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).
