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You would turn to your spectator and announce, or write down on a bit of paper, “I’ve got as many matches as you, then an extra 3 more. And then just enough left to make your number up to 14.” With the prediction made, it’s time for the spectator to count their matches. You then count yours out and your prediction is proven true. Say they have seven matches. From your pile, you count out seven matches and put them aside. Part one of your prediction is true: you have as many matches as them. You then count off your. | Mathematics for Computer Science Eric Lehman and Tom Leighton 2004 2 Contents 1 What is a Proof 15 1.1 Propositions. 15 1.2 Axioms. 19 1.3 Logical Deductions. 20 1.4 Examples of Proofs. 20 1.4.1 A Tautology. 21 1.4.2 A Proof by Contradiction . 22 2 Induction I 23 2.1 A Warmup Puzzle. 23 2.2 Induction. 24 2.3 Using Induction. 25 2.4 A Divisibility Theorem. 28 2.5 A Faulty Induction Proof. 30 2.6 Courtyard Tiling. 31 2.7 Another Faulty Proof. 33 3 Induction II 35 3.1 Good Proofs and Bad Proofs . 35 3.2 A Puzzle . 36 3.3 Unstacking . 40 3.3.1 Strong Induction.40 3.3.2 Analyzing the Game.41