orthogonal Latin squares
Given two Latin squares and of the same order , we can combine them coordinate-wise to form a single square, whose cells are ordered pairs of elements from and respectively. Formally, we can form a function given by
This function says that we have created a new square , whose cell contains the ordered pair of values, the first coordinate of which corresponds to the value in cell of , and the second to the value in cell of . We may write the combined square .
In general, the combined square is not a Latin square unless the original two squares are equivalent: iff . Nevertheless, the more interesting aspect of pairing up two Latin squares (of the same order) lies in the function :
Since there are cells in the combined square, and , the function is a bijection if it is either one-to-one or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.
The combined square is usually known as a Graeco-Latin square, originated from statisticians Fischer and Yates.
- 1 H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 1963
|Title||orthogonal Latin squares|
|Date of creation||2013-03-22 16:04:47|
|Last modified on||2013-03-22 16:04:47|
|Last modified by||CWoo (3771)|
|Synonym||mutually orthogonal Latin squares|
|Synonym||pairwise orthogonal Latin squares|
|Defines||complete set of Latin squares|