Steiner triple system S(2,3,n): n points, blocks of size 3, every pair in exactly one block. Exists iff n≡1,3(mod6). Number of triples: n(n-1)/6. S(2,3,7)=Fano plane (7 points, 7 triples). S(2,3,9)=affine plane AG(2,3).
How to Use
Enter n:
- Triples: All blocks
- Existence: n≡1,3(mod6)?
- Count: n(n-1)/6
Fano Plane
S(2,3,7): the smallest non-trivial Steiner triple system. 7 points, 7 triples: {1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3}. This is also the projective plane of order 2, PG(2,2). It has beautiful symmetry: any permutation of {1,...,7} that preserves the triples is an automorphism.
Construction
For n≡1(mod6): Bose construction using GF(n). For n≡3(mod6): Skolem construction. For n=2^k-1: use binary representations (Hamming codes give Steiner systems). Recursive doubling constructions also work.
Step-by-Step Instructions
- 1Enter n.
- 2Check existence.
- 3Generate triples.
- 4Verify pair coverage.
- 5Count blocks.