## The Enjoyment of MathWhat is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through.What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through. |

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### Contents

The Sequence of Prime Numbers | 9 |

Traversing Nets of Curves | 13 |

Some Maximum Problems | 17 |

Incommensurable Segments and Irrational Numbers | 22 |

A Minimum Property of the Pedal Triangle | 27 |

A Second Proof of the Same Minimum Property | 30 |

The Theory of Sets | 34 |

Some Combinatorial Problems | 43 |

The Spanning Circle of a Finite Set of Points | 103 |

Approximating Irrational Numbers by Means of Rational Numbers | 111 |

Producing Rectilinear Motion by Means of Linkages | 119 |

Perfect Numbers | 129 |

Eulers Proof of the Infinitude of the Prime Numbers | 135 |

Fundamental Principles of Maximum Problems | 139 |

The Figure of Greatest Area with a Given Perimeter | 142 |

Periodic Decimal Fractions | 147 |

On Warings Problem | 52 |

On Closed SelfIntersecting Curves | 61 |

Is the Factorization of a Number into Prime Factors Unique? | 66 |

The FourColor Problem | 73 |

The Regular Polyhedrons | 82 |

Pythagorean Numbers and Formats Theorem | 88 |

The Theorem of the Arithmetic and Geometric Means | 95 |

A Characteristic Property of the Circle | 160 |

Curves of Constant Breadth | 163 |

The Indispensability of the Compass for the Constructions of Elementary Geometry | 177 |

A Property of the Number 30 | 187 |

An Improved Inequality | 192 |

Notes and Remarks | 197 |

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### Common terms and phrases

alternating knot angle balls bisector boundary chapter chord circle K circle of radius circumference closed route colors common multiple cone consider constant breadth construction convex corner countries curve of constant decimal fraction denominator diagonal digits distance divide divisible divisors double point enclosing circle equal equation Euclid Euler's theorem exactly example fact figure finite number formula four squares fourth powers geometric given H. A. Schwarz hence inequality infinite inscribed intersection larger least least common multiple lemma length less linkage mathematics obtain original pair parallel pedal triangle perfect number perimeter period perpendicular plane polygon positive whole numbers possible prime factors prime numbers problem proof rational numbers regular polyhedrons relatively prime remainder Reuleaux triangle rhomboid rhombus right side segment smaller smallest solution straight line supporting lines torus true urns vertex vertices

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