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 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, who constructed Week 1 as follows: 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 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 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. Properties It is clear from the definition of that . (Equalities, while technically possible, lead to trivial systems.) If exists, then taking all blocks containing a specific element and discarding that element gives a derived system . Therefore, the existence of is a necessary condition for the existence of . The number of -element subsets in is , while the number of -element subsets in each block is . Since every -element subset is contained in exactly one block, we have , or where is the number of blocks. Similar reasoning about -element subsets containing a particular element gives us , or = where is the number of blocks containing any given element. From these definitions follows the equation . It is a necessary condition for the existence of that and are integers. As with any block design, Fisher's inequality is true in Steiner systems. Given the parameters of a Steiner system and a subset of size ,