Concept

Block design

A collection of blocks drawn from a set of points, arranged so that every pair of points lies in the same number of blocks. The balance is what makes it a design rather than a list: no pair is favoured, which is why it is the arrangement an experiment wants when every treatment must be compared with every other equally often.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as steiner triple system — the same set of essays touches all of them, so they are one junction rather than several.

A schedule on 9 points where every pair meets exactly once. Points around a circle with the triples of a Steiner system drawn between them, beside the list of triples.

A schedule where every pair meets once

Sort n people into groups of three so that every two of them share a group exactly once. Two divisions have to come out whole, that rules out most sizes — and at every size the divisions permit, a schedule exists.

computation · Finite geometry

A design on 7 points cannot have fewer than 7 blocks. The incidence matrix of a design on 7 points and 7 blocks beside the product of it with its own transpose, which has a constant off the diagonal and a determinant computed exactly.

More blocks than points

A schedule in which every pair meets once cannot use fewer groups than it has people. Nothing about the counting conditions says so, and the proof is not combinatorial at all — it is a determinant, computed over a field the schedules have nothing to do with.

computation · Finite geometry

Named alongside it

The objects these essays reach for when they reach for this one.

Counting argument Existence proof Incidence Projective plane Steiner triple system Divisibility Fano plane Rank

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