About the problem

Steiner systems are highly-symmetrical combinatorial objects with applications in experimental design and error-correcting codes. They have been actively studied since the mid-1800s.

The formal definition is simple: given a set \(S\) of size \(n\), an \((n,q,r)\)-Steiner system is a set of \(q\)-sized subsets of \(S\) such that every \(r\)-sized subset of \(S\) is contained in exactly one of the \(q\)-sized subsets.

Perhaps surprisingly, no example of a Steiner system with \(r > 5\) is known, despite a theorem, proven in 2014, that many such examples do exist. While there is no guarantee that an example exists with \(n < 200\) and \(5 < r < 10\), there is also no reason to expect this not to be the case. It seems unlikely that even an optimized brute-force search could solve this problem, with a novel conceptual approach likely required.

Read more here.

Prompt

Let [n] be the elements {1,...,n}. Call T a q-subset of [n] if T⊆[n] and the size of T is q.

A (n,q,r)-Steiner system S is a set of q-subsets of [n] such that every r-subset of [n] is contained in exactly 1 element of S.

Your task is to find a (n,q,r)-Steiner system for any n > q > r > 5, where r < 10 and n < 200.

Return the Steiner system as a multi-line string. In the first line, specify n, q, and r exactly as #n,q,r. Each subsequent line lists the elements of a q-subset in S separated by whitespace. For example, the below is a valid (7,3,2)-Steiner system.

#7,3,2
1 2 4
2 3 5
3 4 6
4 5 7
5 6 1
6 7 2
1 3 7