E    
  D  
    T

Encyclopaedia of DesignTheory: Latin squares

General Partition Incidence Array
Experimental Other designs Math properties Stat properties
Server External Bibliography Miscellanea

Latin squares as partitions

Given a set X of n2 plots, two partitions P1 and P2, each having n parts, of size n, are said to be orthogonal if every part of the first partition meets every part of the second partition in a single plot.

If two such partitions are orthogonal, then they give X the structure of a square grid, where the parts of P1 are the rows of the grid and the parts of P2 are the columns.

Now let P3 be a third partition orthogonal to both P1 and P2. Then we obtain a Latin square by associating a symbol with each part of P3, and placing in each plot the symbol of the part containing it.

This works the other way too. Given a Latin square, take the plots to be the cells of the array and the partitions to correspond to rows, columns, and symbols.

Example

In the following Latin square, the cells are numbered from 1 to 9; the colours are the symbols.

1 2 3
4 5 6
7 8 9
The three partitions are

Table of contents | Glossary | Topics | Bibliography | History

Peter J. Cameron
16 April 2002