Conjecture and Proof (Classroom Resource Materials) - Miklós Laczkovich

A Book from Hungary, an Inventive Country, for New Ways of Thinking about Mathematical Problems

"Conjecture and Proof" is a collection of the lecture notes designed for a one-semester course in Hungary for American and Canadian students. The course was intended for creative problem solving and for conveying the tradition of Hungarian mathematics. Other purpose of the book includes showing the spirit of mathematics.

Hungary is a country well-known for inventions. Her inventions include Rubik cube, fuel injector, helicopter, stereo, television, transformer, generator, ball pen, telephone switch, neutron bomb, and contact lens. Her scientists have been awarded for Nobel prizes at the least 11 times. With knowing the history of Hungary, the expectation of the book is logical. Even though the book is intended for undergraduate students, it exposure the readers on many deep, interesting, and important theorems with completed proofs but easily accessible methods on
1. the irrationality of the number of the square root of 2, of the number e, and of the number pi,
2. the three classical geometric construction problems,
3. constructible regular polygons (Gauss Theorem),
4. e is transcendental,
5. Banach-Tarski Paradox.

The book also comes with a sketch of the proof of Hilbert's third problem, which states that the regular tetrahedron is not equidecomposable to the cube nor to any rectangular box of the same volume. The proof is based on an additive and invariant function. The value of the function at a regular tetrahedron is nonzero, but the value of the function at the cube is zero.

The first time I read the book while I was an undergraduate. Most of the material covers on the book were new to me other than the irrationality of square root of 2, the pigeonhole principle, countable and uncountable sets. The chapters 2, 3, 6, 7, 11, 13, 15, 16 impressed me the most. They were geometry related, especially the proofs on the three classical geometric construction problems (doubling the cube, trisection of angles, and squaring the circle). Banach-Tarski Paradox was another shock. If a solid ball can be broken down into infinite points and "then be put back together in a different way," two identical copies of the original ball can be yield.

Anyway, I own the book now for the provided proofs.

Good problems, beautiful proofs

I have recently acquired a copy of the Hungarian edition (apparently a translation of this English edition) and I can tell you, these guys have selected some of the best and most representative problems in Mathematics. The proofs are very concise and you really do not need much more than high school/early college math to follow them. There are a number of exercises for each chapter (topic) and some of them also come with hints. I would also consider these exercises to tease brighter high school math wizards. For those who have already seen them during their studies, it is surely worth another look. I have not seen the English edition, but reading the Hungarian version, I assume it must also be very well written.

The basics of proof techniques covered in sufficient depth

As the title indicates, the legendary Paul Erdos was involved in the creation of this book. In 1983, Erdos and other Hungarian mathematicians started the Budapest Semester in Mathematics (BSM), a program for American and Canadian undergraduate students. One of the courses in this program involves creative problem solving, which was the motivation for the material in this book. As is the case with books on problem solving, no particular area of mathematics is examined. The emphasis is on proof techniques, which are largely independent of the mathematical topic.
Of course, the quality of any book of this type is largely dependent on the choice of problems that are described, and in this case the chosen topics are excellent. The book is split into two main sections, which are further split into the following subsections:

I) Proofs of Impossibility, Proofs of Nonexistence.
1) Proofs of Irrationality.
2) The Elements of the Theory of Geometric Constructions.
3) Constructible Regular Polygons.
4) Some Basic Facts About Linear Spaces and Fields.
5) Algebraic and Transcendental Numbers.
6) Cauchy's Functional Equation.
7) Geometric Decompositions.

II) Constructions, Proofs of Existence.
8) The Pigeonhole Principle.
9) Liouville Numbers.
10) Countable and Uncountable Sets.
11) Isometries of R^n.
12) The Problem of Invariant Measures.
13) The Banach-Tarski Paradox.
14) Open and Closed Sets in R. The Cantor Set.
15) The Peano Curve.
16) Borel Sets.
17) The Diagonal Method.

While each of these topics is introduced, that does not mean that the coverage is superficial. The book is advertised as having more than elementary coverage, and I concur with that assessment. Detailed proofs of the main ideas are included with exercises at the end of each section. Hints for the solution of many of the problems are included in an appendix.
This is an excellent short introduction to many of the proof techniques that are the staple of working mathematicians. I strongly recommend it as a primary or secondary text for any course where the goal is to teach basic proof techniques to advanced undergraduates.