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Euler Squares

Author(s): 
Elaine Young

With the recent popularization of Sudoku, interest in related mathematical games such as magic squares and Latin squares has also been revived. Sudoku puzzles are a special case of Latin squares; in fact any solution to a Sudoku puzzle is a Latin square. A Latin square is a square grid filled with symbols in such a way that each symbol occurs once and only once in each row or column. For example, a 3x3 Latin square would have nine cells in which three distinct symbols would be arranged in a way such that no symbol is repeated horizontally or vertically (see Figure 1).

 

A

B

C

B

C

A

C

A

B


Figure 1: 3x3 Latin square

 

Latin squares were known by early Arabic numerologists. These mystical squares, known as wafq majazi, were found on 13th century Islamic amulets [1] and sketched in the margins of a 16th century Arabic medical text [2]. The famous Swiss mathematician Leonhard Euler wrote about Latin squares in his paper Recherches sur une nouvelle espece de Quarres Magiques in 1782. More recently, Arthur Cayley (1821-1892), Ronald A. Fisher (1890-1962), and others have applied Latin squares in the fields of agronomy, computer science, number theory, graph theory, coding theory, and the design and statistical analysis of scientific experiments [3] [4] [5].

Elaine Young, "Euler Squares," Loci (May 2011)

Dummy View - NOT TO BE DELETED