A Kirkman triple system of order is a Steiner triple system with parallelism (Ball and Coxeter 1987), i.e., one with the following additional stipulation: the set of triples is partitioned into components such that each component is a -subset of triples and each of the elements appears exactly once in each component. The Steiner triple systems of order 3 and 9 are Kirkman triple systems with and 1. Solution to Kirkman's schoolgirl problem requires construction of a Kirkman triple system of order .
Ray-Chaudhuri and Wilson (1971) showed that there exists at least one Kirkman triple system for every nonnegative order . Earlier editions of Ball and Coxeter (1987) gave constructions of Kirkman triple systems with . For , there is a single unique (up to an isomorphism) solution, while there are 7 different systems for (Mulder 1917, Cole 1922, Ball and Coxeter 1987).