GenerateRandomSearch

Steiner Triple System Generator

Arrange a group into threes so that every possible pair of people shares a group exactly once — not more, not less. For seven people this takes seven groups and produces the Fano plane, one of the most-drawn objects in mathematics. For eight people it is impossible, and so is ten, and so is twelve: it works only when the group size leaves remainder 1 or 3 on division by six, because otherwise the pairs simply cannot be divided evenly into threes.

What this generator does

Takes the first pair not yet covered, tries every third person who could join them, and backtracks whenever a choice makes the rest impossible — until every pair is covered exactly once.

How to use this tool

  1. Enter how many people are in the group.
  2. Search, and read the list of threes.
  3. Pick any two people and find the single group that holds them both.
  4. Try eight or ten to see why those sizes fail.

Understanding the controls

How many items
Three to fifteen. Nineteen was measured too and took over twelve seconds to settle, which is why the offer stops at fifteen.

Common use cases

  • Arranging discussion groups so everybody works with everybody once
  • Designing a round of a tournament with threes rather than pairs
  • Setting up a balanced experiment where each pair of treatments meets once
  • Explaining why some group sizes simply cannot be balanced
  • Producing the Fano plane's seven triples for teaching

How this generator works

Every triple is broken back into its three pairs and those pairs are counted across the whole system. Each of them must appear exactly once — a pair covered twice fails as surely as a pair missed. The number of triples is also checked against what the pair count demands, which catches a system that is internally consistent but incomplete.

Randomness and fairness

Nothing is random. The search runs in a fixed order, so the same size always gives the same system.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

  • Three to fifteen items, a ceiling set by how long the search takes.
  • Only sizes leaving remainder 1 or 3 on division by six have a system at all; the rest are reported as impossible.
  • One system is shown, though many exist for larger sizes.
  • Arranging the triples into rounds where everybody appears once per round — the schoolgirl problem — is a further constraint and is not applied.

Privacy and your data

The search runs in your browser and nothing about your group is transmitted or stored.

Related generators