CHAPTER I MATHEMATICAL GAMES 


The fascination of ordinary numbers 


Why fifteen Fellows of the Royal Society? 


The scale of ten 


A problem involving ordinary numbers 


A very long division 


A much shorter solution of the digital problem 


"Sixteen months in the year, and their names" 


"The binary scale, or scale of two" 


A magic table of numbers 


The game of Nim 


As played by an electronic brain against humans 


The theory behind the game 


Winning positions in the game 


Punched cards and automatic rearrangement of twelve cards 


The twelvecoin problem 


Can it be done without the use of mathematics? 


"The ternary scale, or scale of three" 


A solution of the twelvecoin problem 


Weighing up to forty pounds with only four weights 


There is an infinity of prime numbers 


The square root of two is not a rational number 

CHAPTER II CHANCE AND CHOICE 


A coin is spun 


Dr. Joad and the law of averages 


Historical background to theory of probability 


What is random behaviour? 


Scattering seed at random 


Urns and dice 


Addition law of probabilities 


Multiplication law 


Errors of mathematicians 


Eliza Doolittle 


Odds that a head turns up in tossing a penny 


A problem of Samuel Pepys 


Two letters from Isaac Newton to Samuel Pepys 


Expectation of a prize in a football pool 


Expectation of eternal bliss 


The St. Petersburg problem 


Moral criticism of mathematical results 


"Buffon's test, using child labour" 


The courageous Bertrand 


Anything which can happen will happen 


Buffon's needle theorem and the evaluation of p 


The giddy Lazzerini 


Extrasensory perception and psychokinesis 


Why does heads turn up when you pray for tails? 

CHAPTER III WHERE DOES IT END? 


Is infinity greater than infinity? 


Can you count? 


Definition of an infinite class 


Countable infinities 


The positive rationals can be counted 


The decimals greater than zero and less than one cannot be counted 


A great unsolved problem of mathematics 


The terrible Cantor 

CHAPTER IV AUTOMATIC THINKING 


Classes 


One class contained in another 


Syllogisms 


Socrates was mortal 


Universal class and null class 


Some laws are unsatisfactory 


Writers and Shakespeare 


Another Lewis Carroll teaser 


Algebra of classes and propositions 


"Alice, Brenda, Cissie and Doreen" 


Who won the scholarship? 

CHAPTER V TWOWAY STRETCH 


Ballon d'essai 


Rubbersheet geometry 


Topological transformation defined 


Deformations 


The escapeartist's trick 


Supplying three houses with main services 


Is topology worth while? 


Multiplyconnected figures 


Sphere and torus 


The Moebius band 


"Fun with paper, gum and scissors" 


Rotating ring of tetrahedra 


Modern art and the Klein bottle 


Simple polyhedra and Euler's formula 


The fourcolour theorem 


Can you prove it? 


Disdainful doggerel 

CHAPTER VI RULES OF PLAY 


Laws of addition 


A double negative gives a positive 


Additive groups 


"What every airman knows, or how to add vectors" 


Rotation is addition 


Finite groups 


How to multiply 


Rings (not of commercial firms) 


The Pascal triangle 


The binomial theorem 


"Perms. and combs., or how to arrange and select" 


No help with footballpools 


How to divide 


Why exclude division by zero? 


The group postulates 


Do you put your shirt on before your tie? 


A plane slides over itself 


Symmetry investigated 


Inkblots rationalised 


Rotational symmetry 


Ornaments 


Pointlattices and curtain materials 


The symmetries in Arabic art 

CHAPTER VII AN ACCOUNTANT'S NIGHTMARE 


The gullible Emperor 


A fable of a slowly but surely divergent series 


A wellbehaved series 


Can you rub out this line? 


Decimals which come to an end 


Those which do not 


What kind of decimals arise from rational numbers 


The uniqueness of infinite decimals 


Irrational numbers 


The number p 


Shanks and p 


A mystic rhyme for p 


Why should seven suffer? 


"Sir your superior mathematics" 


Trouble with series 


Pinning them down 


More fuss and bother 


Safety first 


Achilles and the tortoise 


Is he still running? 

CHAPTER VIII DOUBLE TALK 


Mathematicians not logical 


The uncertainty of logic 


Paradoxes galore 


Class of all classes paradox 


A humble mathematician 


Mathematics not logic 


Infinite collections of shoes 


Of socks 


Can you choose? 


Intuitionism 


Law of the excluded middle 


Right or wrong? 


Formalist view 


No neurosis amongst mathematicians 

CHAPTER IX WHAT IS MATHEMATICS? 


What mathematicians do 


International conferences 


Mathematicians as human beings 


What mathematics is not 


Poincaré to the rescue 


