Latin Squares and their Applications

Latin Squares and their Applications

New Developments in the Theory and Applications

2nd Edition - July 24, 2015

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  • Authors: A. Donald Keedwell, József Dénes
  • Hardcover ISBN: 9780444635556
  • eBook ISBN: 9780444635587

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Latin Squares and Their Applications, Second edition offers a long-awaited update and reissue of this seminal account of the subject. The revision retains foundational, original material from the frequently-cited 1974 volume but is completely updated throughout. As with the earlier version, the author hopes to take the reader ‘from the beginnings of the subject to the frontiers of research’. By omitting a few topics which are no longer of current interest, the book expands upon active and emerging areas. Also, the present state of knowledge regarding the 73 then-unsolved problems given at the end of the first edition is discussed and commented upon. In addition, a number of new unsolved problems are proposed. Using an engaging narrative style, this book provides thorough coverage of most parts of the subject, one of the oldest of all discrete mathematical structures and still one of the most relevant. However, in consequence of the huge expansion of the subject in the past 40 years, some topics have had to be omitted in order to keep the book of a reasonable length. Latin squares, or sets of mutually orthogonal latin squares (MOLS), encode the incidence structure of finite geometries; they prescribe the order in which to apply the different treatments in designing an experiment in order to permit effective statistical analysis of the results; they produce optimal density error-correcting codes; they encapsulate the structure of finite groups and of more general algebraic objects known as quasigroups. As regards more recreational aspects of the subject, latin squares provide the most effective and efficient designs for many kinds of games tournaments and they are the templates for Sudoku puzzles. Also, they provide a number of ways of constructing magic squares, both simple magic squares and also ones with additional properties.

Key Features

  • Retains the organization and updated foundational material from the original edition
  • Explores current and emerging research topics
  • Includes the original 73 ‘Unsolved Problems’ with the current state of knowledge regarding them, as well as new Unsolved Problems for further study


Researchers in combinatorial design theory and areas of statistics such as design and analysis of experiments. The book may also be of interest to amateur mathematicians interested in magic squares, in designing games tournaments and/or in latin squares related to Sudoku puzzles

Table of Contents

  • 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

Product details

  • No. of pages: 438
  • Language: English
  • Copyright: © North Holland 2015
  • Published: July 24, 2015
  • Imprint: North Holland
  • Hardcover ISBN: 9780444635556
  • eBook ISBN: 9780444635587

About the Authors

A. Donald Keedwell

He has spent almost the whole of his teaching career at the University of Surrey. Since retirement from teaching he has been an Honorary Senior Research Fellow at that Institution. He is the author of some ninety research papers and two books, most related in some way to latin squares. He is a Foundation Fellow of the Institute of Combinatorics and their Applications and a member of the Editorial Board of \Quasigroups and Related Systems". He was the rst secretary of the British Combinatorial Committee and continued to serve in that capacity for 8 years. Also, he was one of the organizers of the 1991 British Combinatorial Conference and editor of its Proceedings. He is probably best known internationally as one of the authors of \Latin Squares and their Applications" of which the present book is a re-written and updated new edition.

Affiliations and Expertise

Department of Mathematical and Computing Science, University of Surrey, Guildford, Surrey, UK

József Dénes

Affiliations and Expertise

Industrial and Scientific Consultant, Formerly Head of Mathematics, Institute for Research and Co-ordination of Computing Techniques (SZKI), Budapest, Hungary

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