Transversal
Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
One cell short of a transversal
A transversal of a Latin square picks one cell in every row and every column with every symbol different. The cyclic squares of even order have none, and that was settled by a parity argument centuries old. Whether every square of odd order has one is a conjecture from 1967 that nobody has proved; whether every square comes within one cell of having one was settled only in 2023, and only for squares large enough.
A mate is rare until order ten
Euler asked whether a Latin square can have an orthogonal mate, and for every order but two and six the answer is yes for some square. For a square chosen at random the answer is different. Exactly 6 of the 56 reduced squares of order five have a mate and none of order six does; of squares drawn at random, about one in a hundred of order seven has one, one in three hundred of order eight, one in seventy of order nine — and three in five of order ten.
Ten digits need a symmetry that does not commute
A check digit should catch one wrong digit and two neighbours swapped. Over eleven symbols a weighted sum does both; over the ten decimal digits no weighted sum can, no scheme of any shape built on adding modulo ten can, and the reason is the same as the reason Euler's thirty-six officers cannot be paraded. What works is the ten symmetries of a pentagon, which do not commute.
The best of thirty-four thousand scramblings
Verhoeff's check digit multiplies digits in the symmetry group of a pentagon after scrambling each one a different number of times, and 34,040 scramblings make it catch every swap of neighbours. Graded on the rarer errors, none of them catches everything, the best catch 86 of 90 doubled-digit slips and 848 of 900 swaps across a digit — and the scrambling Verhoeff published in 1969 is one of the forty that are best at both.
Named alongside it
The objects these essays reach for when they reach for this one.
Latin squareExhaustive searchCounting argumentOrthogonal latin squaresCheck digitCounterexampleDihedral groupExistence proofMatchingPermutationAssociativityCommutativity