Latin squares — the series

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.

part 1 · computation

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.

part 2 · computation

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.

part 3 · computation

Nine thousand four hundred and eight

There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.

part 4 · computation

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.

part 5 · computation

The cyclic square of order 6: no transversal, and one of 5 cells. A Latin square of order 6 with a partial transversal of 5 cells shaded — one cell in each row and column but one, each with a different symbol. No full transversal exists.

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.

part 6 · computation

Latin squares of order 4, and the ones that are also Sudoku grids. Two 4×4 Latin squares with their 2×2 boxes outlined: one in which every box also holds 1 to 4, and the cyclic square, whose top-left box repeats a symbol. Of all 576 Latin squares of order 4, 288 pass the box test.

A Latin square with boxes

A finished Sudoku is a Latin square of order nine with one extra rule: each 3×3 box holds every digit once. At order four the extra rule keeps exactly half of the 576 Latin squares, the 288 survivors are two grids in disguise, and no puzzle can be pinned down by fewer than four clues. At order nine every one of those questions needed a computer, and the answers are 6.67 × 10²¹ grids, 5.47 billion essentially different ones, and seventeen clues.

part 7 · computation

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.

part 8 · computation