Games and Mathematics by Wells, David -- Read -- Imperial Library of Trantor

Index

Acknowledgements Part i Mathematical recreations and abstract games Introduction Everyday puzzles 1 Recreations from Euler to Lucas

Euler and the Bridges of Königsberg Euler and knight tours Lucas and mathematical recreations Lucas's game of solitaire calculation

2 Four abstract games

From Dudeney's puzzle to Golomb's Game Nine Men's Morris Hex Chess Go

3 Mathematics and games: mysterious connections

Games and mathematics can be analysed in the head… Can you ‘look ahead’? A novel kind of object They are abstract They are difficult Rules Hidden structures forced by the rules Argument and proof Certainty, error and truth Players make mistakes Reasoning, imagination and intuition The power of analogy Simplicity, elegance and beauty Science and games: let's go exploring

4 Why chess is not mathematics

Competition Asking questions about Metamathematics and game-like mathematics Changing conceptions of problem solving Creating new concepts and new objects Increasing abstraction Finding common structures The interaction between mathematics and sciences

5 Proving versus checking

The limitations of mathematical recreations Abstract games and checking solutions How do you ‘prove’ that 11 is prime? Is ‘5 is prime’ a coincidence? Proof versus checking Structure, pattern and representation Arbitrariness and un-manageability Near the boundary

Part ii Mathematics: game-like, scientific and perceptual Introduction 6 Game-like mathematics

Introduction Tactics and strategy Sums of cubes and a hidden connection A masterpiece by Euler

7 Euclid and the rules of his geometrical game

Ceva's theorem Simson's line The parabola and its geometrical properties Dandelin's spheres

8 New concepts and new objects

Creating new objects Does it exist? The force of circumstance Infinity and infinite series Calculus and the idea of a tangent What is the shape of a parabola?

9 Convergent and divergent series

The pioneers The harmonic series diverges Weird objects and mysterious situations A practical use for divergent series

10 Mathematics becomes game-like

Euler's relation for polyhedra The invention-discovery of groups Atiyah and MacLane disagree Mathematics and geography

11 Mathematics as science

Introduction Triangle geometry: the Euler line of a triangle Modern geometry of the triangle The Seven-Circle Theorem, and other New Theorems

12 Numbers and sequences

The sums of squares Easy questions, easy answers The prime numbers Prime pairs The limits of conjecture A Polya conjecture and refutation The limitations of experiment Proof versus intuition

13 Computers and mathematics

Hofstadter on good problems Computers and mathematical proof Computers and ‘proof’ Finally: formulae and yet more formulae

14 Mathematics and the sciences

Scientists abstract Mathematics anticipates science and technology The success of mathematics in science How do scientists use mathematics? Methods and technique in pure and applied mathematics Quadrature: finding the areas under curves The cycloid Science inspires mathematics

15 Minimum paths: elegant simplicity

A familiar puzzle Developing Heron's theorem Extremal problems Pappus and the honeycomb

16 The foundations: perception, imagination, insight

Archimedes' lemma and proof by looking Chinese proofs by dissection Napoleon's theorem The polygonal numbers Problems with partitions Invented or discovered? (Again)

17 Structure

Pythagoras' theorem Euclidean coordinate geometry The average of two points The skew quadrilateral

18 Hidden structure, common structure

The primes and the lucky numbers Objects hidden behind a veil Proving consistency Transforming structure, transforming perception

19 Mathematics and beauty

Hardy on mathematics and chess Experience and expectations Beauty and Brilliancies in chess and mathematics Beauty, analogy and structure Beauty and individual differences in perception The general versus the specific and contingent Beauty, form and understanding

20 Origins: formality in the everyday world

The psychology of play The rise and fall of formality Religious ritual, games and mathematics Formality and mathematics Hidden mathematics Style and culture, style in mathematics The spirit of system versus problem solving Visual versus verbal: geometry versus algebra Women, games and mathematics Mathematics and abstract games: an intimate connection

References Index