Do you love combinatorial designs? If so, this is a book for you! This volume is an updated and greatly expanded version of the 1996 first edition. With more pages than Euler had publications and more than 211 references, it is just what you expect in a handbook — a comprehensive guide to Everything You Always Wanted To Know About Combinatorial Designs, but didn’t know where to look. The editors have done this by assembling 109 chapters within seven main sections, written individually and collectively by 110 contributors, all experts in the field. The contributors clearly took considerable care in their work: for the most part, this vast compendium is well written. The book has lists, tables, examples, diagrams, theorems, constructions, and even three pictures. (You guess who.)
Section 1 begins, most appropriately with the Fano Plane, also known as the (7,3,1) design. For, this beautiful and deceptively simple combinatorial object illustrates so many different ideas from the world of combinatorial designs: symmetry, balance, sets, arrays, graphs, finite geometries, Latin Squares, matrices, error-correcting codes, number theory, finite fields, groups, and even an optimal strategy for the Seven Hats Problem. They proceed immediately to one of the highlights, a section on the history of designs from ancient times up to 1950. Here is where you learn just how pivotal Thomas Kirkman was to the field, and how R. C. Bose’s 1939 paper found the field a collection of results and left it an important mathematical discipline.
Next, there are four sections devoted to four major areas of study. Section 2 treats block designs, and their generalizations, t-designs. A t-(v,k,λ) design is a v-set V together with a set B of k-subsets of V (the blocks) such that every t-subset of V is contained in exactly λ blocks. A block design is a 2-(v,k,λ) design. Section 2 also treats triple systems, Steiner systems, symmetric designs, and resolvable designs, including the famous Kirkman Schoolgirls Problem. Section 3 deals with the construction, enumeration and existence of Latin squares, mutually orthogonal Latin squares (MOLS) and orthogonal arrays, including an account of the history and eventual disproof of Euler’s Conjecture on MOLS.
Section 4 is concerned with partially balanced designs (PBD) and group divisible designs (GDD). A running example through this section is the famous 3x3 magic square — yes, that one, and you know how to construct it. Its twelve rows, columns and extended diagonals form a PBD, a GDD, a 2-(9,3,1) design, a resolvable design, and an affine plane of order three. In Section 5, we meet Hadamard matrices, Hadamard designs, and various generalizations of these. (The incidence matrix of the (7,3,1) design gives rise to a Hadamard matrix of order 8.)
Section 6, the longest section, consists of 65 short chapters on other types of combinatorial designs from Association Schemes to Youden Designs. There are chapters on magic squares, difference sets, Skolem and Langford sequences, Whist tournaments, Room squares, and designs intriguingly called SAMDRR (Spouse-Avoiding Mixed Doubles Round Robin) tournaments. Finally, Section 7 contains a wealth of information about other mathematical areas and their relation to combinatorial designs. These include such areas as number theory, graph theory, linear algebra, codes, finite groups, finite fields, and matroids.
The book contains some fascinating tidbits along the way. For example, the section on scheduling a tournament contains examples from Major League Baseball, the NFL, World Cup Soccer, the Czech National Hockey League, the World Motorcycle Speedway Championships, and jai-alai. Chapters 52 and 58 of Section 6 contain information on authentication codes, secrecy codes and secret-sharing schemes. Finally, the chapter on Latin Squares has a brief section on that most tantalizing and addictive game, Sudoku.
In their introduction, the editors state that the handbook is not designed to be used sequentially, and they are absolutely right. This vast book should not be read like a novel. Researchers in the field will find a chapter on their particular area of interest. If you are new to the area, read the introduction and the historical background first, then dive in. Order a copy for your school’s library, or (if you are a combinatorialist) for yourself. But be careful: combinatorial designs can be addictive!
Ezra Brown (firstname.lastname@example.org) is Alumni Distinguished Professor of Mathematics at Virginia Tech, with degrees from Rice and LSU. He is a number theorist by trade, but his first publication was about tournaments and Hadamard matrices. He is a fairly regular contributor to the MAA journals. He sings (everything from blues to opera), plays a tolerable jazz piano, and his wife Jo is teaching him to be a gardener. He occasionally bakes biscuits for his students.