Mathemagical Cruise

簡介

本書不僅收集了有趣的問題及巧妙的解答，更提供了思考與解決問題的絕佳案例。各章節可按任意順序獨立閱讀，藉由巡遊書中蘊藏著各式珍寶的島嶼，讀者將跨越數學的邊界，航行在浩瀚海洋，探索地平線外的無限可能。

Mathemagical Cruise is not a mere collection of fun problems with clever solutions. It offers shining examples of how to approach problem solving.

Each chapter is independent and can be read in any order by everyone with a basic background in high school mathematics. Some highlights of the excursion are:

● Slick Solutions of Double Sequence, Klarner’s Puzzle, Cube Tour, etc.
● Easy Proofs of Bolyai-Gerwin Theorem, Problem by P. Erdös and more
● New Year Puzzles (Especially, Year 2021 & 2022)
● Twelve Points on the Nine-Point Circle
● What's a Point in a Square?
● Five Circles through a 5x6 Grid
● Generalization of Ceva's Theorem
● Easy Approach to Coaxal Circles
● Inversion and its Applications
● Lattice Integer Triangles
● Isbell's Problem
● Sequence of Theorems of Simson & Cantor
● Miscellaneous Problems with Solutions

By cruising through these treasure islands, the reader will traverse mathematical boundaries. Be adventurous and inspired to explore the seas beyond the horizon.

more...

目次

Contents
Preface iv
1 Puzzles 1
1.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Double Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 15-Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Klarner’s Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 A Cube Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Safe Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 A Problem on Weighted Trees . . . . . . . . . . . . . . . . . . . . . 21
2 The Bolyai-Gerwin Theorem 25
2.1 Baby Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 A Triangular Carpet . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 The Bolyai-Gerwin Theorem . . . . . . . . . . . . . . . . . . . . . . 32
3 New Year Puzzles 40
3.1 New Year Puzzle 2014 . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 New Year Puzzle 2015 . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Heron’s Formula Revisited . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 New Year Puzzle 2016 . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 New Year Puzzle 2017 . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 New Year Puzzle 2018 . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 New Year Puzzle 2019 . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.8 New Year Puzzle 2020 . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.9 New Year Puzzle 2021 . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.10 New Year Puzzle 2022 . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.11 New Year Puzzle 2023 . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.12 New Year Puzzle 2024 . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.13 New Year Puzzle 2025 . . . . . . . . . . . . . . . . . . . . . . . . . 70
iii
iv
4 In Remembrance of Professor Ross Honsberger 72
4.1 The Bulging Semicircle . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 The Last Digits of 79999 . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 A Diophantine Equation . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Sum of the Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Gaps between Consecutive Primes . . . . . . . . . . . . . . . . . . . 76
4.6 Triangle Numbers That Are Perfect Squares . . . . . . . . . . . . . 77
4.7 A Problem by Erd˝os . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Triangles 86
5.1 Medians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Orthocenter and Circumcenter . . . . . . . . . . . . . . . . . . . . . 87
5.3 Incenter and Excenters . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 From the Desks of My Friends 97
6.1 From Dean Ballard . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.1 What’s a Point in a Square? . . . . . . . . . . . . . . . . . . 97
6.1.2 Wythoff’s Game . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.3 The Game of Nim . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 From Tien-Sheng Hsu . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7 How Many Interior Right Angles Can a Polygon Have? 120
8 Ceva and Menelaus Revisited 127
9 Circles 133
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.2 Radical Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.3 Coaxal Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.4 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9.5 Theorems of Ptolemy, Steiner and Poncelet . . . . . . . . . . . . . . 150
9.6 An Old Japanese Theorem . . . . . . . . . . . . . . . . . . . . . . . 156
9.7 With Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10 Lattice Points 162
10.1 The Schinzel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 162
10.2 Lattice Integer Triangles . . . . . . . . . . . . . . . . . . . . . . . . 168
10.3 The Isbell Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11 On the Theorems of Simson and of Cantor 186
11.1 The Simson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A Problems 198
B Solutions and Hints 206