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Skills are to mathematics what scales are to music or spelling is
to writing. The objective of learning is to write, to
play music, or to solve problems --- not just to master skills.
--- Everybody Counts
For students of mathematics, science, and engineering, the name of the game is problem solving. Whether it's proving a theorem, writing a computer program, designing a statistical experiment, or solving for the stresses in a bridge, the essential challenge is problem solving.
Mathematical problem solving has been the subject of countless books, magazines, anthologies, and more recently, web sites. It has been promoted and enjoyed by diverse audiences that include students, teachers, and recreational aficionados. Given the immense scope of problem solving, it's clear that one book cannot possible do the subject justice.
To explain why this book may be different than many others, let's attempt a brief taxonomy of mathematical problem solving. Greatly simplifying the matter, we might identify the following categories of problems that mathematics students typically encounter. The list has no special order, and the categories certainly overlap.
1. Recreational problems are concise intellectual challenges often
associated with puzzle connoisseurs such as Sam Loyd and Martin Gardner.
These problems may or may not be mathematical in nature, but they generally
require keen critical thinking and ingenious strategies.
Books of many flavors and difficulties are devoted to recreational problems,
and any serious problem solver should be familiar with them. If nothing
else, they provide excellent mind calisthenics and occasionally come in
handy as party tricks.
2. Contest problems are precisely formulated mathematical problems that often appear in formal exams and competitions such as the American High School Math Exam (AHSME), the USA Mathematical Olympiad (USAMO), the International Mathematical Olympiad (IMO), and the \index{Putnam Exam}Putnam Exam. Such contest problems generally requires a fair amount of mathematical background and sophistication.
3. Logic problems are generally qualitative in nature, and often take the form of a story. Their solution requires organized thinking and often formal logic. Collections of logic problems abound; in fact, monthly magazines of logic problems can be found in supermarkets. These problems provide excellent thinking exercises and are often used as training for standardized exams such as the GRE, LSAT, and MCAT.
4. Modeling or story problems are quantitative problems that are posed in a realistic context. A key distinction of these problems is that they are not posed explicitly as mathematical problems. For this reason, their solution requires an essential preliminary step that may be the crux of the solution. That step, often called modeling, is to transform the stated problem from words to mathematics. Having formulated the problem in mathematical terms, it must still be solved! The only formal exam that emphasizes such problems is the Mathematical Contest in Modeling.
5. The previous four categories require analytical techniques --- traditional
methods carried out with pencil and paper (and brain). Twenty-five years
ago, the list would have ended here. However, we must now acknowledge that
there is another tool in the problem-solving arsenal; it is the computer.
With its numerical, graphical, and symbolic capabilities, the computer is
the laboratory of mathematics; it is a tool of exploration and discovery.
Without diminishing the role of analytical methods, it is fair to say that
fluency with a programming language (for example, C++ or Java) or a mathematical
environment (for example, MATLAB, Maple, Derive, or Mathematica) is an essential
skill for all mathematics students. This observation leads to another problem
category: Computational problems are those problems for which the
powers of a computer are used for insight, exploration, or solution.
Even if oversimpified, these problem categories certainly demonstrate
that problem solving is a diverse and complex enterprise. In choosing the
focus of this book, choices had to be made. Because problems posed in a
realistic context are common and important, and because solving such problems
is a valuable skill, this book highlights modeling or story problems.
This is not
to say that the other categories are neglected, but the emphasis of the
book is decidedly on the following two-step process:
If you have gotten this far, you have read about 1% of this book! Before going any further you should know a few honest facts. The first is that this book (or any problem-solving book) will not give you a universal formula for problem solving; it just doesn't exist. However, this book (and many others) can help you become a better problem solver.
For everyone, problem solving power comes with practice. Through practice, you see similar problems and patterns reappear. Through practice, you master techniques and variations on those techniques. And, perhaps most important of all, through practice, you gain confidence.
It's easy to say that practice leads to mastery, but practice is not always easy. A pianist cannot practice enough to improve unless she finds some enjoyment in practice. Analogously, to improve at problem solving, you really must find some enjoyment in mathematics and problem solving. And just as a competitive cyclist cannot seriously train unless he has a desire to win, proficiency in problem solving will be hard-earned without the desire to be a better problem solver. Hopefully, this book can provide enjoyment and instill desire.
A word about answers and solutions is in order. A lone answer, which is typically a numerical result, such as 8.23 miles, is never acceptable. It must always be accompanied by a complete solution, which is a full account of how you arrived at the answer. One goal of the book is to improve mathematical communication, both written and verbal. With mathematical proofs, the standards of exposition are fairly clear. With the more open-ended story problems that appear in this book, the rules may not be so evident. But here are some guidelines.
Solutions must be compelling and convincing to others who have never seen the problem, but have the mathematical background needed to understand the solution. Graphs, tables, and figures can often make a solution much more digestible. You should write a solution so that if you were to read it five years from now, you could make sense of your own writing.
Throughout the book, you are encouraged to focus on the process of problem solving as much as the final answer. It often helps to step back and watch yourself in the act of problem solving. Are certain environments more conducive to successful problem solving? For example, do you do better on a bus, in the shower, surrounded by silence, or inside a set of vibrating headphones? Are certain times of day more effective? Are you more successful after a big pasta dinner or on an empty stomach? What strategies worked? Were there breakthrough moments when you experienced a key Aha! or Eureka! moment? Does working in groups help or hinder your problem solving? Try to follow your efforts, both the victories and the frustrations. You will learn a lot about problem solving by watching yourself in the act.
Having said that the end result isn't everything, it is always rewarding to know when you devise a correct solution. In this book, you will find hints and answers for most problems at the end of each chapter. Solutions to selected odd problems (marked with a diamond) appear at the back of the book. Remember that very little is gained by reading the solution to a problem before seriously attempting to solve it.
Finally, mathematical problems are like folklore: The origins of many problems are lost in the shadows of time. Problems fall into obscurity and are rediscovered; story lines change as problems are passed on. As an inveterate collector, I have regrettably forgotten where I first saw or heard many problems; and even if I could remember the source, it may not be the primary source. I have done my best to give credit for problems used in this book by citing my primary source. I apologize (and would like to be informed) if I have failed to recognize the creator of a good problem.
Acknowledgements: I am grateful to four years of problem solving students at the University of Colorado at Denver who patiently worked through early drafts of this book. Thanks go to Mike Kawai of the Mathematics Department at CU-Denver for doing an accuracy check of many of the solutions. John Rogosich of Techsetters, Inc. provided valuable advice with the LaTeX preparation of the manuscript; I am grateful for his assistance. Thanks also go to Linda Thiel, Donna Witzleben, Simon Dickey, and Andrea Missias who offered editorial guidance that made the book much better than it would have been otherwise. Finally, this book is dedicated to my father, who solved many problems in his lifetime.
Exercise 9.17. The statement of the problem should make it clear that the ice cream sphere is tangent to the cone and to the extension of the cone. More seriously, the solution is dead wrong! It does not suffice to consider a two dmensional cross-section of the problem. The problem must be solved in three dimensions, but can be done without calculus provided the formula for the volume of a spherical cap is known. The optimal radius is approximately 0.229 units and the corresponding volume is approximately 0.0426.
Exercise 9.20 In part (iv), the question should ask for the minimum initial speed such that (B) gives the faster travel time for all trip lengths.
Second Printing
In the second printing (spring of 2007) the above errors and a few other ambiguities were corrected. Here are errors, notes and clarifications for the second printing!
p. 23. Under Hint 6, in the second paragraph, change 450-mile trip to 400-mile trip.
p. 38 and 134, Exercise 4.1(ii). Professor Michael Erickson in the Psychology Department at UC Riverside kindly pointed out that the "jealousy condition" in Tartaglia's Brides Problem needs to be stated very clearly. The condition stated in the book allows two brides and one man, who is the husband of one of the brides, to be alone during a transition on a shore. This interpretation leads to the nine-crossing solution given in the book. A more interesting problem and solution arises if the condition is "no bride may be present on a shore or on the boat with another husband unless her husband is also present." This condition leads to an 11-crossing solution (that I will leave to the reader!).
p. 39, Exercise 4.2(iii). All units should be gallons.
p. 78. Figure 8.2b should consist of discrete dots (not a continuous curve).
p. 96. In the solution to Example 9.4, delete the reference to Chapter 11.
p. 100, Exercise 9.18. There should be only one male in this problem; let's call him George.
p. 130, Solution to 2.19. In the last bullet change the first 5 to 8.
p. 133, Solution to 3.33. Change "volume of the sphere is r inches" to "radius of the sphere is r inches."
p. 135, Solution to 4.2(iii). The second to last line of the table should have (1,11) in the first column.
p. 136, Solution to 4.3(ii). The last line of the table should have (8,8,8,0) in the first column.
p. 136, Solution to 4.4(v). I am suspicious of this solution, but haven't time to check it again.