The problem has a long and storied history. This section is based on historical work done at different times by
Robin Wilson 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 \frac{n!}{q! (n-q)!} \div \frac{p!}{q! (p-q)!} . 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. (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 {15 \choose 3} = 455 triples exactly once overall, observing that 455 = 13 \times 35. 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). 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. 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.
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 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==