Mathematical Problems - An Essay on Their Nature and Importance

✍ Scribed by Smorynski, Craig

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 410
• Edition: 1
• Category: Library

✦ Synopsis

The life and soul of any science are its problems. This is particularly true of mathematics, which, not referring to any physical reality, consists only of its problems, their solutions, and, most excitingly, the challenges they pose. Mathematical problems come in many flavours, from simple puzzles to major open problems. The problems stimulate, the stories of their successful solutions inspire, and their applications are wide.

The literature abounds with books dedicated to mathematical problems — collections of problems, hints on how to solve them, and even histories of the paths to the solutions of some famous ones. The present book, aimed at the proverbial “bright high-school student”, takes a different, more philosophical approach, first dividing mathematical problems into three broad classes — puzzles, exercises, and open problems — and discussing their various roles in one’s mathematical education. Various chapters are devoted to discussing examples of each type of problem, along with their solutions and some of the developments arising from them. For the truly dedicated reader, more involved material is offered in an appendix.

Mathematics does not exist in a vacuum, whence the author peppers the material with frequent extra-mathematical cultural references. The mathematics itself is elementary, for the most part pre-calculus. The few references to the calculus use the integral notation which the reader need not truly be familiar with, opting to read the integral sign as strange notation for area or as operationally defined by the appropriate buttons on his or her graphing calculator. Nothing further is required.

Advance praise for Mathematical Problems

"There are many books on mathematical problems, but Smoryński’s compelling book offers something unique. Firstly, it includes a fruitful classification and analysis of the nature of mathematical problems. Secondly, and perhaps most importantly, it leads the reader from clear and often amusing accounts of traditional problems to the serious mathematics that grew out of some of them." - John Baldwin, University of Illinois at Chicago

"Smoryński manages to discuss the famous puzzles from the past and the new items in various modern theories with the same elegance and personality. He presents and solves puzzles and traditional topics with a laudable sense of humor. Readers of all ages and training will find the book a rich treasure chest." - Dirk van Dalen, Universiteit Utrecht

Craig Smoryński got his PhD at the University of Illinois at Chicago Circle with a dissertation on Kripke models for intuitionistic systems. This served as good background when he got involved in the birth and early development of provability logic, publishing the first mathematical textbook on the subject (Self-Reference and Modal Logic, Springer, 1985). He has written a number of books on mathematics and its history, most notably Logical Number Theory (Springer, 1991), Adventures in Formalism (College Publications, 2012), and MVT: A Most Valuable Theorem (Springer, 2017). In addition, he has contributed chapters to the Handbook of Mathematical Logic and the Handbook of Philosophical Logic.

✦ Table of Contents

Contents
1 Introduction
2 Logic Puzzles
Traditional Logic Puzzles
Knowledge-Based Puzzles
Liars and Truth-Tellers
Concluding Remarks
3 Some Basic Mathematical Exercises
Drill Exercises
Challenge Exercises
Exploratory Exercises: The Fibonacci Sequence
Properties of the Fibonacci Sequence
Digression on Induction
Number-Theoretic Properties of the Fibonacci Sequence
Growth and Complexity
Programming Issues and the Tower of Hanoi
Return to the Fibonacci Numbers: The Golden Ratio
The Golden Ratio in Art and Nature
Concluding Remarks
4 Probability
The Problem of Points
The Petersburg Problem
Bayes's Theorem
What is Probability?
Concluding Remarks
5 Graph Theory
The Seven Bridges of Königsberg
Proof of Euler's Theorem
Dudeney's Eulerian Puzzles
Wolf, Goat, and Cabbage
Knight's Tours
Hamiltonian Circuits in General
Planar Graphs
Graph Colouring
Concluding Remarks
Further Explorations
The Tower of Hanoi; A Humanly Doable Solution
The Problem of Points; Exploring the Given Solution
The Problem of Points; Lagrange's Method
The Problem of Points; The Modern Method
The Tower of Hanoi Revisited
Index