Beasley, John D.

The Mathematics of Games

1. Introduction 1(5) 2. The luck of the deal Counting made easy 6(3) 4-3-3-3 and all that 9(2) Shuffle the pack and deal again 11(6) 3. The luck of the die Counting again made easy 17(2) The true law of averages 19(3) How random is a toss? 22(1) Cubic and other dice 23(1) The arithmetic of dice games 24(4) Simulation by computer 28(3) 4. To err is human Finding a hole in the ground 31(5) Finding a hole in the defence 36(6) A game of glorious uncertainty 42(5) 5. If A beats B, and B beats C... The assessment of a single player in isolation 47(2) The estimation of trends 49(2) Interactive games 51(3) Grades as measures of ability 54(2) The self-fulfilling nature of grading systems 56(3) The limitations of grading 59(2) Cyclic expectations 61(3) 6. Bluff and double bluff I've got a picture 64(2) An optimal strategy for each player 66(2) Scissors, paper, stone 68(1) You cut, I'll choose 69(2) The nature of bluffing 71(1) Analysing a game 72(5) 7. The analysis of puzzles Black and white squares 77(5) Divisibility by three 82(3) Positions with limited potential 85(3) Systematic progress within a puzzle 88(2) Systematic progress between puzzles 90(9) 8. Sauce for the gander A winning strategy at nim 99(2) Nim in disguise 101(5) All cul-de-sacs lead to nim 106(2) Grundy analysis in practice 108(6) Some more balancing acts 114(4) Playing to lose 118(2) 9. The measure of a game Nim with personal counters 120(2) Games of fractional measure 122(3) General piles 125(4) The nature of a numeric game 129(2) The measure of a feeble threat 131(2) Infinite games 133(4) 10. When the counting has to stop The symptoms of a hard game 137(4) When you know who, but rot how 141(4) The paradox underlying games of pure skill 145(4) Round and round in circles Driving the old woman to bed 149(4) Turing games 153(4) Turing's paradox 157(2) The hole at the heart of mathematics 159(3) Further reading 162(5) Index 167