Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Penyngton Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary (pg.48). The problem states:
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.
Solutions A solution to this problem is an example of a Kirkman triple system, 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. Such Steiner systems that have a parallelism are also called resolvable.
There are exactly seven non-isomorphic solutions to the schoolgirl problem, as originally listed by Frank Nelson Cole in Kirkman Parades in 1922. The seven solutions are summarized in the table below, denoting the 15 girls with the letters A to O.
Solution classAutomorphism groupDay 1Day 2Day 3Day 4Day 5Day 6Day 7Solution IOrder 168, generated by(A K G E I L B)(C H M J N O D)and(A M L K O C D)(B H N G I E J).Related to PG(3,2).ABCDEFGHIJKLMNOADGBEHCJMFKNILOAEOBIJCDNFHLGKMAIMBDLCEKFGOHJNAFJBKOCGLDHMEINAHKBGNCFIDJOELMALNBFMCHODIKEGJSolution IIOrder 168, generated by(A B I M F C J)(D N H K O L E)and(A J M I B F C)(D H G N K E O).Related to PG(3,2).ABCDEFGHIJKLMNOADGBEHCJMFKNILOAEOBIJCDNFHLGKMAFJBGNCHODIKELMAHKBFMCGLDJOEINAIMBDLCEKFGOHJNALNBKOCFIDHMEGJSolution IIIOrder 24, generated by(A H E)(B O K)(C F I)(D J L)(G N M)and(A J B M)(D L E O)(F I)(G K H N)ABCDEFGHIJKLMNOADGBEHCJMFKNILOAEOBIMCDKFGLHJNAFMBGNCHLDJOEIKAHKBFJCGODINELMAIJBDLCENFHOGKMALNBKOCFIDHMEGJSolution IVOrder 24, generated by(A J M)(C F I)(D E K)(H O L)and(A L B O)(C I)(D K E N)(G J H M)ABCDEFGHIJKLMNOADGBEHCJMFKNILOAEOBIMCDKFGLHJNAFMBKOCHLDINEGJAHKBGNCFIDJOELMAIJBDLCENFHOGKMALNBFJCGODHMEIKSolution VTetrahedral group of order 12,generated by(A L)(B G)(E O)(F J)(H K)(I M)and(A B C)(D L G)(F J I)(E K H)ABCDEFGHIJKLMNOADGBEJCHMFKNILOAEMBDLCIKFGOHJNAFHBKMCGLDJOEINAIJBGNCEODHKFLMAKOBFICDNEHLGJMALNBHOCFJDIMEGKSolution VITetrahedral group of order 12,generated by(A L)(B G)(E O)(H K)(F J)(I M)and(A B C)(D L G)(E K H)(F J I)ABCDEFGHIJKLMNOADGBEJCHMFKNILOAEMBDLCIKFGOHJNAFHBKMCGLDJOEINAIJBHOCDNEGKFLMAKOBGNCFJDIMEHLALNBFICEODHKGJMSolution VIIOrder 21, generated by(A B L C G D N)(E H K I O J F)and(B G L)(C D N)(E F K)(H I O)ABCDEFGHIJKLMNOADGBEJCHMFKNILOAEIBDNCJOFHLGKMAFOBIKCGNDHJELMAHKBFMCDLEGOIJNAJMBGLCFIDKOEHNALNBHOCEKFGJDIM
From the number of automorphisms for each solution and the definition of an automorphism group, the total number of solutions including isomorphic solutions is therefore:
.
History
The problem has a long and storied history. This section is based on historical work done at different times by Robin WilsonThe Early History of Block Designs by Robin Wilson, Dept of Pure Mathematics, The Open University, UK and by Louise Duffield Cummings. The history is as follows:
In 1844, Wesley Woolhouse, the editor of The Lady's and Gentleman's Diary at the time, asked the general question: "Determine the number of combinations that can be made out of n symbols, p symbols in each; with this limitation, that no combination of q 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 . As the question did not ask for anything more than the number of combinations, nothing was received about the conditions on n, p, or q when such a solution could be achieved. In 1846, Woolhouse asked: "How many triads can be made out of n 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 p = 3 and q = 2. In 1847, at age 41, Thomas Kirkman published his paper titled On a Problem in Combinations which comprehensively described and solved the problem of constructing triple systems of order n where n = 1 or 3 (mod 6). He also considered other values of n even though perfect balance would not be possible. He gave two different sequences of triple systems, one for n = 7, 15, 19, 27, etc., and another for n = 9, 13, 25, etc. Using these propositions, he proved that triple systems exist for all values of n = 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 n = 9 and 15; resolvable triple systems are now known as Kirkman triple systems. He could not conclusively say for what other values of n would resolvable triple systems exist; that problem would not be solved until the 1960s (see below). 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 solution but that was incorrect. Arthur Cayley's solution would be later found to be isomorphic to Solution II. Both solutions could be embedded in PG(3,2) 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 below. Also in 1850, James Joseph Sylvester asked if there could be 13 different solutions to the 15-schoolgirl problem that would use all triples exactly once overall, observing that . 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 RHF Denniston (see below). In 1852, Robert Richard Anstice provided a cyclic solution, made by constructing the first day's five triples to be 0Gg, AbC, aDE, cef, BdF on the 15 symbols 0ABCDEFGabcdefg 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 (AbC, aDE, cef, BdF) are taken and uppercase converted to lowercase (abc, ade, cef, bdf) they form what would later be called the Pasch configuration. The Pasch configuration would become important in isomorph rejection techniques in the 20th century. In 1853, Jakob Steiner, completely unaware of Kirkman's 1847 paper, published his paper titled Combinatorische Aufgabe which reintroduced the concept of triple systems but did not mention resolvability into separate parallel classes. Steiner noted that it is necessary for n 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 Steiner triple systems. In 1859, Michel Reiss 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 plagiarism. Kirkman himself expressed his bitterness. In 1860, Benjamin Peirce 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 Cayley's. In 1861, James Joseph Sylvester revisited the problem and tried to claim that he 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 Kirkman. In 1861-1862, Kirkman had a falling out with Arthur Cayley over an unrelated matter (Cayley's choosing not to publish a series of papers by Kirkman on group theory and polyhedra 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 history. 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 Édouard Lucas, Rouse Ball, Wilhelm Ahrens, and Henry Dudeney. In his lifetime, Kirkman would complain about his serious mathematical work being eclipsed by the popularity of the schoolgirl problem. Kirkman died in 1895. In 1918, Kirkman's serious mathematical work was brought back to wider attention by Louise Duffield Cummings in a paper titled An Undervalued Kirkman Paper which discussed the early history of the field and corrected the historical omission. At about the same time, Cummings was working with Frank Nelson Cole and Henry Seely White on triple systems. This culminated in their famous and widely cited 1919 paper Complete classification of triad systems on 15 elements 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. In 1922, Cole published his paper Kirkman Parades 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 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. In the 1960s, it was proved that Kirkman triple systems exist for all orders n = 3 (mod 6). This was first proved by Lu Jiaxi () in 1965, and he submitted it to Acta Mathematica Sinica 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 Cultural Revolution and rejected again. In 1968, the generalized theorem was proven independently by D. K. Ray-Chaudhuri and R. M. Wilson. In 1974, RHF Denniston 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.
Sylvester's problem