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Games and Mathematics

Book Description

The appeal of games and puzzles is timeless and universal. In this unique book, David Wells explores the fascinating connections between games and mathematics, proving that mathematics is not just about tedious calculation but imagination, insight and intuition. The first part of the book introduces games, puzzles and mathematical recreations, including knight tours on a chessboard. The second part explains how thinking about playing games can mirror the thinking of a mathematician, using scientific investigation, tactics and strategy, and sharp observation. Finally the author considers game-like features found in a wide range of human behaviours, illuminating the role of mathematics and helping to explain why it exists at all.

This thought-provoking book is perfect for anyone with a thirst for mathematics and its hidden beauty; a good high school grounding in mathematics is all the background that is required, and the puzzles and games will suit pupils from 14 years.

Table of Contents

  1. Coverpage
  2. Games and Mathematics
  3. Title page
  4. Copyright page
  5. Contents
  6. Acknowledgements
  7. Part I Mathematical recreations and abstract games
    1. Introduction
    2. Everyday puzzles
    3. 1 Recreations from Euler to Lucas
      1. Euler and the Bridges of Königsberg
      2. Euler and knight tours
      3. Lucas and mathematical recreations
      4. Lucas's game of solitaire calculation
    4. 2 Four abstract games
      1. From Dudeney's puzzle to Golomb's Game
      2. Nine Men's Morris
      3. Hex
      4. Chess
      5. Go
    5. 3 Mathematics and games: mysterious connections
      1. Games and mathematics can be analysed in the head...
      2. Can you ‘look ahead’?
      3. A novel kind of object
      4. They are abstract
      5. They are difficult
      6. Rules
      7. Hidden structures forced by the rules
      8. Argument and proof
      9. Certainty, error and truth
      10. Players make mistakes
      11. Reasoning, imagination and intuition
      12. The power of analogy
      13. Simplicity, elegance and beauty
      14. Science and games: let's go exploring
    6. 4 Why chess is not mathematics
      1. Competition
      2. Asking questions about
      3. Metamathematics and game-like mathematics
      4. Changing conceptions of problem solving
      5. Creating new concepts and new objects
      6. Increasing abstraction
      7. Finding common structures
      8. The interaction between mathematics and sciences
    7. 5 Proving versus checking
      1. The limitations of mathematical recreations
      2. Abstract games and checking solutions
      3. How do you ‘prove’ that 11 is prime?
      4. Is ‘5 is prime’ a coincidence?
      5. Proof versus checking
      6. Structure, pattern and representation
      7. Arbitrariness and un-manageability
      8. Near the boundary
  8. Part II Mathematics: game-like, scientific and perceptual
    1. Introduction
    2. 6 Game-like mathematics
      1. Introduction
      2. Tactics and strategy
      3. Sums of cubes and a hidden connection
      4. A masterpiece by Euler
    3. 7 Euclid and the rules of his geometrical game
      1. Ceva's theorem
      2. Simson's line
      3. The parabola and its geometrical properties
      4. Dandelin's spheres
    4. 8 New concepts and new objects
      1. Creating new objects
      2. Does it exist?
      3. The force of circumstance
      4. Infinity and infinite series
      5. Calculus and the idea of a tangent
      6. What is the shape of a parabola?
    5. 9 Convergent and divergent series
      1. The pioneers
      2. The harmonic series diverges
      3. Weird objects and mysterious situations
      4. A practical use for divergent series
    6. 10 Mathematics becomes game-like
      1. Euler's relation for polyhedra
      2. The invention-discovery of groups
      3. Atiyah and MacLane disagree
      4. Mathematics and geography
    7. 11 Mathematics as science
      1. Introduction
      2. Triangle geometry: the Euler line of a triangle
      3. Modern geometry of the triangle
      4. The Seven-Circle Theorem, and other New Theorems
    8. 12 Numbers and sequences
      1. The sums of squares
      2. Easy questions, easy answers
      3. The prime numbers
      4. Prime pairs
      5. The limits of conjecture
      6. A Polya conjecture and refutation
      7. The limitations of experiment
      8. Proof versus intuition
    9. 13 Computers and mathematics
      1. Hofstadter on good problems
      2. Computers and mathematical proof
      3. Computers and ‘proof’
      4. Finally: formulae and yet more formulae
    10. 14 Mathematics and the sciences
      1. Scientists abstract
      2. Mathematics anticipates science and technology
      3. The success of mathematics in science
      4. How do scientists use mathematics?
      5. Methods and technique in pure and applied mathematics
      6. Quadrature: finding the areas under curves
      7. The cycloid
      8. Science inspires mathematics
    11. 15 Minimum paths: elegant simplicity
      1. A familiar puzzle
      2. Developing Heron's theorem
      3. Extremal problems
      4. Pappus and the honeycomb
    12. 16 The foundations: perception, imagination, insight
      1. Archimedes' lemma and proof by looking
      2. Chinese proofs by dissection
      3. Napoleon's theorem
      4. The polygonal numbers
      5. Problems with partitions
      6. Invented or discovered? (Again)
    13. 17 Structure
      1. Pythagoras' theorem
      2. Euclidean coordinate geometry
      3. The average of two points
      4. The skew quadrilateral
    14. 18 Hidden structure, common structure
      1. The primes and the lucky numbers
      2. Objects hidden behind a veil
      3. Proving consistency
      4. Transforming structure, transforming perception
    15. 19 Mathematics and beauty
      1. Hardy on mathematics and chess
      2. Experience and expectations
      3. Beauty and Brilliancies in chess and mathematics
      4. Beauty, analogy and structure
      5. Beauty and individual differences in perception
      6. The general versus the specific and contingent
      7. Beauty, form and understanding
    16. 20 Origins: formality in the everyday world
      1. The psychology of play
      2. The rise and fall of formality
      3. Religious ritual, games and mathematics
      4. Formality and mathematics
      5. Hidden mathematics
      6. Style and culture, style in mathematics
      7. The spirit of system versus problem solving
      8. Visual versus verbal: geometry versus algebra
      9. Women, games and mathematics
      10. Mathematics and abstract games: an intimate connection
  9. References
  10. Index