1st Edition

Puzzles, Paradoxes, and Problem Solving An Introduction to Mathematical Thinking

By Marilyn A. Reba, Douglas R. Shier Copyright 2015
    608 Pages 201 B/W Illustrations
    by Chapman & Hall

    A Classroom-Tested, Alternative Approach to Teaching Math for Liberal Arts

    Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking uses puzzles and paradoxes to introduce basic principles of mathematical thought. The text is designed for students in liberal arts mathematics courses. Decision-making situations that progress from recreational problems to important contemporary applications develop the critical-thinking skills of non-science and non-technical majors.

    The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. The text’s five units focus on graphs, logic, probability, voting, and cryptography. The authors also cover related areas, such as operations research, game theory, number theory, combinatorics, statistics, and circuit design.

    The text uses a core set of common representations, strategies, and algorithms to analyze diverse games, puzzles, and applications. This unified treatment logically connects the topics with a recurring set of solution approaches.

    Requiring no mathematical prerequisites, this book helps students explore creative mathematical thinking and enhance their own critical-thinking skills. Students will acquire quantitative literacy and appreciation of mathematics through the text’s unified approach and wide range of interesting applications.

    Graphs: Puzzles and Optimization
    Graphical Representation and Search
    Greedy Algorithms and Dynamic Programming
    Shortest Paths, DNA Sequences, and GPS Systems
    Routing Problems and Optimal Circuits
    Traveling Salesmen and Optimal Orderings
    Vertex Colorings and Edge Matchings

    Logic: Rational Inference and Computer Circuits
    Inductive and Deductive Arguments
    Deductive Arguments and Truth-Tables
    Deductive Arguments and Derivations
    Deductive Logic and Equivalence
    Modeling Using Deductive Logic

    Probability: Predictions and Expectations
    Probability and Counting
    Counting and Unordered Outcomes
    Independence and Conditional Probabilities
    Bayes’ Law and Applications of Conditional Probabilities
    Expected Values and Decision Making

    Counting: Voting Methods and Apportionment
    Voting Methods
    Fairness Criteria and Arrow’s Impossibility Theorem
    Weighted Voting Systems and Voting Power
    Assessing Apportionment Methods

    Numbers: Cryptosystems and Security
    Modular Arithmetic and Cryptography
    Binary Representation and Symmetric Cryptosystems
    Prime Numbers and Public-Key Cryptosystems

    Appendix A: Applications
    Appendix B: Classifications



    Marilyn A. Reba is a senior lecturer and Douglas R. Shier is a professor emeritus, both in the Department of Mathematical Sciences at Clemson University. The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. Selected material from this book is currently being used in the Department of Mathematical Sciences’ liberal arts mathematics course and in a problem-solving course in the Honors College.

    "… an interesting approach to the world of mathematics. … The layout is good, as is the coverage. … The reinforcing exercises are excellent. Using logic and other techniques, the text lays out methods to help students learn to think in a mathematical manner. Summing up: Recommended. General readers, lower- and upper-division undergraduates."
    —M. D. Sanford, Felician College, Lodi, New Jersey, USA for CHOICE, November 2015