although the problem was already solved by the Chinese mathematician Liu Jaxi (陆 家 羲) in 1965 [19] , but at that time the solution was still not published [20] .

Several variants of the main task were considered. Alan Hartman solved a problem of this type with the requirement that no three walk in rows of four more than once [21] , using the Steiner system of fours.

Recently, a similar problem has become known, known as the “golf course challenge,” in which there are 32 golfers who want to play with different people every day in groups of 4 for 10 consecutive days.

Since this is a regrouping strategy, when all groups are orthogonal, this process of forming from a large group of small groups in which no two people fall simultaneously into more than one group can be considered as orthogonal rearrangement. However, this term is rarely used and it can be considered that there is no generally accepted term for this process.

The Oberwolf problem of decomposing a complete graph into disjoint copies of a given 2-regular graph also generalizes the Kirkman schoolgirl problem. The Kirkman problem is a special case of the Oberwolf problem, in which a 2-regular graph consists of five disjoint triangles [22] .

Other Applications

  • Cooperative learning - a strategy for increasing student collaboration
  • Sport competitions

Notes

  1. ↑ Rouse Ball, Coxeter, 1987 , p. 287–289.
  2. ↑ Weisstein, Eric W. Kirkman's Schoolgirl Problem (English) on Wolfram MathWorld .
  3. ↑ Cole, 1922 , p. 435–437.
  4. ↑ Falcone, Pavone, 2011 , p. 887–900.
  5. ↑ Cayley, 1850 , p. 50–53.
  6. ↑ Kirkman, 1850 .
  7. ↑ 1 2 Kirkman, 1847 .
  8. ↑ Lucas, 1883 , p. 183–188.
  9. ↑ Rouse Ball, 1892 .
  10. ↑ Ahrens, 1901 .
  11. ↑ Dudeney, 1917 .
  12. ↑ Cummings, 1918 .
  13. ↑ Conwell, 1910 , p. 60–76.
  14. ↑ Conwell, 1910 , p. 67.
  15. ↑ Conwell, 1910 , p. 69
  16. ↑ Conwell, 1910 , p. 74.
  17. ↑ Tarakanov, 1985 , p. 109.
  18. ↑ Ray-Chaudhuri, Wilson, 1971 .
  19. ↑ Lu, 1990 .
  20. ↑ Colbourn, Dinitz, 2007 , p. 13.
  21. ↑ Hartman, 1980 .
  22. ↑ Bryant, Danziger, 2011 .

Literature

  • Cole FW Kirkman parades // Bulletin of the American Mathematical Society. - 1922. - T. 28 . - DOI : 10.1090 / S0002-9904-1922-03599-9 .
  • Giovanni Falcone, Marco Pavone. Kirkman's Tetrahedron and the Fifteen Schoolgirl Problem // American Mathematical Month . - 2011. - T. 118 . - DOI : 10.4169 / amer.math.monthly.118.10.887 .
  • George M. Conwell. The 3-space PG (3,2) and its Groups // Annals of Mathematics . - 1910. - T. 11 . - DOI : 10.2307 / 1967582 .
  • Cayley A. On the triadic arrangements of the seven and fifteen things // Philosophical Magazine . - 1850. - V. 37 . - DOI : 10.1080 / 14786445008646550 .
  • Hirschfeld JWP Finite Projective Space of Three Dimensions. - Oxford University Press , 1985. - ISBN 0-19-853536-8 .
  • Ahrens W. Mathematische Unterhaltungen und Spiele. - Leipzig: Teubner, 1901.
  • Darryn Bryant, Peter Danziger. On bipartite 2-factorizations ofKn-I {\ displaystyle K_ {n} -I}   and the Oberwolfach problem // Journal of Graph Theory . - 2011. - Vol. 68 , no. 1 . - pp . 22–37 . - DOI : 10.1002 / jgt.20538 .
  • Charles J. Colbourn, Jeffrey H. Dinitz. Handbook of Combinatorial Designs. - 2nd. - Boca Raton: Chapman & Hall / CRC, 2007. - ISBN 1-58488-506-8 .
  • Cummings LD An undervalued Kirkman paper // Bulletin of the American Mathematical Society. - 1918. - T. 24 . - p . 336–339 . - DOI : 10.1090 / S0002-9904-1918-03086-3 .
  • Dudeney HE Amusements in Mathematics. - New York: Dover, 1917.
    • Dudeney HE [ Amusements in Mathematics in Google Books Amusements in Mathematics]. - Mineola, New York : Dover, 1958. - (Dover Recreational Math). - ISBN 978-0-486-20473-4 .
  • Ronald L. Graham, Martin Grötschel, László Lovász. Handbook of Combinatorics, Volume 2. - Cambridge, MA : The MIT Press, 1995. - ISBN 0-262-07171-1 .
  • Alan Hartman. Kirkman's trombone player problem // Ars Combinatoria . - 1980. - Vol . 10 . - p . 19–26 .
  • Jiaxi Lu. Collected Works of Lu Jiaxi on Combinatorial Designs. - Huhhot: Inner Mongolia People's Press, 1990.
  • Thomas P. Kirkman. On the Problem in Combinations // The Cambridge Mathematical Journal . - Macmillan, Barclay, and Macmillan, 1847. - T. II . - p . 191–204 .
  • Thomas P. Kirkman. Note on an unanswered prize // The Cambridge Mathematical Journal . - Macmillan, Barclay and Macmillan, 1850. - Vol . 5 . - p . 255-262 .
  • Lucas É. [ Récréations mathématiques, 2 in Google Books Récréations Mathématiques]. - Paris: Gauthier-Villars, 1883. - Vol. 2. - p. 183–188.
  • Ray-Chaudhuri DK, Wilson RM Solution of Kirkman's schoolgirl problem, in Combinatorics, University of California, Los Angeles, 1968 . - Proceedings Symposisa Pure Mathematics. - 1971. - T. XIX. - p. 187–203. - ISBN 978-0-8218-1419-2 . - DOI : 10.1090 / pspum / 019/9959 .
  • Rouse Ball WW Mathematical Recreations and Essays. - London: Macmillan, 1892.
    • Rouse Ball WW, Coxeter HSM [ Mathematical Recreations and Essays in Google Books , Mathematical Recreations and Essays]. - 13th. - Dover, 1987. - p. 287−289. - ISBN 0-486-25357-0 . Original edition: 1974
  • Tarakanov V. Е. Combinatorial problems and (0,1) matrices. - Moscow: "Science", 1985. - (Problems of science and technical progress).

Links

  • Erica Klarreich. A design dilemma solved, minus designs. // Quanta Magazine . - 2015. - June.
Source - https://ru.wikipedia.org/w/index.php?title=Task_Kirkman_o_schoolgirls&oldid=100904323


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