Latin square

A Latin square is an n-by-n array of n symbols in which each symbol occurs exactly once in every row and column.

Permutation: A permutation is a rearrangement of a finite set’s elements. Each row and column can be viewed as a permutation of the symbols.
Cyclic Latin square: A cyclic Latin square is formed by repeatedly shifting the symbols in the first row by one position. This simple construction satisfies the Latin condition at every order.
Latin square design: A Latin square design is an experimental layout that controls two nuisance factors by assigning each treatment once per row and column. It turns the array’s balance into a method for separating treatment effects from two sources of variation.
Leonhard Euler: Leonhard Euler was an eighteenth-century Swiss mathematician whose work shaped number theory, analysis, and combinatorics. Euler introduced the Latin-square terminology and studied their orthogonality.
Sudoku: Sudoku is a number-placement puzzle requiring each symbol once in every row, column, and designated subgrid. Sudoku adds subgrid constraints absent from the Latin-square definition.
Transversal (Latin square): A transversal is a selection of n cells, one from each row and column, containing every symbol exactly once. Transversals reveal additional structure inside a Latin square.
Backtracking algorithm: A backtracking algorithm builds a solution incrementally and abandons partial choices that cannot succeed. It can fill a Latin square cell by cell while rejecting row or column conflicts.
Graeco-Latin square: A Graeco-Latin square is a pair of orthogonal Latin squares used together to balance two symbol factors. The superimposed symbols let designs control an additional factor.
Euler's officers problem: Euler’s officers problem asks whether 36 officers from six regiments and six ranks can be arranged in a 6-by-6 square with each rank and regiment occurring once per row and column. The problem became a famous test case for orthogonal Latin squares.
Magic square: A magic square is a square array of numbers whose rows, columns, and usually diagonals have equal sums. Its defining condition concerns sums, not the once-per-row-and-column symbol rule.

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