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Orthogonal latin squares

Two Latin squares of the same order such that superimposing them produces every ordered pair of symbols exactly once. Families in which every pair is orthogonal exist at every prime power order, are the same objects as finite planes, and are what Euler's officer problem asks for.

Named by 6 essays across one field — 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 The 3 mutually orthogonal squares of order 4. Every Latin square built from the field of order 4 as a·i + j, one for each non-zero multiplier, with every pair checked orthogonal.

A field's worth of squares

Two orthogonal squares of order five are easy to stumble on. Four of them, every pair orthogonal, is not a stumble — it is one line of arithmetic over a field, and the field supplies as many as the order allows.

computation · Latin squares The affine plane of order 3, one parallel class at a time. The n² cells of a complete set of orthogonal Latin squares of order 3, with the rows, the columns and each square's symbol classes drawn as lines of a plane.

The plane hiding in the squares

A complete family of orthogonal squares is not a collection of squares that happen to agree nowhere. It is a geometry — a plane with n² points in which every two points lie on exactly one line — and reading it that way is how the impossible orders were found.

computation · Latin squares 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 3 orthogonal squares of order 4, read as a code. A table of 16 words of length 5 over 4 symbols, one per cell of 3 orthogonal Latin squares of order 4: row, column and the entry in each square. Any two words agree in at most one position.

Orthogonal squares are a code

Write down each cell of a set of orthogonal Latin squares as a word — its row, its column, and its entry in each square — and no two words agree in more than one place. That is not a pleasant accident of the squares. It is exactly what being Latin and being orthogonal say, it makes the list an error-correcting code as good as any code of its size can be, and the squares a field builds turn out to be a Reed–Solomon code, the one on every compact disc.

computation · Latin squares A random Latin square of order ten with an orthogonal mate. A 10 by 10 grid with a uniformly random Latin square and an orthogonal mate written into each cell; the square has 848 transversals.

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.

computation · Latin squares

Named alongside it

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

Latin squareCounting argumentFinite fieldProjective planeExhaustive searchTransversalAffine planeAssociativityBijectionConstructionCounterexampleErasure

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