The Art and Craft of Problem Solving

44 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS theory problems in this and the next chapter. First let 's introduce some ext ...

Mathematical Induction 2.3 METHODS OF ARGUMENT 45 This is a very powerful method for proving assertions that are "indexed" by in ...

46 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Let S be an arbitrary (n + 1 )-gon with vertices VI, V2,"., Vn+l. Decompose S ...

2.3 METHODS OF ARGUMENT 47 natural thing to "induct on" is the number of lines we draw. In other words, we shall prove the state ...

48 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Now our stronger inductive hypothesis allows us to use the truth of both P(u) ...

2.3 METHODS OF ARGUMENT 49 R (abbreviating left and right, respectively). It may tum out that one of Lor R doesn't exist (this w ...

50 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Extracting the quantity in the large brackets, and using the inductive hypoth ...

be compared and it can be decided that one is "bigger" or one is "smaller" or they are both equal? (Using the norm function Ix + ...

52 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS previous one, as follows. For each location in the grid, examine that square, ...

2.4 OTHER IMPORTANT STRATEGIES 53 mind during any investigation. We will also discuss more advanced strategies in later chapters ...

54 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS ............................................... ....................... ·.... ...

2.4 OTHER IMPO RTANT STRATEGIES 55 is placed, will occupy exactly one black and one white square. The 31 dominos thus require 31 ...

56 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS This procedure works for all three colors. One color must be used at l n/3 J ...

2.4 OTHER IMPORTANT STRATEGIES 57 Now we introduce a simple trig tool: Given an expression of the form acos 0 + bsin O, write ac ...

58 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS the planets are not touching each other.) It is easy to check that if n = 2, ...

2.4 OTHER IMPORTANT STRATEGIES 59 bystander tells her that her hat fell into the river as she originally dived. The swimmer cont ...

60 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Prove that if T is a triangular number, then 8T + 1 is a perfect square. You ...

Chapter 3 Tactics for Solving Problems Now we tum to the tactical level of problem solving. Recall that tactics are broadly appl ...

62 CHAPTER 3 TACTICS FOR SOLVING PROBLEMS using) order can quickly simplify such problems. Consequently, we will begin by studyi ...

3.1 SYMMETRY 63 Why is symmetry important? Because it gives you "free" information. If you know that something is, say, symmetri ...