Kirkman's schoolgirl problem

Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. 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.

If the girls are numbered from 01 to 15, the following arrangement is one solution:

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.

There are seven non-isomorphic solutions to the schoolgirl problem. Two of these can be visualized as relations between a tetrahedron and its vertices, edges, and faces. An approach using projective geometry of three dimensions over GF(2) is given below.

The first solution was published by Arthur Cayley. This was shortly followed by Kirkman's own solution which was given as a special case of his considerations on combinatorial arrangements published three years prior.J. J. Sylvester also investigated the problem and ended up declaring that Kirkman stole the idea from him. The puzzle appeared in several recreational mathematics books at the turn of the century by Lucas, Rouse Ball, Ahrens, and Dudeney.

Kirkman often complained about the fact that his substantial paper (Kirkman 1847) was totally eclipsed by the popular interest in the schoolgirl problem.

In 1910 the problem was addressed using Galois geometry by George Conwell

The Galois field GF(2) with two elements is used with four homogeneous coordinates to form PG(3,2) which has 15 points, 3 points to a line, 7 points and 7 lines in a plane. A plane can be considered a complete quadrilateral together with the line through its diagonal points. Each point is on 7 lines, and there are 35 lines in all.

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