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Latin Squares and Their Applications

Donald Keedwell and József Dénes
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Lauren Keough
, on

The game of Sudoku can be found in many newspapers and airplane magazines. It’s possible that this game is the reason that Latin Squares are a popular topic for undergraduate research. Sudoku is definitely the reason that I chose to read this text. But the research on Latin squares is far reaching, deep, and can be dated back at least as far as Euler. Latin Squares and Their Applications is an attempt at an exhaustive study of the subject. There is enough to say to fill 325 pages plus a 60-page bibliography (and the authors admit that even this bibliography is not comprehensive).

In the preface to the first edition the authors state, “We hope this book will be found intelligible to any reader whose standard of mathematical attainment is equivalent to that of a third year mathematics undergraduate of a British or Hungarian University”. (I think this line is because the authors are British and Hungarian.) I read this book with a bright junior undergraduate at a US institution. The book will certainly stretch a student, but it is for the most part accessible. The authors don’t call too many things “clear” that aren’t actually clear. The proofs of reasonable length are included and can be followed. The exposition is also very good and I was pleasantly surprised by the amount of historical context included. I would say that this book is accessible to strong American undergraduates who have had a course in abstract algebra. As a side note, since we are speaking of algebra, I was surprised by the many connections to everything from group theory to algebraic geometry.

When reading this text we wished for a glossary (so many group-related objects!) and practice problems. Of course, this book is not intended to be used as a textbook, but rather an attempt at a nearly exhaustive study of the research that has been done on the subject. And the book does achieve that. Overall, it was an excellent way to dive deep into Latin Squares research. After reading the first 2.5 chapters (which is over 100 pages of the book) we could read papers that were recently posted to the arXiv and understand the terms used and the motivation for the research. The later chapters are shorter and more specialized.

In addition to the extensive reference list, there are lists of open problems. The problems from the first edition are classified as solved, partially solved, or unsolved. Those that are partially solved include descriptions of the progress and references. In addition, there are about 50 open problems that have been added since the first edition. Many of the problems from the first edition in 1974 remain unsolved.

The book’s title may be a bit of a misnomer, in the sense that the number of included applications (in the true sense of applied mathematics) is small. There are references to some applications in experiment building for statistics (on page 73) and creating tournaments for the card game bridge (on page 225). In Chapter 11, “Miscellaneous Topics” there is a short section on Latin Squares’ application to coding theory and a more extensive discussion of creating tournaments. It is possible that the authors did not intend to mean “applications” in the sense of applied math, but rather in applications to other fields. Another book on Latin Squares, Discrete Mathematics Using Latin Squares by Laywine and Mullen. seems (at least according to the table of contents) to discuss a few more applications.

Latin Squares and Their Applications is far from the leisurely read about Sudoku that I thought it might be. What I did get is an in-depth summary of the extensive research about Latin Squares. I now have a possible new research direction that can provide many questions for undergraduates and the book will be a good resource in the future.

Lauren Keough is an Assistant Professor of Mathematics at Grand Valley State University in Allendale, Michigan. Her mathematical interests include discrete mathematics, undergraduate research, and mathematics teaching.

  • Foreword to the First Edition
  • Preface to the First Edition
  • Acknowledgements (First Edition)
  • Preface to the Second Edition
  • Chapter 1: Elementary properties
    • 1.1 The multiplication table of a quasigroup
    • 1.2 The Cayley table of a group
    • 1.3 Isotopy
    • 1.4 Conjugacy and parastrophy
    • 1.5 Transversals and complete mappings
    • 1.6 Latin subsquares and subquasigroups
  • Chapter 2: Special types of latin square
    • 2.1 Quasigroup identities and latin squares
    • 2.2 Quasigroups of some special types and the concept of generalized associativity
    • 2.3 Triple systems and quasigroups
    • 2.4 Group-based latin squares and nuclei of loops
    • 2.5 Transversals in group-based latin squares
    • 2.6 Complete latin squares
  • Chapter 3: Partial latin squares and partial transversals
    • 3.1 Latin rectangles and row latin squares
    • 3.2 Critical sets and Sudoku puzzles
    • 3.3 Fuchs’ problems
    • 3.4 Incomplete latin squares and partial quasigroups
    • 3.5 Partial transversals and generalized transversals
  • Chapter 4: Classification and enumeration of latin squares and latin rectangles
    • 4.1 The autotopism group of a quasigroup
    • 4.2 Classification of latin squares
    • 4.3 History of the classification and enumeration of latin squares
    • 4.4 Enumeration of latin rectangles
    • 4.5 Enumeration of transversals
    • 4.6 Enumeration of subsquares
  • Chapter 5: The concept of orthogonality
    • 5.1 Existence questions for incomplete sets of orthogonal latin squares
    • 5.2 Complete sets of orthogonal latin squares and projective planes
    • 5.3 Sets of MOLS of maximum and minimum size
    • 5.4 Orthogonal quasigroups, groupoids and triple systems
    • 5.5 Self-orthogonal and other parastrophic orthogonal latin squares and quasigroups
    • 5.6 Orthogonality in other structures related to latin squares
  • Chapter 6: Connections between latin squares and magic squares
    • 6.1 Diagonal (or magic) latin squares
    • 6.2 Construction of magic squares with the aid of orthogonal latin squares
    • 6.3 Additional results on magic squares
    • 6.4 Room squares: their construction and uses
  • Chapter 7: Constructions of orthogonal latin squares which involve rearrangement of rows and columns
    • 7.1 Generalized Bose construction: constructions based on abelian groups
    • 7.2 The automorphism method of H.B. Mann
    • 7.3 The construction of pairs of orthogonal latin squares of order ten
    • 7.4 The column method
    • 7.5 The diagonal method
    • 7.6 Left neofields and orthomorphisms of groups
  • Chapter 8: Connections with geometry and graph theory
    • 8.1 Quasigroups and 3-nets
    • 8.2 Orthogonal latin squares, k-nets and introduction of co-ordinates
    • 8.3 Latin squares and graphs
  • Chapter 9: Latin squares with particular properties
    • 9.1 Bachelor squares
    • 9.2 Homogeneous latin squares
    • 9.3 Diagonally cyclic latin squares and Parker squares
    • 9.4 Non-cyclic latin squares with cyclic properties
  • Chapter 10: Alternative versions of orthogonality
    • 10.1 Variants of orthogonality
    • 10.2 Power sets of latin squares
  • Chapter 11: Miscellaneous topics
    • 11.1 Orthogonal arrays and latin squares
    • 11.2 The direct product and singular direct product of quasigroups
    • 11.3 The Kézdy-Snevily conjecture
    • 11.4 Practical applications of latin squares
    • 11.5 Latin triangles
    • 11.6 Latin squares and computers
  • Comment on the Problems
    • Chapter 1
    • Chapter 2
    • Chapter 3
    • Chapter 4
    • Chapter 5
    • Chapter 6
    • Chapter 7
    • Chapter 8
    • Chapter 9
    • Chapter 10
    • Chapter 11
    • Chapter 12
    • Chapter 13
  • New Problems
    • Chapter 1
    • Chapter 2
    • Chapter 3
    • Chapter 4
    • Chapter 5
    • Chapter 6
    • Chapter 8
    • Chapter 9
    • Chapter 10
    • Chapter 11
  • Bibliography
  • Index