## Multicolor Problems ( Mathematical Conversations Part 1) – Dynkin, Uspenskii

In this post, we will see the book Multicolor Problems ( Mathematical Conversations Part 1) by E. B. Dynkin and V. A. Uspenskii. Multicolor Problems is a translation of Part One of Mathe­matical Conversations by E. B. Dynkin and V. A. Uspenskii, which was published in the Russian series. Library of the Mathematics Circle. The originality of the exposition and the variety of the problems presented here make this booklet especially useful in stimulating an inventive approach to mathematics.
This booklet deals with several of the classical map-coloring problems. The technique is one of developing an ordered presentation of problems and extensive solutions to them. A discussion of the famous four-color problem, which has puz­zled mathematicians for nearly a century, is included.
The booklet is designed for the reader’s active participation, as the problems are carefully integrated with the text and should be solved in sequence. The reader should have a back­ ground of high school algebra and should also be acquainted with the method of mathematical induction.
E. B. DYNKIN, a Professor at Moscow State University, is an eminent mathematician and author, whose specialties are higher algebra, topology, and probability theory. V. A. USPENSKII, a Lecturer at Moscow State University, spe­ cializes in mathematical logic.

The book was translated from Russian by was published in 1962. There is a recent volume by Dover which has all three parts in one book.

You can get the book here.

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# Contents

Introduction 1

## CHAPTER 1. Coloring with Two Colors 3

1. Simple two-color problems 3
2. Problems on square boards 5
3. Problems involving even and odd numbers 6
4. Networks and maps 7
5. General two-color problems 9

## CHAPTER 2. Coloring with Three Colors 12

6. A simple three-color problem 12
7. Problems on hexagonal boards 12
8. Dual diagrams 14
9. Triangulation 16
10. Dual maps 19
11. Normal maps in three colors 23

## CHAPTER 3. The Four-Color Problem 24

12. Normal maps in four colors 24
13. Volynskii’s theorem 25

## CHAPTER 4. The Five-Color Theorem 27

14. Euler’s theorem 27
15. The five-color theorem 32

Concluding Remarks 33

Appendix 34
Coloring a sphere with three colors 34

Solutions to Problems 40

Bibliography 66 