Games and Mathematics

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Introduction; Part I. Mathematical recreations and abstract games: 1. Recreations from Euler to Lucas; 2. Four abstract games; 3. Mathematics and games: mysterious connections; 4. Why chess is not mathematics; 5. Proving versus checking; Part II. Mathematics: game-like, scientific and perceptual: 6. Game-like mathematics; 7. Euclid and the rules of his geometrical game; 8. New concepts and new objects; 9. Convergent and divergent series; 10. Mathematics becomes game-like; 11. Maths as science; 12. Numbers and sequences; 13. Computers and mathematics; 14. Mathematics and the sciences; 15. Minimum paths from Heron to Feynmann; 16. The foundations: perception, imagination and insight; 17. Structure; 18. Hidden structure, common structure; 19. Mathematics and beauty; 20. Origins: formality in the everyday world; Bibliography; Index.

A unique book providing a tour through the fascinating connections between mathematics and games.

David Wells is the author of more than a dozen books on popular mathematics, puzzles and recreations. He has written many articles on mathematics teaching and a secondary mathematics course based on problem-solving. A former British under-21 chess champion, he has also worked as a game inventor and puzzle editor.

'One of the wellsprings out of which the discipline of mathematics
has developed is human delight in intellectual play, manifest in
the ubiquity of abstract games across millennia and cultures. The
author of this fascinating book is expert in both domains, and in
the art of clearly explaining significant aspects of mathematics in
ways both accessible to non-experts and illuminating to experts.
Through a delightfully rich variety of historical and multicultural
examples, he unveils the intimate relationship between abstract
games and mathematics as the study of structures, and, in so doing,
illuminates much more about mathematical behaviour and cognition.
At a time when too much of mathematics education in school seems
designed to squeeze out every last drop of playfulness, we are
reminded that mathematics can, and should, be an intellectual
playground.' Brian Greer, Portland State University

'This is a very approachable yet erudite book. Wells' game is to
turn Plato's theory of forms on its head: instead of starting with
physical examples and imagining their ideal forms, we should take
our cue from the abstract laws and intuitions of games. Chess
pieces are defined by their powers rather than their physical
forms, and so are many situations in mathematics ... The book
illustrates this thesis with fascinating context: from Ulam's
'lucky' numbers ... to the symmetry of the theorem discovered by
Emperor Napoleon and the mathematics which arises in Go, Hex and
chess ... Games and Mathematics makes an important advance in
communicating the nature of mathematics. It contains a profound
message for philosophers of mathematics, but all
mathematically-inclined readers will find [it] as compelling as
Wells' excellent 'Curious and Interesting' books.' Paul Brown,
author of Proof: Interesting Activities in Conjecture and
Mathematical Proof

'This is no ordinary compilation of recreational problems in
mathematics ... the text reads well and is something of a
page-turner ... this is not a mere compilation of problems, but a
guided tour, and one would hope that it would reach a wider
audience, so the authors' expressed intention, of showing that
mathematics is not merely computation, but actually foremost an
imaginative play, should become effective ... All in all this could
be a delightful volume in every aspect and I find myself
recommending the work ... with warmth and enthusiasm and with no
qualms.' Ulf Persson, Chalmers University of Technology,
Gothenburg

'Complete with a consistent argument and a wealth of supportive
references, this is a fun work for both game players and
mathematicians to explore. Highly recommended.' J. Johnson,
Choice

"One of the wellsprings out of which the discipline of mathematics
has developed is human delight in intellectual play, manifest in
the ubiquity of abstract games across millennia and cultures. The
author of this fascinating book is expert in both domains, and in
the art of clearly explaining significant aspects of mathematics in
ways both accessible to non-experts and illuminating to experts.
Through a delightfully rich variety of historical and multicultural
examples, he unveils the intimate relationship between abstract
games and mathematics as the study of structures, and, in so doing,
illuminates much more about mathematical behavior and cognition. At
a time when too much of mathematics education in school seems
designed to squeeze out every last drop of playfulness, we are
reminded that mathematics can, and should, be an intellectual
playground." Brian Greer, Portland State University

"This is a very approachable yet erudite book. Wells' game is to
turn Plato's theory of forms on its head: instead of starting with
physical examples and imagining their ideal forms, we should take
our cue from the abstract laws and intuitions of games. Chess
pieces are defined by their powers rather than their physical
forms, and so are many situations in mathematics. The book
illustrates this thesis with fascinating context: from Ulam's
"lucky" numbers which share many of the qualities of primes, to the
symmetry of the theorem discovered by Emperor Napoleon and the
mathematics which arises in Go, Hex and chess. Wells notes that
mathematicians use analogy and other play techniques as they
construct proof. He draws the reader to a new appreciation of proof
- not mere certification of correctness but a deeper exploration of
the mathematical world. Games and Mathematics makes an important
advance in communicating the nature of mathematics. It contains a
profound message for philosophers of mathematics, but all
mathematically-inclined readers will find Games and Mathematics as
compelling as Wells' excellent "Curious and Interesting" books."
Dr. Paul Brown, Carmel School, Perth, Western Australia Author of
Proof: Interesting Activities in Conjecture and Mathematical
Proof

"This is no ordinary compilation of recreational problems in
mathematics, it differs from the usual ones in several aspects. 1.
The author makes a well argued philosophical point about
mathematics and games, what they have in common and in what they
differ. 2. The problems are chosen to illustrate his philosophical
ideas and although overlaps with standard examples are inevitable
the author has gone out of his way to seek out original and
striking problems, whose interest goes beyond the merely curious or
recreational. In his selection he shows exquisite mathematical
taste, and even as a professional mathematician I found several of
his examples eye-openers. 3. In addition to his general
philosophical framework he presents insightful comments on what it
means to be a mathematician, how mathematical taste can differ and
the crucial, if elusive. notion of mathematical beauty being an
integral part of any true mathematical activity. 4. Finally his
presentation displays erudition, with many examples ranging over
cultures and historical epochs. In addition I would say that the
text reads well and is something of a page-turner. As noted this is
not a mere compilation of problems, but a guided tour, and one
would hope that it would reach a wider audience, so the authors
expressed intention, of showing that mathematics is not merely
computation, but actually foremost an imaginative play, should
become effective. The only criticism I would make is that many of
the figures the author has drawn are substandard and do not measure
up to the standards of the text. I believe that CUP will remedy
this and in general design a lay-out that the text deserves. All in
all this could be a delightful volume in every aspect and I find
myself recommending the work, if such are the expectations, with
warmth and enthusiasm and with no qualms." Ulf Persson, Chalmers
University of Technology, Gothenburg

"Complete with a consistent argument and a wealth of supportive
references, this is a fun work for both game players and
mathematicians to explore. Highly recommended." J. Johnson, Western
Washington University for Choice Magazine

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