Latin squares and experimental design

University of Puerto Rico
Río Piedras Campus
College of Natural Sciences
Department of Mathematics

Latin Squares & Experimental Designs

I. Introduction:
A Latin Square is a square (nxn)array of numbers, letters or items, in which each article appears exactly one time in each row and column, different to magic squares in which there is no repetition. There has been record of theirexistence since medieval times. In the 13th Century, Arabic Manuscripts seemed to feature the first Latin Square and their where given Kabblahlic significance. Also, a mathematician who was deeplyinvolved in the study of Latin Squares was Leonhard Euler (1707 -1783). “The name Latin Squares originates from a paper from Euler about Latin Squares titled “Investigations on a New Species of MagicSquare”, where Euler put Latin letters into grid, and called it a Latin Squared. Later on, he added Greek letters and called them Greco-Latin Square.” [1]

A special case of Latin squares is theSudoku puzzle since any solution to a Sudoku puzzle is a Latin square.  The only difference in the original definition of Latin Squares is that Sudoku imposes the additional restriction that nineparticular 3×3 adjacent sub-squares must also contain the digits 1–9.

Even though, a Latin Square is a mathematical object it has a lot of applications. Arthur Cayley continued work on Latin squares andin the 1930s the concept arose again in the guise of multiplication tables when the theory of quasi-groups and loops began to be developed as a generalization of the group concept. Latin squaresplayed an important role in the foundation of finite geometries, a subject which was also in development at this time.
Sets of Latin squares that are orthogonal to each other have found an applicationas error-correcting codes in situations where communication is disturbed by noise besides simple white noise, such as when attempting to transmit broadband internet over powerlines. Latin squares are...