There are numerous problem collections on the market in different parts of the world, many of them excellent. is there a need for one more collection of problems? Perhaps not, but I felt tempted to convey some of my views and suggestions on learning about mathematics through problem solving. To discuss mathematical ideas without presenting a variety of problems seemed pointless. Since I had already gathered a fair number of problems from 'problem seminars' at international camps for young mathematicians which I have run in the past years, I decided to compile a book of problems and solutions from the camps. The arrangement of the text, the selection and grouping of the questions, the comments, references to related mathematical topics, and hints for further reading should illustrate my ideas about studying - or educating - through problem solving.
At the camps the age of the participants ranged from 13 to 19 and their mathematical backgrounds were tremendously varied. Consequently problems at the problem seminars varied in their degrees of difficulty. the same applies to the problems of this collection. It is hoped that any interested reader, aged 13 and above, might benefit from 'bits and pieces' of the text.
The composition of the book reflects the philosophy of the problem seminars at the camps. The campers were led step by step through four stages of problem solving:
Stage 1: Encouraging independent investigation.Finding an answer to a question by one's own means brings about a pleasure known to problem solvers of all ages. We recommended that all participants, especially beginners, first attempted all the investigations of easier problems which could be tackled alone, without hints and guidance.
Stage 2: Demonstrating approaches to problem solving.Having tested the joy of independent discovery, the youngsters became more critical towards their own achievements and wondered: "Is there a better (that is, quicker, or more elementary, or more elegant) solution to this problem?" By this time the campers were motivated, and were given the opportunity to learn more about techniques for problem solving.
Stage 3: Discussing solutions of famous problems from past centuries.Detecting problems and attempting their solution has been the lifetime occupation of professional mathematicians throughout the centuries. Advanced problem solvers were encouraged to study famous problems, their role in the development of mathematics and their solutions by celebrated thinkers. As well as improving their problem-solving skills they learnt to appreciate mathematics as part of our culture.
Stage 4: Describing questions considered by eminent contemporary mathematicians.Extensive study of research problems of modern mathematics is, generally, not possible at pre-university level. nevertheless there are a number of questions, e.g. in number theory, geometry, or modern combinatorics, which can be understood without much previous knowledge of 'higher mathematics'. The aim at the last stage at the seminars was to describe a selection of questions which have attracted the attention of eminent twentieth-century mathematicians.
Following this pattern of four stages of problem solving, the treatment of the problems in this book is divided into four chapters. These bear the names and are guided by the ideas of the corresponding stages at the seminars.
Even at a very early stage, investigating a pattern of numbers or shapes can lead to intriguing discoveries (no matter how small) and may raise a number of challenging questions. Therefore investigative work seems to provide a suitable introduction to the art of problem solving. What should one investigate? And how?
This chapter contains a selection of problems, based on some popular types of investigation, such as:
1. iterating a certain procedure and analysing the results;
2, search for patterns;
3. looking for exceptions, or special cases in a pattern;
4. generalising given problems;
5. studying converse problems.
'Devising a plan, conceiving the idea of an appropriate action, is the main achievement in the solution of a problem', claims Polya in his famous book on problem solving, How to Solve It.
According to Polya, a good idea is a piece of good fortune and we have to deserve it by perseverance:
"An oak is not felled at one stroke. If at first you don't succeed, try, try again. It is not enough to try repeatedly. We must try different means, vary our trials."
Problem solving requires a versatile mind, but one cannot be versatile without a fair knowledge of techniques and methods of discovery. The aim of this chapter is to present a selection of approaches to problem solving, applied to problems in the corresponding sections.
There are eight sections in this chapter, concentrating on the following hints for problem solvers:
1. Express the problem in a 'different language'.
2. Extend the field of investigation.
3. Find out: Is some mathematical transformation involved in a given problem? Do any properties of the objects considered remain invariant under this transformation? If so, make use of the invariants.
4. Make use of extremal (minimal or maximal) elements.
5. Try the method of infinite descent.
6. Try mathematical induction.
7. Attempt proof by contradiction.
8. Employ physics.
This chapter presents problems treated by eminent mathematicians in the past. Our selection aims to provide readers with additional information on topics encountered in the school syllabus and to raise interest in the History of Mathematics.
Chapter III consists of five sections:
1. Problems on prime numbers.
2. The number π.
3. Applications of complex numbers and quaternions.
4. On Euclidean and non-Euclidean geometries.
5. The art of counting.
What kinds of research work are professional mathematicians concerned with nowadays? Answers to this question are beyond the scope of our book. Nevertheless, we shall be able to discuss here a variety of elementary, beautiful and intriguing problems which have caught the attention of eminent contemporary scholars. In addition to being intrinsically interesting, the problems chosen for this chapter are connected with important branches of modern mathematics.
The geometrical problem of Sylvester-Gallai, described in Section 1, leads to a generalisation in the theory of block designs (a field of mathematics with various applications, e.g. in coding theory). The competition problems described in section 2 are related to deep questions in combinatorics of Ramsey numbers. Section 3 contains problems on lattice points; the latter have been used since Minkowski in number theory and are encountered in a wide range of mathematical topics. Finally, Section 4 is devoted to special cases and generalisations of Fermat's Last Theorem, a statement which has puzzled generations of mathematicians from the seventeenth century until the present day.