A Kirkman triple system of order v = 6n + 3 is a Steiner triple system with parallelism, i.e., one with the following additional stipulation: the set of b = (2n + 1)(3n + 1) triples is partitioned into 3n + 1 components such that each component is a (2n + 1)-subset of triples and each of the v elements appears exactly once in each component. The Steiner triple systems of order 3 and 9 are Kirkman triple systems with n = 0 and 1. Solution to Kirkman's schoolgirl problem requires construction of a Kirkman triple system of order n = 2.

Steiner triple system

Thomas Penyngton Kirkman