# Euler Squares - Euler Squares

Author(s):
Elaine Young

Euler squares are created when two orthogonal Latin squares are overlaid to include two attributes in each cell of the array. Orthogonal Latin squares ensure that each and every possible pairing of the two attributes appears exactly once in the array. Now known as Graeco-Roman squares or Euler squares, these arrays are classified by their “order”, or the number of items along one side of the square array. For example, a 3x3 Euler square has order three.

In 1694, Jacques Ozanam [6] posed the problem of arranging 16 playing cards in a 4 x 4 array such that no row or column contained more than one card of each suit and each rank (Figure 2).  The solution forms an Euler square of order four with the attributes ofcard rank and suit.  There are 144 possible solutions, not including symmetrical transformations [7].

Figure 2:  One solution to the playing card problem

Elaine Young, "Euler Squares - Euler Squares," Convergence (May 2011)