Woolhouse in 1844 in the Prize question #1733 of Lady's and Gentlemen's Diary.
In 1850 Kirkman posed a variation of the problem known as Kirkman's schoolgirl problem, which asks for triple systems having an additional property (resolvability).
The corresponding question for finding thirteen different disjoint S(2,3,15) systems was asked by James Sylvester in 1860 and answered by RHF Denniston in 1974.
If the column sums' parities don't match the row sum parity, or each other, or if there do not exist a, b, c such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S(5,8,24). The MOG is based on creating a bijection (Conwell 1910, "The three-space PG(3,2) and its group") between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the Fano 3-space PG(3,2).
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.
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 Sylvester in 1860 and answered by RHF Denniston in 1974.
It is also geometrically related (Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A193-194, Feb 1979) to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite affine space, then the groups form a set of parallel subspaces. == See also == Constant weight code Kirkman's schoolgirl problem Sylvester–Gallai configuration ==Notes== ==References== . .
The smallest order for which the existence is not known (as of 2011) is 21. ===Steiner triple systems=== An S(2,3,n) is called a Steiner triple system, and its blocks are called triples.
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