Information AboutLatin Square |
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| latin squares | |
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Latin squares occur as the Multiplication Table s ( Cayley Table s) of Quasigroup s. They have applications in the Design Of Experiments . The name Latin square originates from Leonhard Euler , who used Latin characters as symbols. A Latin square is said to be ''reduced'' (also, ''normalized'' or ''in standard form'') if its first row and first column are in natural order. For example, the first Latin square above is reduced because both its first row and its first column are 1,2,3 (rather than 3,1,2 or some other order). We can make any Latin square reduced by permuting (reordering) the rows and permuting the columns. ORTHOGONAL ARRAY REPRESENTATION If each entry of an ''n'' × ''n'' Latin square is written as a triple (''r'',''c'',''s''), where ''r'' is the row, ''c'' is the column, and ''s'' is the symbol, we obtain a set of ''n''2 triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the first Latin square displayed above is : { (1,1,1),(1,2,2),(1,3,3),(2,1,2),(2,2,3),(2,3,1),(3,1,3),(3,2,1),(3,3,2) }, where for example the triple (2,3,1) means that in row 2 and column 3 there is the symbol 1. The definition of a Latin square can be written in terms of orthogonal arrays as follows:
The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below. EQUIVALENCE CLASSES OF LATIN SQUARES 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. Isotopism is an Equivalence Relation , so the set of all Latin squares is divided into subsets, called ''isotopy classes'', such that two squares in the same class are isotopic and two squares in different classes are not isotopic. Another type of operation is easiest to explain using the orthogonal array representation of the Latin square. 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. Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also Parastrophe s) 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. This is again an equivalence relation, with the equivalence classes called Main Class es, ''species'', or Paratopy Classes . Each main class contains up to 6 isotopy classes. THE NUMBER OF LATIN SQUARES There is no known easily-computable formula for the number of ''n'' × ''n'' Latin squares with symbols 1,2,...,''n''. The most accurate upper and lower bounds known for large ''n'' are far apart. Here we will give all the known exact values. It can be seen that the numbers grow exceedingly quickly. For each ''n'', the number of Latin squares altogether is ''n''! (''n''-1)! times the number of reduced Latin squares . For each ''n'', each isotopy class contains up to (''n''!)3 Latin squares (the exact number varies), while each main class contains either 1, 2, 3 or 6 isotopy classes. EXAMPLES We give one example of a Latin square from each main class up to order 5. They present, respectively, the multiplication tables of the following groups:
LATIN SQUARES AND MATHEMATICAL PUZZLES The popular '' Sudoku '' puzzles are a special case of Latin squares; any solution to a ''Sudoku'' puzzle is a Latin square. ''Sudoku'' imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version). The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of "orthogonal squares" ( Diamond Theory , 1976) or "orthogonal matrices"-- orthogonal, that is, in a combinatorial, not a linear-algebra sense ( A. E. Brouwer , 1991). For a connection to Finite Geometry , see Latin-Square Geometry . SEE ALSO EXTERNAL LINKS
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