Euler Squares

Editor’s note: This article was published in March 2007.

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).

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].