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mathematics, in pictures
Concept

Transversal

A choice of one member from each of several sets, with all the choices distinct. Whether one exists is decided by comparing the size of every subfamily with the size of the union of its sets.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders.

The thirty-six officers

Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.

computation · latin squares A matching that covers all 5 of one side. A bipartite graph with every possible pairing drawn thin and one complete matching drawn thick, so that each vertex on the left is joined to a distinct vertex on the right.

One bottleneck and nothing else

A set of jobs can be filled by distinct people unless some group of jobs has too few candidates between them — and that single obstruction is the only one there is, which is what makes the theorem worth having.

discrete · halls theorem The 576 squares of order 4, sorted by whether they associate. Every Latin square of order 4, counted by whether it associates and by which group it is when it does.

Sixteen of five hundred and seventy-six

A Latin square is a multiplication table in which every equation has exactly one solution. Ask it to be associative as well and almost every square drops out — sixteen of the five hundred and seventy-six of order four survive, and they are the two groups.

computation · latin squares

Named alongside it

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

Counting argumentLatin squareExistence proofOrthogonal latin squaresAssociativityCounterexampleDeficiencyExhaustive searchFinite fieldGraphGroupGroup action

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