Ian Stewart
Profile Books Ltd 2014, 307 pages
PRICE (HARDBACK) £12.99 ISBN 978-1-846-68347-3
Professor Stewart’s Casebook of Mathematical Mysteries is the third in a series of popular mathematics books. It contains some 125 short articles some featuring two fictional characters: Victorian detective Hemlock Soames and Dr John Watsup who may bear some resemblance to the creations of Sir Arthur Conan Doyle. This actually works quite well as a device to make the stories readable. There is a mix of serious mathematics, anecdotes, mnemonics and jokes.
A few merit particular mention:
Prime Number Mysteries gives an interesting analysis of unsolved problems in Number Theory. There is discussion of computer verification for large numbers of specific cases including computer produced graphs, but not of computer aided proofs.
Sign of One is a series of four articles on the problem of expressing integers using only the digit 1 a fixed number of times (akin to the more familiar ‘four fours’ puzzle). Which integers can be expressed in this way depends on which symbols are allowed. By making use of the exotic ‘floor’ and ‘ceiling’ functions, familiar to spreadsheet users, along with a combination of factorials and square roots expressions, many integers can be derived. A missing square root sign in two of the expressions (pp. 116) makes the reasoning a bit tricky to follow. There is also use of the ‘double factorial’ symbol with which readers may not be familiar although this is explained on pp. 108. Eventually it is shown that every integer can be expressed using just a single 1 – but only by repeatedly using the natural log, exponential (ex) and ceiling functions.
The Wave of Translation gives an introduction to solitons (solitary waves), recounting the experience of the Scottish civil engineer John Scott Russell, who observed an instance of the phenomena on a canal in 1834 and chased it for several miles on horseback.
A Tiling That Is Not Periodic starts with considering shapes that can be used to tile the plane periodically (repeating indefinitely) or non-periodically. But questions about non-periodic tiling lead into deep areas of Mathematical Logic. In particular it has been proved that there is no general algorithm that can determine whether a given set of shapes can tile the plane (the domino problem), making it undecidable in the sense of Gödel’s Theorem.
How to Write Very Big Numbers explores the names and symbols used for expressing large numbers starting with some historical background, from Roman numerals to the different British and American ‘billion’. Eventually we get to Knuth’s arrow notation with applications in string theory and cosmology.
There is a large section The Mysteries Demystified (58 pages) of detailed answers and explanations to many of the puzzles posed. There is no index or formal bibliography, but there are in-text references including web sites and the work of other mathematicians is acknowledged throughout. Some problems are solvable with the help of simple programming tools or spreadsheets. The articles are well-illustrated in black-and-white.
This book is first and foremost entertaining, and will appeal to the mathematically-inclined reader at all levels. Professor Stewart uses mysteries and puzzles as a vehicle on a journey into deeper understanding and further enquiry.
Francis McGonigal CMath MIMA
Birmingham City University
Book review published directly onto IMA website (December 2015)
