From the author The book is written on the materials of one of the sections of the school mathematical circle at Moscow State University . The proposed book includes three topics: problems about multicolor coloring maps, problems from number theory, solved with the help of arithmetic deductions, and problems from probability theory associated with the so-called random wanderings. For reading the first two sections, knowledge of mathematics in the amount of 8 years of secondary school is enough; the third section requires a slightly greater mathematical culture. The book is designed mainly for high school students, but can also be used in student circles in junior courses. Foreword by . Instructions for use of the book. SECTION FIRST. PROBLEMS OF MULTI-CULTURAL SCREENING . ' 1. The problem of two colors . ' 2. Three-color coloring . ' 3. On the problem of four colors . Volynsky Theorem . ' 4. Euler's theorem . The Five Colors Theorem . Conclusion. Appendix to Section I. On the tricolor coloring of the sphere . SECTION TWO . CHALLENGES FROM THE NUMBER THEORY . Chapter I. Arithmetic of deductions. ' 1. Arithmetic of deductions modulo m, or m-arithmetic. ' 2. Deduction arithmetic modulo p, or p-arithmetic. ' 3. Extraction of square root. Square Equations. ' 4. Extraction of cubic root. Simple divisors of numbers of the form a2+3. ' 5. polynomials and equations of higher degrees . Chapter II. m-adic and p-adic numbers. ' 1. Application of 10-arithmetic to the division of multivalued numbers. ' 2. Endless Numbers . ' 3. m-adic and p-adic numbers. Chapter III. Applications of m-arithmetic and p-arithmetic to number theory. ' 1. Fibonacci Series. ' 2. Pascal's Triangle . ' 3. Fractional-linear functions. Chapter IV. More information about the Fibonacci series and Pascal's triangle. ' 1. Application of p-adic numbers to the Fibonacci series. ' 2. The relationship between the Pascal triangle and the Fibonacci series. ' 3. Fibonacci members multiples of a given number. Chapter V. Equation x2-5y21. Conclusion. SECTION THIRD . Occasional Wanderings. (MARKOVA CHECKS). ' 1. Basic Properties of Probability. ' 2. The task of wandering in an infinite straight line . The Probability Triangle . ' 3. The Law of Large Numbers . ' 4. Wanderings with a finite number of states. ' 5. Wanderings with an infinite number of states. Conclusion. SOLUTIONS . Section 1. Tasks about multicolor coloring. Section 2. Problems in Number Theory . Section 3. Random Wanderings (Markov chains). Other issues of the series on the website Vyp. 1 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Arithmetic and Algebraic . 2 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Geometry (planimetry) Vyp. 3 - Chentsov N. N. , Shklyarsky D. About. , Yaglom I. M. Selected Problems and Theorems of Elementary Mathematics . Geometry (stereometry) Vyp. 4 - Boltyansky V. G. , Yaglom I. M. Convex Figures Issue. 5 - Yaglom I. M. , Yaglom A. M. Non-elementary tasks in the elementary presentation More on the topic'coloring maps? Donets G. A. , Shore N. Z. Algebraic approach to the problem of coloring flat graphs Smirnov S. G. Walks on closed surfaces More on the subject'theory of numbers? Buchstab A. A. Number theory More on the subject'random wanderings? Sosinsky A. B. Soap films and occasional wolf wanderings and. K. , Zuev S. M. , Tsvetkova G. M. Random processes Buy book: ozon. en, my-shop. ru