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# How to Guard an Art Gallery and Other Discrete Mathematical Adventures

T.S. Michael
THE JOHNS HOPKINS UNIVERSITY PRESS 2009, 257 PAGES
PRICE (PAPERBACK) £13.00 ISBN 978-0-801-89299-8

‘How to Guard an Art Gallery and Other Discrete Mathematical Adventures’ models solutions to a variety of problems - what is the largest number of pizza slices that we can make with n straight lines, how does a computer configure the best arrangement of pixels to represent a straight line, what is the minimum number of guards needed to guard an art gallery?

The book is intended for the general reader and counting problems are introduced that lead to a finite evaluated set. It contains seven chapters and a new question is developed in each. Michael gives clear solutions to each problem. He follows the maxim of Sir Michael Atiyah, “Any good theorem should have several proofs, the more the better”. There is a strong visual element to his explanations, which readily increases the book’s suitability to non-specialists. He takes time to describe pitfalls and shortcomings associated with discrete mathematics in determining proof. A strength of the book is in the approach taken in the development of well constructed algorithms which “hones our problem solving skills” and “focuses our attention on the essential aspects of a mathematical problem”.

A strong historical thread runs through the book in detailing the mathematicians who set or contributed to the problems under consideration: Sylvester, Pick, Chvatal etc. This aspect is worthy of study in its own right. Michael blends the historical aspect with the contemporary. He updates Jakob Steiner’s problem of n lines forming a region in a plane by adding a boundary circle to form a pizza. Essentially the same problem, but in a modern context. Michael cultivates a strong interactive element throughout the book. After deriving a solution to the pizza cutters problem in Chapter 1 he adds further value by introducing the reader to the On-Line Encyclopedia of Integer Sequences, where the sequence can be found with other contexts that would give rise to it. At the end of each chapter there are problems to try. Applets are available for the Measuring Water and How to Guard an Art Gallery sections.

Michael interestingly considers a variety of polygons to represent a gallery, including crown shaped galleries. He is dealing with a computational geometry problem, linking theory to geometric configurations to algorithm. Although he does not mention the film ‘The Thomas Crown Affair’ he directly refers to the water jug problem in ‘Die Hard’. Bruce Willis uses a fountain, two unmarked three and five gallon jugs to find four gallons of water to disarm a bomb. Michael initially derives an algorithmic solution to this problem leading to an arithmetic array. This is one of the examples in the book that would transfer well to the secondary mathematics classroom (without the bomb!). In a wider sense, discrete mathematics provides an interesting contrast to the more abstract nature of school mathematics to GCSE level, certainly as functional mathematics is included in the curriculum.

I would strongly recommend the book to anyone that has an interest in discrete mathematics or would like to develop one.

Wallace A Ferguson CMath MIMA CSci
Chatham House Grammar School
Mathematics Today December 2010

How to Guard an Art Gallery and Other Discrete Mathematical Adventures can be purchased at  Amazon.co.uk