Let X be a set of v>=3 elements together with a set B of 3-subset (triples) of X such that every 2-subset of X occurs in exactly one triple of B. Then B is called a Steiner triple system and is a special case of a Steiner system with t = 2 and k = 3. A Steiner triple system S(v) = S(v, k = 3, λ = 1) of order v exists iff v congruent 1, 3 (mod 6). In addition, if Steiner triple systems S_1 and S_2 of orders v_1 and v_2 exist, then so does a Steiner triple system S of order v_1 v_2. Examples of Steiner triple systems S(v) of small orders v are S_3 | = | {{1, 2, 3}} S_7 | = | {{1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {4, 5, 7}, {5, 6, 1}, {6, 7, 2}, {7, 1, 3}} S_9 | = | {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {1, 4, 7}, {2, 5, 8}, {3, 6, 9}, {1, 5, 9}, {2, 6, 7}, {3, 4, 8}, {1, 6, 8}, {2, 4, 9}, {3, 5, 7}}.

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