The Art and Craft of Problem Solving

24 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS [37]. These puzzles use mostly everyday ideas, but always require a good amou ...

(a) One day Martha said, "I have been alive during all or part of five decades." Rounded to the near­ est year, what is the youn ...

26 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Begin to classify: is it a "to find" or "to prove" problem? Is the problem s ...

2.2 STRATEGIES FOR GETTING STARTED 27 regarded as one of the greatest mathematicians in history (see page 67), was a big fan of ...

28 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS which yields ao + ao = ao, so ao = O. Now use the known indices 0, 1: plug m ...

2.2 STRATEGIES FOR GETTING STARTED 29 possible GCD value for pairs of consecutive terms. Neither of these items is hard to prove ...

30 CHAPTER^2 STRATEGIES FOR INVESTIGATING PROBLEMS Hence we have restated our conjecture into an unrecognizably different, yet e ...

2.2 STRATEGIES FOR GETTING STARTED 31 1 � 1 � 1 ; 1 ;^1 � 1 (�) 1 j 1!^1 ;^1 ;^1 ; 1 (��9 1 �^1!^1 ;^1 ;^1 ; I· At this point, w ...

32 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS The pattern seems simple: f(m) increases by 1 as m increases, until m is a pe ...

2.2 STRATEGIES FOR GETTING STARTED 33 In other words, g (n^2 + b) = n if and only if 0 :S b < 2n + 1. Now let us look at the ...

34 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS Example 2.2.6 (Putnam 19 91) Let A and B be different n x n matrices with rea ...

2.2 STRATEGIES FOR GETTING STARTED 35 In other words, if we let q(x) = p(x) - p"(x) , then p(x) - p' (x) - p" (x) + pili (x) = q ...

36 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS • The graph of y = q(x) stays below (or just touches) the x-axis. Then, since ...

Problems and Exercises 2.2 STRATEGIES FOR GETTING STARTED 37 For most of these, your task is just to experiment and get your han ...

38 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS with integer coordinates). Let A,B and I respec­ tively denote the area, numb ...

Can you discover a general fonnula for the num­ ber of guards, as a function of n? 2.2.27 The Josephus Problem. A group of n peo ...

40 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS more on its intended audience than anything else. For example, many professio ...

2.3 METHODS OF ARGUMENT 41 the ":=" is the new definition (usually a simple variable) and the thing on the right is an expressio ...

42 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS has no positive integer solutions. Solution: We wish to show that the equalit ...

2.3 METHODS OF ARGUMENT 43 has at least one real root. Let us assume that the conclusion is false, i.e., that Q(x) has no real r ...