contained in at least one block, one can compute the number of blocks intersecting that subset in a fixed number of elements by constructing a Pascal triangle. In particular, the number of blocks intersecting a fixed block in any number of elements is independent of the chosen block. The number of blocks that contain any i-element set of points is: It can be shown that if there is a Steiner system , where is a prime power greater than 1, then 1 or . In particular, a Steiner triple system must have . And as we have already mentioned, this is the only restriction on Steiner triple systems, that is, for each natural number , systems and exist. History Steiner triple systems were defined for the first time by Wesley S. B. Woolhouse in 1844 in the Prize question #1733 of Lady's and Gentlemen's Diary. The posed problem was solved by . In 1850 Kirkman posed a variation of the problem known as Kirkman's schoolgirl problem, which asks for triple systems having an additional property (resolvability). Unaware of Kirkman's work, reintroduced triple systems, and as this work was more widely known, the systems were named in his honor. Mathieu groups Several examples of Steiner systems are closely related to group theory. In particular, the finite simple groups called Mathieu groups arise as automorphism groups of Steiner systems: The Mathieu group M11 is the automorphism group of a S(4,5,11) Steiner system The Mathieu group M12 is the automorphism group of a S(5,6,12) Steiner system The Mathieu group M22 is the unique index 2 subgroup of the automorphism group of a S(3,6,22) Steiner system The Mathieu group M23 is the automorphism group of a S(4,7,23) Steiner system The Mathieu group M24 is the automorphism group of a S(5,8,24) Steiner system. The Steiner system S(5, 6, 12) There is a unique S(5,6,12) Steiner system; its automorphism group is the Mathieu group M12, and in that context it is denoted by W12. Projective line construction This construction is due to Carmichael (1937). Add a new element, call it , to the 11 elements of the finite field 11 (that is, the integers mod 11). This set, , of 12 elements can be formally identified with the points of the projective line over 11. Call the following specific subset of size 6, a "block" (it contains together with the 5 nonzero squares in 11). From