Mathematical Mysteries by Calvin C. Clawson, Plenum, US,
$27.95, ISBN 0 306 45404 1

MATHEMATICS is full of beauty and surprises, and many an author has tried to
convey this sense of wonder to the general reader. As a teenager considering a
career in mathematics, my enthusiasm was fired by Edward Kasner and James
Newman’s Mathematics and the Imagination. After a long career in higher
education, my excitement and energy are a little blunted, but reading Calvin
Clawson’s Mathematical Mysteries reminded me of the joy of discovering
the secrets of numbers.

Clawson covers a predictable group of topics: number systems, infinite
series, Fibonacci numbers, prime numbers and so on, much of which was in Kasner
and Newman’s book. But Clawson’s book differs in two important ways. First, he
discusses the history and motivation behind some great discoveries in the
evolution of mathematics. I would usually applaud such detail, but his approach
is a little too rich for my taste. For example, I found the opening chapter on
stone markings and the naming of numbers too long and subjective. When the
author says “one is tempted to go out into the backyard and toe over a few rocks
to see what is underneath”, I definitely did not agree.

The second major difference is that he attempts to explain difficult ideas to
the nonmathematician, and apparently succeeds. As a mathematician, it is hard
for me to judge what the general reader will make of these, but his introduction
to the logarithm and exponential functions, the work of Srinivasa Ramanujan and
(remarkably) his description of the Riemann hypothesis will excite a reader even
if he or she does not understand all the details.

Contrary to popular belief, mathematics is a living subject that changes
constantly, and this becomes apparent when you compare this book with those of a
decade ago. The search for new primes and their uses in codes, for example, has
been transformed by the advent of powerful computers. Clawson makes an excellent
job of bringing us up to date.

I would certainly recommend this book to anyone embarking on a life in
mathematics, or to someone with a fairly scientific mind who felt that they had
missed out on some of its mysteries. Clawson’s enthusiasm is infectious, and
with such gems on display who could fail to be dazzled?

Puzzles can range from trivial riddles which you see within seconds
(or fail to see and then groan when you look up the answers) to mind-bending
exercises requiring time and inspiration to solve. New ĐÓ°ÉÔ­´´â€™s popular
Enigmas tend to fall into this latter category, and indeed Christopher Maslanka
sets an occasional Enigma. But readers of his puzzle column in The Guardian
on Saturdays will know that he is capable of setting puzzles right across
the range, from the silly to the serious.

For the riddle-type of puzzle try these two geographical samples: I
was born in March and only have one birthday every four years – explain;
and how long can one’s birthday be made to last?

The next example is more meaty. The scientific readers of this magazine
will probably know the idea, but it was new to me when I first saw it in
The Guardian: How long must a mirror be in order for a six-foot tall person
(standing upright) to be able to see themself from top to toe? It took a
great deal of discussion and experiment for me to convice my son that a
three-foot mirror was needed. Any puzzle which creates that sort of response
is obviously a great success. But the question goes on to ask how high up
the wall the mirror must be mounted, and at one stage gives the answer of
one foot six inches. That is clearly not the case (indeed, a better answer
is given elsewhere). This was not the only occasion on which the solutions
were irritating and sometimes misleading.

The collection also includes a selection of middle-range numerical problems,
such as counting three-figure palindromes or the number of noughts at the
end of the answer to: 100 times 99 times 98 times 97 times . . . times 2
times 1. These are passably engaging for those people (like me) who find
such things entertaining, but again I found that the solutions sometimes
had ends that the means did not justify.

For example, when asked to use the digits 0 to 9 once each to make a
proper fraction (that is, less than one) as large as possible, the solution
gives the explanation that the fraction will be ABCDE/FGHIJ, where A is
one less than F, BCDE is as large as possible and GHIJ is as small as possible.
But that logic would lead to 49876/50123, and not the correct answer 69854/70123.
The correct answer was given, but it certainly was not deduced in the way
Maslanka stated.

He also includes a large selection of entertaining word puzzles, some
of which are simply rehashes of old nursery favour1Cites. For example, can
you place the same two letters in the same order before and after RI to
give a word – – RI – -? Or can you, by careful punctuation, make sense of
a piece of prose where the word ‘had’ appears 11 times consecutively?

Of course, in a book of 140 short puzzles, especially from an author
under pressure to produce a constant stream of ideas for a newspaper column,
many are going to be familiar. But surely the idea of making five equilateral
triangles by laying out nine matches on a table is as cliched as they come.
(And to use the idea twice is a bit cheeky.)

That same pressure to produce puzzles also means that some of the ideas
are a bit thin: for example, the question concerning which way to drop a
piece of paper so that it falls straight down is out of place, unsatisfactorily
answered, and is followed by an excruciatingly unfunny punch-line. Similarly,
the final puzzle of the collection, concerning an ambiguous layout of light-bulbs,
was a disappointing way to end.

In a preface to an earlier book Maslanka’s style was likened to that
of Franz Kafka, and here he is described as ‘a modern-day Lewis Carroll’.
These publisher’s attempts to raise this puzzle book to a higher plane must
surely embarrass the author. If you are willing to ignore these pretentious
claims and to accept that you may have seen a lot of these ideas before,
then there is plenty in this latest book of puzzles to divert you for several
happy hours.

Victor Bryant is in the department of pure mathematics, University of
Sheffield.

PROPOSITION 1, Victor Bryant is honest.

2. If Raymond Smullyan is a liar, then his book To Mock a Mockingbird
is worth buying.

The assistant editor of New ĐÓ°ÉÔ­´´ (knowing the above two facts)
asked me or Raymond Smullyan if this book was worth buying. From the answer
she could deduce whether it was worth buying or not. Then the editor himself
(knowing all the above facts, just like you do, but not knowing which of
the two she had asked nor what the answer was) asked whether she would have
definitely been able to draw that conclusion if she had asked the other
of the two instead. From her answer to this question, he (like you) could
work out whether this book is worth buying.

So is it worth buying? The first quarter of the book consists of puzzles
about knights (who always tell the truth) and knaves (who always lie), barbers
who shave various combinations of people who don’t shave themselves, and
other miscellaneous deluded characters. This part of the book pursues well-worn
tracks and gives variations of them almost ad nauseam. But the climax of
this first part is what the author rather grandly describes as a ‘profound
metapuzzle’ concerning the fountain of youth: the problem which I gave you
above is a variation of that puzzle and should enable you to decide whether
the book is worth buying. One thing is for certain: it will surely amuse
those who are amused by this sort of thing.

Then the fun really starts: or does it? The rest of the book is a logical
tale of a weird selection of birds and their cries: it is meant to be an
introduction to combinatory logic, for the general reader, a central theme
of the theory of computer science and artificial intelligence. But it’s
basically a piece of abstract algebra concerning making deductions from
axioms. Removing the birds’ disguises, the first exercise turns out to be
as follows: you are told these two facts about an operation * on a set (with
answers back in the set).

1. Given any w and x in the set there exists a y in the set such that
y * z = w * (x * z) for all z in the set.

2. There exists an x in the set such that x * y = y * y for all y in
the set.

Your job is to deduce from those two facts that for each x in the set
there exists a y in the set with x * y = y.

The birds are an ingenious way of dressing this up and they enable that
problem to be set in a slightly more accessible form, but they do not make
it any more interesting or its solution very much simpler.

It is almost impossible to make such an abstract piece of mathematics
appealing and understandable to a wide range of readers, but Smullyan does
make a very brave attempt. For a book to start with a trivial problem about
three flowers and to end up using Godel numbers to show that there cannot
exist a purely mechanical device that can decide the truth or otherwise
of any mathematical statement is an amazing feat. To Mock A Mockingbird
has been well received in academic circles, but perhaps it is only people
in those circles who are interested in such deductions, no matter how brilliantly
they are dressed up.

Victor Bryant is in the Department of Pure Mathematics, University of
Sheffield.