A Primer for Mathematics Competitions by Alexander Zawaira and Gavin Hitchcock, Oxford University Press, 2009, 344 pp, £50.00 ISBN 978-0-19-953987-1 (hbk), £22.50 ISBN 978-0-19-953988-8 (pbk).
I approach the daunting but enjoyable prospect of reviewing this book as someone who is a passionate follower of mathematics, and who relishes any challenge or interesting problem that may crop up. On the other hand, I am only a year into the A-Level course. As such, I will probably not be able to give this book the constructive analysis it deserves, but I shall do my best to deliver a thorough and fair description of what I consider to be a captivating read.
The opening chapter on geometry has both a clever and imaginative blend of familiar formulae and new, exciting little rules that can be applied to an interesting variety of problems. The first half of this chapter is very fluent and easy to follow. The second half is somewhat more complex! This is an effective start to the book, providing a hook to encourage the ’must read on’ mania. The following chapter on inequalities and induction introduces interesting and extremely powerful ways of making algebraic proofs that can be applied to anything, simply from a few basic axioms and seemingly ’elementary facts’. However, I remain rather baffled by Section 2.3 on harder inequalities.
Chapter Three, on Diophantine Equations, is the most difficult chapter of the book, and yet the most gripping. It is extremely well-constructed; refreshing alongside the more ’traditional’ mathematical topics such as Geometry, Algebra and Number Theory.
Chapter Four covers Number Theory. This particular aspect of mathematics is the reason why I fell in love with the subject. It was exciting to see the wide extent of theorems here – particularly the elegant Chinese Remainder Theorem – and to sample the wide range of problems given at the end.
I have used trigonometry a great deal during my mathematical journey. I was, therefore, pleasantly surprised to find myself ignorant of many of the new trigonometric identities in the fifth chapter. At around page 199, the chapter quickly becomes more difficult, with quite a few identities to try and memorise! (I don’t think I will make it to paradise! – see page 203.) The next chapter, on sequences and series, is quite brilliant. It is well structured, has great depth and is one of the easiest chapters to understand. It is packed with a wide variety of juicy problems which one felt almost ’compelled’ to solve.
Chapter Seven, on the Binomial Theorem takes a rather difficult topic and makes it easy to interpret. My only struggle was in understanding the proof of the Binomial Theorem; however, I had no trouble applying it afterwards to the problems offered. The chapter has left me ’wanting more’! The next chapter, on combinatorics, is one of the most interesting chapters of the whole book. The depth to which the reader is plunged in this chapter is not only justified, but welcome. The way in which it is presented and employed here has given me an entirely new look on the subject helping me to solve problems which have been irking my mind for some time. The most interesting part of the chapter concerned the pigeon-hole principle which was presented as a fun and dynamic concept.
The final chapter is entitled ’Miscellaneous Problems’. Why isn’t every mathematical examination paper like this? The huge range of subjects covered in this last section did not perturb me in the slightest; on the contrary, I found it to be a nice touch after going into so much depth on a variety of subjects, although Problem 7 did catch me out!
This book is brilliant. I enjoyed its structure, and how it all seems to fit together, so that readers can make their own interpretation of it, even if the mathematics is unfamiliar to them in certain places. The cartoons are a welcome interlude from the hard-core mathematics, giving the reader a funny take on an aspect of mathematical history or culture. Speaking of mathematical culture, one of the key authors of this book is Zimbabwean, showing that mathematical education there is still strong despite the political turmoil. Finally this book will prepare Olympiad hopefuls for ’The Intermediate Challenge of Mathematical Olympiads’. The reader will certainly be ready. But, most importantly, primed!
The Embalmer’s Book of Recipes by Ann Lingard, IndePenPress, 2009, 310 pp, £8.99, ISBN 978-1-90-671017-0.
This is an intricate and absorbing novel, set mainly in Cumbria, about time, memory and prejudice. It is being mentioned here because I felt it worth drawing attention to this sympathetic and convincing portrayal of one of the main characters, Lisa Wallace, a mathematician (she is also achondroplasic). The author acknowledges guidance from mathematicians including Ian Stewart, Uwe Grimm and Ian Porteous: Lisa as mathematician rings true, as does the mathematics in the novel (which includes a conference on mathematics and art), and it is refreshing to read a novel which integrates mathematics seamlessly into its themes. The story is not always comfortable but I found it rewarding and Lisa has become one of my favourite fictional mathematicians. The author runs SciTalk – a web resource for connecting writers and scientists – and this novel, very much the author’s own but influenced by fruitful conversations with practitioners, exemplifies the value of such a resource. Background information on the novel can be found at www.annlingard.com/ebor.htm.
Venn That Tune by Andrew Viner, Hodder & Stoughton, 2008, 128 pp, ISBN 978-0-34-095567-3.
The cover for this book holds a Venn diagram containing three intersecting sets: Males, My Siblings and Things That Are Heavy, with the appropriate intersection marked. That is,
Males ∩ My Siblings ∩ (Things That Are Heavy)c
If that raised a chuckle then this book is for you.
The book is a collection of 112 charts and diagrams, featuring Venn diagrams and several other types, each of which can be interpreted as the title of a pop song. In this book ’pop song’ means a song that has featured in the UK Top 40 chart and this provides a broad mix of songs ranging from the 1950s to today. The solutions to the puzzles are, as tradition dictates, in the back of the book. As well as the answer, each solution gives the artist and date that the song was in the charts, often more than one artist and date. Several also give some explanation of the mathematics behind the diagram.
The result is a collection of light logic puzzles using gentle lateral thinking which are mostly witty and amusing. This is not a book for reading from cover to cover, although I diligently did so for the purposes of this review (I was the annoying person in the train carriage with the giggles). Rather, this is an excellent book to dip into. I have showed this book to a few friends and most have responded very positively to it. One friend, who I know to dislike ’pointless logic puzzles’ (and a physicist, to boot), responded very badly to the book, regarding it as silly and pointless, so I would not advise it for people who don’t like a bit of lateral thinking. Most people (mathematicians and non-mathematicians alike) who have seen my copy of the book responded very well.
In many cases, a song I remember from childhood was revealed to be a cover version of an older song. Featuring songs that have charted in multiple decades is a good option for increasing the range of potential audience for the book. Additionally, some of the songs either have been subsequently formed into or are drawn from a popular saying and so I found I was able to answer some of the problems without knowing the songs themselves. Nevertheless, I found there were a number of song titles for which I did not know the song, so no amount of puzzling revealed the answer. A couple of people I showed the book to exhibited an almost complete lack of awareness of popular music and so found the book very unexciting and I relate this as a warning. You do not need an intimate knowledge of music to appreciate this book but if you do not know some of the most well-known songs of the latter half of the 20th Century you may struggle to gain a full appreciation.
Physically, this is small and portable for a hardback book with 128 pages of slightly smaller dimensions than the LMS Newsletter. Well-formatted, clean diagrams give the book an attractive visual appeal.
The author Andrew Viner is a comedy, animation and children’s writer for television and radio and has a degree in electronic engineering. Mathematical consultancy for the book is credited to Dr Nick Gilbert of Heriot-Watt University. At the end of the book the author encourages the reader to draw their own diagrams and indeed people have been doing just that. Seemingly separately from this book there is a phenomenon called Song Charts which also involves charts and diagrams to illustrate songs. There is a Flickr group dedicated to this (www.flickr.com/groups/songchart) and you may have received an email circular featuring these. As for Venn That Tune, there is a website (www.vennthattune.com) and audience participation is encouraged via a Facebook page (search for Venn That Tune). The idea for the book originated on the author’s blog, which is an amusing read (smaller-than-life.blogspot.com).
The website for the book proposes two sets – People Who Like Music and People Who Like Venn Diagrams – and suggests that people who find themselves in the intersection of these sets will like this book. I suggest the second set is a little restrictive; I would suggest that if you like logic puzzles, have a sense of humour and a passing familiarity with some popular music then you should find this book amusing. I certainly enjoyed it thoroughly.