Mathemagical Cruise

Preface
1 Puzzles
1.1 Parity
1.2 Double Sequences
1.3 15-Puzzle
1.4 Klarner’s Puzzle
1.5 A Cube Tour
1.6 Safe Cracking
1.7 Tilings
1.8 A ProblemonWeighted Trees

2 The Bolyai-Gerwin Theorem
2.1 Baby Pythagoras
2.2 A Triangular Carpet
2.3 The Bolyai-Gerwin Theorem

3 New Year Puzzles
3.1 New Year Puzzle 2014
3.2 New Year Puzzle 2015
3.3 Heron’s Formula Revisited
3.4 New Year Puzzle 2016
3.5 New Year Puzzle 2017
3.6 New Year Puzzle 2018
3.7 New Year Puzzle 2019
3.8 New Year Puzzle 2020
3.9 New Year Puzzle 2021
3.10 New Year Puzzle 2022
3.11 New Year Puzzle 2023
3.12 New Year Puzzle 2024
3.13 New Year Puzzle 2025

4 In Remembrance of Professor Ross Honsberger
4.1 The Bulging Semicircle
4.2 The Last Digits of 79999
4.3 A Diophantine Equation
4.4 Sumof the Digits
4.5 Gaps between Consecutive Primes
4.6 Triangle Numbers That Are Perfect Squares
4.7 A Problemby Erdӧs

5 Triangles
5.1 Medians
5.2 Orthocenter and Circumcenter
5.3 Incenter and Excenters

6 From the Desks of My Friends
6.1 FromDean Ballard
6.1.1 What’s a Point in a Square?
6.1.2 Wythoff’s Game
6.1.3 The Game of Nim
6.2 From Tien-Sheng Hsu

7 How Many Interior Right Angles Can a Polygon Have?

8 Ceva and Menelaus Revisited

9 Circles
9.1 Preliminaries
9.2 Radical Axes
9.3 Coaxal Circles
9.4 Inversion
9.5 Theorems of Ptolemy, Steiner and Poncelet
9.6 An Old Japanese Theorem
9.7 With Coordinates

10 Lattice Points
10.1 The Schinzel Theorem
10.2 Lattice Integer Triangles
10.3 The Isbell Problem

11 On the Theorems of Simson and of Cantor

Appendix A Problems
Appendix B Solutions and Hints