Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks

Persi Diaconis

Princeton University Press | 2011 | 258 páginas | epub | 3,49 Mb

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Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks--and the profound mathematical ideas behind them--that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today's mathematical knowledge. For example, the Gilbreath Principle--a fantastic effect where the cards remain in control despite being shuffled--is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat's last theorem.

Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories--and reveals the best tricks--of the eccentric and brilliant inventors of mathematical magic. Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick--and much more.

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

William Cook

Princeton University Press | 2012 | PDF | 272 páginas | 7,5Mb

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What is the shortest possible route for a traveling salesman seeking to visit each city on a list exactly once and return to his city of origin? It sounds simple enough, yet the traveling salesman problem is one of the most intensely studied puzzles in applied mathematics and it has defied solution to this day.
In this book, William Cook takes readers on a mathematical excursion, picking up the salesman's trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today's state-of-the-art attempts to solve it.
Cook examines the origins and history of the salesman problem and explores its many important applications, from genome sequencing and designing computer processors to arranging music and hunting for planets. He looks at how computers stack up against the traveling salesman problem on a grand scale, and discusses how humans, unaided by computers, go about trying to solve the puzzle. Cook traces the salesman problem to the realms of neuroscience, psychology, and art, and he also challenges readers to tackle the problem themselves. The traveling salesman problem is--literally--a $1 million question. That's the prize the Clay Mathematics Institute is offering to anyone who can solve the problem or prove that it can't be done.
In Pursuit of the Traveling Salesman travels to the very threshold of our understanding about the nature of complexity, and challenges you yourself to discover the solution to this captivating mathematical problem.

TABLE OF CONTENTS:

Preface xi

Chapter 1: Challenges 1
Tour of the United States 2
An Impossible Task? 6
One Problem at a Time 10
Road Map of the Book 16

Chapter 2: Origins of the Problem 19
Before the Mathematicians 19
Euler and Hamilton 27
Vienna to Harvard to Princeton 35
And on to the RAND Corporation 38
A Statistical View 39

Chapter 3: The Salesman in Action 44
Road Trips 44
Mapping Genomes 49
Aiming Telescopes, X-rays, and Lasers 51
Guiding Industrial Machines 53
Organizing Data 56
Tests for Microprocessors 59
Scheduling Jobs 60
And More 60

Chapter 4: Searching for a Tour 62
The 48-States Problem 62
Growing Trees and Tours 65
AlterationsWhile You Wait 75
Borrowing from Physics and Biology 84
The DIMACS Challenge 91
Tour Champions 92

Chapter 5: Linear Programming 94
General-Purpose Model 94
The Simplex Algorithm 99
Two for the Price of One: LP Duality 105
The Degree LP Relaxation of the TSP 108
Eliminating Subtours 113
A Perfect Relaxation 118
Integer Programming 122
Operations Research 125

Chapter 6: Cutting Planes 127
The Cutting-Plane Method 127
A Catalog of TSP Inequalities 131
The Separation Problem 137
Edmonds's Glimpse of Heaven 142
Cutting Planes for Integer Programming 144

Chapter 7: Branching 146
Breaking Up 146
The Search Party 148
Branch-and-bound for Integer Programming 151

Chapter 8: Big Computing 153
World Records 153
The TSP on a Grand Scale 163

Chapter 9: Complexity 168
A Model of Computation 169
The Campaign of Jack Edmonds 171
Cook's Theorem and Karp's List 174
State of the TSP 178
Do We Need Computers? 184

Chapter 10: The Human Touch 191
Humans versus Computers 191
Tour-finding Strategies 192
The TSP in Neuroscience 196
Animals Solving the TSP 197

Chapter 11: Aesthetics 199
Julian Lethbridge 199
Jordan Curves 201
Continuous Lines 205
Art and Mathematics 207
Chapter 12: Pushing the Limits 211

Notes 213
Bibliography 223
Index 225

Convex figures

(Library of the mathematical circle)

I. M. Yaglom, V. G. Boltyanskii

Holt, Rinehart and Winston | 1961 | 301 páginas | djvu | 3,86 Mb

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CONTENTS 

Foreword to the American Edition 

Authors' Preface 

Suggestions for Using This Book 

Exercises 

I. General properties of convex figures 

2. Helly's Theorem and its applications 

3. A property of continuous functions 

4. The addition of convex figures and curves 

5. The isoperimetric problem 

6. Various exercises on maxima and minima 

7. Curves of constant width 

8. Curves which can be rotated in an equilateral 

triangle ( -curves) and related curves 

Appendix I. Limit Theorem for curves 

Appendix II. On the concepts of convex and non- 

convex figures 

Translators ' Note: Similitudes 

Solutions 

Bibliography 

Index 

The Decomposition of Figures Into Smaller Parts

 (Popular Lectures in Mathematics)

Vladimir Grigor'evich Boltyanskii, Izrail' Tsudikovich Gohberg, Henry Christoffers, Thomas P. Branso
University Of Chicago Press | 1980 | 80 páginas | djvu | 935 kb

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In contrast to the vast literature on Euclidean geometry as a whole, little has been published on the relatively recent developments in the field of combinatorial geometry. Boltyanskii and Gohberg's book investigates this area, which has undergone particularly rapid growth in the last thirty years. By restricting themselves to two dimensions, the authors make the book uniquely accessible to interested high school students while maintaining a high level of rigor. They discuss a variety of problems on figures of constant width, convex figures, coverings, and illumination. The book offers a thorough exposition of the problem of cutting figures into smaller pieces. The central theorem gives the minimum number of pieces into which a figure can be divided so that all the pieces are of smaller diameter than the original figure. This theorem, which serves as a basis for the rest of the material, is proved for both the Euclidean plane and Minkowski's plane.

Preface 
1. Division of Figures into Pieces of Smaller Diameter 
1.1. The Diameter of a Figure 
1.2. Formulation of the Problem 
1.3. Borsuk's Theorem 
1.4. Convex Figures 
1.5. Figures of Constant Width 
1.6. Embedding in a Figure of Constant Width 
1.7. For Which Figures is a(F) = 3? 
2. Division of Figures in the Minkowski Plane 
2.1. A Graphic Example 
2.2. The Minkowski Plane 
2.3. Borsuk's Problem in Minkowski Planes 
3. The Covering of Convex Figures by Reduced Copies 
3.1. Formulation of the Problem 
3.2. Another Formulation of the Problem 
3.3. Solution of the Covering Problem 
3.4. Proof of Theorem 2.2 
4. The Problem of Illumination 
4.1. Formulation of the Problem 
4.2. Solution of the Problem of Illumination 
4.3. The Equivalence of the Last Two Problems 
4.4. Division and Illumination of Unbounded Convex Figures 
Remarks