math problem

Exploring the Beauty of Mathematical Problem Solving

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This work is dedicated to everyone who really enjoys teaching mathematics and cannot have a course of good problems in mathematics without trying to solve them all. Our aim is to offer here some activities to those whose job includes mathematical problem solving. This work is not contracted to follow established programs and is only partially concerned with selecting the best exercises or the largest collection of problems. However, after searching around, we believe it is adequate for use as a supplement for any course in mathematical problem solving. We apologize in advance for any omission that occurs. Practical usefulness was our only criterion when collecting the problems and examples presented throughout this work. We would like to make it clear that we have made no attempt here to provide a summary of the many heuristic and pedagogical methods recommended in books of mathematical problem solving. But of course, those who enjoy this book should succeed in discussing many of these problems. Finally, we believe that if you are someone who loves to solve a problem and someone who loves to teach this, this book will be useful.

While technology is transforming and revolutionizing the use of mathematics in many scientific, industrial, and commercial fields, it is also transforming the nature of mathematics itself. Computers have become an indispensable tool for researchers and teachers, while at the same time technology is playing an increasing role in motivating youngsters to study and enjoy mathematics. Open-ended problems bring to the classroom the good qualities of true research and attract unconditionally the curiosity of the pupils. This chapter introduces an extraordinary collection of open-ended research-style problems. Some of these problems are very elementary, but all of them are paradoxical and not considered yet. The great challenge here is pedagogical: To find ways of communicating these ideas to students at various levels of mathematical development.

Guess and check is a fundamental problem-solving strategy in the field of mathematics. It is as simple as it sounds: just guess and then check. Sometimes numbers can be plugged in for variables in the problem statement. It is often helpful to look at the problem and see if you can find a pattern. Pattern problems don’t necessarily have to have a pattern that repeats, but finding a number pattern problem can often help to find a solution to the original problem. You will sometimes be able to solve a problem by mapping out the relationships of objects or ideas with one another. Then you will draw the connections visually. It will tell you directly if one problem will relate to the other, especially if it is a function. There are various forms of logic, and they can usually be used to help solve mathematics problems.

Analyzing problems is often a good strategy for getting started with mathematical problem solving. The process of breaking down a problem into simpler subproblems can sometimes suggest new approaches to the problem. Furthermore, considering special cases can help illuminate key features of a problem. Strategies for analyzing problems might include looking at small examples, looking for a pattern, looking for a larger context, and establishing necessary and sufficient conditions.

Every time when an intelligent conversation is held near a blackboard, there is some mathematics involved. In most cases, at least two people who were interested independently researched a certain mathematical problem. These people were very excited when they agreed to meet and tried to convince each other to become aware of the beauty of the mathematical results obtained. Moreover, they were courageous to explain how they solved their particular problems. However, normal human beings try to avoid presenting this ugly process mainly because the final formula or theorem usually looks very nice and elegant. However, in the real-life process of mathematical problem solving, there are a lot of changes, often surprises, “aha” and “eureka” effects, drama, and even romance with the desired proof.

The significance of mathematics lies in its applications. However, mathematical problem solving can be perceived as a highly abstract and sophisticated activity that exists only on the high academic and professional level. Consequently, educational efforts often neglect the value and beauty of mathematics. Students may hesitate to solve mathematical problems and even the average student may have a strange feeling that, for example, a US county scores nearly 100% on the White-Baby Result Paradox. René Deza expresses his sincere feelings: “I used the term beautiful repeatedly in this book. I believe that an outstanding aesthetic experience is derived from absorbing the wisdom, experience, and sensibility conveyed by these works of art. Actually, mathematics is considered an art, an abstract form of art. Clearly, it must be more than a theoretical exercise for the professionals only.

– Not reading/disregarding the whole question. – Getting stuck in a rut without looking back. – Not being familiar with theorems and results. – Complexity blinders: missing the forest for the trees. – Plowing ahead without a plan. – Skimming details or not paying attention to relevant details. – Assuming the entire forest is a single tree. – Tenacity. – Frustration. – Unresolved issue/loose ends. – Escape velocity. – Too fast, too furious. – Over-fragmentation. – Breaking into small pieces. And that’s the problem!

There are countless ways to get better at problem solving. The more challenging approach is to explore one’s own challenges and common pitfalls. We have identified over 20 of these, based on our observations at Math Olympiads and other problem solving contests, and perhaps suffer from one or two of these ourselves. (We are indeed fortunate to have had students who demonstrate several of these pitfalls on a single problem over the years.) Be aware of the following pitfalls:

One of my main breakthroughs in mathematical problem-solving has been studying handouts and books about solving problems in Diophantine equations. Here, given a finite family of smooth functions and a smooth algebraic curve, there are just finitely many rational points on the graph. Thus, if one can find finitely many families of rational points on the graph of some associated Diophantine equation, after applying a bunch of successive steps through which the set of all rational numbers decreases to one which one knows produce a constant family of rational points, through those finitely many steps one has precisely managed to get the job done.

Of course, most of the books I am going to mention can be used before the reader goes further, but, just to start somewhere, it becomes natural to list them here when one is already talking about the techniques required for solving difficult and beautiful problems. In one of my very first international Olympiads, I was very puzzled by two problems which I understand to involve the techniques of descent and induction. As about seven years later I found a book listing in complete detail all used techniques, and three lines were enough to deal with both problems, this gave me additional motivation to study those handouts and books over and over again to truly understand what was going on, rather than simply wanting to get trained for some competitions with a relatively narrow repertoire of questions.

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