condition is that n 4 (mod 5), which comes from the fact that the number of blocks must be an integer. Sufficient conditions are not known. There is a unique Steiner quintuple system of order 11, but none of order 15 or order 17. Systems are known for orders 23, 35, 47, 71, 83, 107, 131, 167 and 243. 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. It is common to see the abbreviation STS(n) for a Steiner triple system of order n. The total number of pairs is n(n-1)/2, of which three appear in a triple, and so the total number of triples is n(n−1)/6. This shows that n must be of the form 6k+1 or 6k + 3 for some k. The fact that this condition on n is sufficient for the existence of an S(2,3,n) was proved by Raj Chandra Bose and T. Skolem. The projective plane of order 2 (the Fano plane) is an STS(7) and the affine plane of order 3 is an STS(9). Up to isomorphism, the STS(7) and STS(9) are unique, there are two STS(13)s, 80 STS(15)s, and 11,084,874,829 STS(19)s. We can define a multiplication on the set S using the Steiner triple system by setting aa = a for all a in S, and ab = c if {a,b,c} is a triple. This makes S an idempotent, commutative quasigroup. It has the additional property that ab = c implies bc = a and ca = b. Conversely, any (finite) quasigroup with these properties arises from a Steiner triple system. Commutative idempotent quasigroups satisfying this additional property are called Steiner quasigroups. Resolvable Steiner systems Some of the S(2,3,n) systems can have their triples partitioned into (n-1)/2 sets each having (n/3) pairwise disjoint triples. This is called resolvable and such systems are called Kirkman triple systems after Thomas Kirkman, who studied such resolvable systems before Steiner. Dale Mesner, Earl Kramer, and others investigated collections of Steiner triple systems that are mutually disjoint (i.e., no two Steiner systems in such a collection share a common triplet). 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