Laywine C.F., Mullen G.L. Discrete Mathematics Using Latin Squares

John Wiley, 1998, -163 p.This book is designed as a textbook. Its aim is to introduce various areas and applications in discrete mathematics via the use of Latin squares. Latin squares have been studied for centuries. They have a very fascinating history and, more important, many practical applications in areas of science, engineering, and statistics as well as being used widely within mathematics itself. In recognition of the significance of the then recent disproof of a famous conjecture concerning Latin squares made by Euler in 1782, an editorial in the New York Times of April 27, 1959, stated that "it would be a serious mistake to suppose that modern mathematics is far from real life. Actually there has never been a time in history when mathematics was so widely applied in so many different fields for so many vital purposes as is true now." What was true in 1959 holds even more so now.
The examples cited in the editorial suggest that at that time important applications of mathematics arose almost exclusively from traditional continuous mathematics rather than discrete mathematics. Almost no one would support that point of view today. An enormous collection of applications of discrete mathematics has developed in the intervening 39 years. Many of these topics are related to Latin squares, and many of these are introduced in this book.
Our book could easily be used to design numerous different courses depending on the interests of the instructor and students. Except for several basic chapters like Chapters 1 and 2 which should be carefully studied by all readers, most of the remaining chapters were written independently of each other. For example, after covering most of the first two chapters, one could choose the more mathematical chapters. Alternatively, one could develop a more applied course of study even going so far as to have a fairly strong orientation to computer science and information science. As a more middle of the road approach, one could build upon various parts of both the theoretical and applied chapters with a combination of the two. The actual number of possibilities is quite large.
Among the many exercises are included some that can be made team pro- projects. In this way the book can be used in a seminar type course where students study a particular topic and then present that topic to the class. Some other exercises are intended for the more computationally oriented student, and their solutions require computer calculations. Finally we give hints and partial solutions for most or the exercises.Latin squares
A brief Introduction to Latin Squares
Mutually Orthogonal Latin Squares
Generalizations
Orthogonal hypercubes
Frequency Squares
Related mathematics
Principle of Inclusion-Exclusion
Groups and Latin Squares
Graphs and Latin Squares
Applications
Affine and Projective Planes
Orthogonal hypercubes and Affine Designs
Magic Squares
Room Squares
Statistics
Error-Correcting Codes
Cryptology
(t,m,s)-Nets
Miscellaneous Applications of Latin SquaresA: algebraic Background
B: hints and Partial Solutions to Selected Exercises