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Displaying a 7 × 7 Latin square, this stained glass window honors Ronald Fisher, whose Design of Experiments discussed Latin squares. An example of a 3x3 Latin square is: A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order.
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column.
If we systematically and consistently reorder the three items in each triple, another orthogonal array (and, thus, another Latin square) is obtained.
For example, we can replace each triple (r,c,s) by (c,r,s) which corresponds to transposing the square (reflecting about its main diagonal), or we could replace each triple (r,c,s) by (c,s,r), which is a more complicated operation.
The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.
The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square.
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Because of the noise, we can no longer tell if the first two slots were 1,1 or 1,2 or 2,1 or 2,2.Many operations on a Latin square produce another Latin square (for example, turning it upside down).If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first.Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also parastrophes) of the original square.Finally, we can combine these two equivalence operations: two Latin squares are said to be paratopic, also main class isotopic, if one of them is isotopic to a conjugate of the other.