| CHAPTER I MATHEMATICAL GAMES |
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The fascination of ordinary numbers |
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Why fifteen Fellows of the Royal Society? |
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The scale of ten |
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A problem involving ordinary numbers |
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A very long division |
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A much shorter solution of the digital problem |
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"Sixteen months in the year, and their names" |
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"The binary scale, or scale of two" |
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A magic table of numbers |
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The game of Nim |
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As played by an electronic brain against humans |
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The theory behind the game |
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Winning positions in the game |
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Punched cards and automatic rearrangement of twelve cards |
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The twelve-coin problem |
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Can it be done without the use of mathematics? |
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"The ternary scale, or scale of three" |
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A solution of the twelve-coin problem |
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Weighing up to forty pounds with only four weights |
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There is an infinity of prime numbers |
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The square root of two is not a rational number |
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CHAPTER II CHANCE AND CHOICE |
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A coin is spun |
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Dr. Joad and the law of averages |
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Historical background to theory of probability |
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What is random behaviour? |
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Scattering seed at random |
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Urns and dice |
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Addition law of probabilities |
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Multiplication law |
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Errors of mathematicians |
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Eliza Doolittle |
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Odds that a head turns up in tossing a penny |
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A problem of Samuel Pepys |
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Two letters from Isaac Newton to Samuel Pepys |
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Expectation of a prize in a football pool |
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Expectation of eternal bliss |
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The St. Petersburg problem |
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Moral criticism of mathematical results |
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"Buffon's test, using child labour" |
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The courageous Bertrand |
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Anything which can happen will happen |
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Buffon's needle theorem and the evaluation of p |
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The giddy Lazzerini |
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Extra-sensory perception and psycho-kinesis |
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Why does heads turn up when you pray for tails? |
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CHAPTER III WHERE DOES IT END? |
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Is infinity greater than infinity? |
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Can you count? |
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Definition of an infinite class |
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Countable infinities |
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The positive rationals can be counted |
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The decimals greater than zero and less than one cannot be counted |
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A great unsolved problem of mathematics |
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The terrible Cantor |
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CHAPTER IV AUTOMATIC THINKING |
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Classes |
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One class contained in another |
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Syllogisms |
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Socrates was mortal |
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Universal class and null class |
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Some laws are unsatisfactory |
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Writers and Shakespeare |
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Another Lewis Carroll teaser |
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Algebra of classes and propositions |
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"Alice, Brenda, Cissie and Doreen" |
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Who won the scholarship? |
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CHAPTER V TWO-WAY STRETCH |
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Ballon d'essai |
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Rubber-sheet geometry |
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Topological transformation defined |
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Deformations |
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The escape-artist's trick |
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Supplying three houses with main services |
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Is topology worth while? |
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Multiply-connected figures |
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Sphere and torus |
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The Moebius band |
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"Fun with paper, gum and scissors" |
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Rotating ring of tetrahedra |
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Modern art and the Klein bottle |
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Simple polyhedra and Euler's formula |
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The four-colour theorem |
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Can you prove it? |
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Disdainful doggerel |
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CHAPTER VI RULES OF PLAY |
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Laws of addition |
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A double negative gives a positive |
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Additive groups |
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"What every airman knows, or how to add vectors" |
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Rotation is addition |
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Finite groups |
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How to multiply |
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Rings (not of commercial firms) |
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The Pascal triangle |
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The binomial theorem |
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"Perms. and combs., or how to arrange and select" |
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No help with football-pools |
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How to divide |
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Why exclude division by zero? |
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The group postulates |
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Do you put your shirt on before your tie? |
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A plane slides over itself |
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Symmetry investigated |
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Inkblots rationalised |
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Rotational symmetry |
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Ornaments |
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Point-lattices and curtain materials |
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The symmetries in Arabic art |
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CHAPTER VII AN ACCOUNTANT'S NIGHTMARE |
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The gullible Emperor |
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A fable of a slowly but surely divergent series |
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A well-behaved series |
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Can you rub out this line? |
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Decimals which come to an end |
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Those which do not |
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What kind of decimals arise from rational numbers |
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The uniqueness of infinite decimals |
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Irrational numbers |
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The number p |
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Shanks and p |
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A mystic rhyme for p |
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Why should seven suffer? |
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"Sir your superior mathematics" |
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Trouble with series |
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Pinning them down |
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More fuss and bother |
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Safety first |
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Achilles and the tortoise |
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Is he still running? |
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CHAPTER VIII DOUBLE TALK |
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Mathematicians not logical |
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The uncertainty of logic |
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Paradoxes galore |
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Class of all classes paradox |
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A humble mathematician |
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Mathematics not logic |
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Infinite collections of shoes |
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Of socks |
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Can you choose? |
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Intuitionism |
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Law of the excluded middle |
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Right or wrong? |
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Formalist view |
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No neurosis amongst mathematicians |
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CHAPTER IX WHAT IS MATHEMATICS? |
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What mathematicians do |
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International conferences |
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Mathematicians as human beings |
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What mathematics is not |
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Poincaré to the rescue |
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