Editing Steiner system (section)

===Resolvable Steiner systems===
Some of the S(2,3,n) systems can have their triples partitioned into (n-1)/2 sets each having (n/3) pairwise disjoint triples. This is called ''resolvable'' and such systems are called ''Kirkman triple systems'' after [[Thomas Kirkman]], who studied such resolvable systems before Steiner. Dale Mesner, Earl Kramer, and others investigated collections of Steiner triple systems that are mutually disjoint (i.e., no two Steiner systems in such a collection share a common triplet). It is known (Bays 1917, Kramer & Mesner 1974) that seven different S(2,3,9) systems can be generated to together cover all 84 triplets on a 9-set; it was also known by them that there are 15360 different ways to find such 7-sets of solutions, which reduce to two non-isomorphic solutions under relabeling, with multiplicities 6720 and 8640 respectively.

The corresponding question for finding thirteen different disjoint S(2,3,15) systems was asked by [[James Joseph Sylvester|James Sylvester]] in 1860 as an extension of the [[Kirkman's schoolgirl problem]], namely whether Kirkman's schoolgirls could march for an entire term of 13 weeks with no triplet of girls being repeated over the whole term. The question was solved by [[RHF Denniston]] in 1974,<ref name="denniston">{{cite journal| title = Denniston's paper, open access| journal = Discrete Mathematics| date = September 1974| volume = 9| issue = 3| pages = 229–233| doi = 10.1016/0012-365X(74)90004-1| last1 = Denniston| first1 = R. H. F.| doi-access = free}}</ref> who constructed Week 1 as follows:

<pre>
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
</pre>

for girls labeled A to O, and constructed each subsequent week's solution from its immediate predecessor by changing A to B, B to C, ... L to M and M back to A, all while leaving N and O unchanged. The Week 13 solution, upon undergoing that relabeling, returns to the Week 1 solution. Denniston reported in his paper that the search he employed took 7 hours on an [[Elliott Brothers (computer company)|Elliott 4130]] computer at the [[University of Leicester]], and he immediately ended the search on finding the solution above, not looking to establish uniqueness. The number of non-isomorphic solutions to Sylvester's problem remains unknown as of 2021.