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.

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

**Preface**

*Denken Sie nach!*(Ferdinand Georg Frobenius)