Puzzles, Games, and Problem Solving

I. Course description

Puzzles Games, and Problem Solving is a course focusing on the techniques used in both mathematical and non-mathematical problem solving.  In this course students will learn how many of the same strategies and techniques used in solving puzzles and playing games transfer to more traditional problem solving applications and, conversely, how techniques and strategies used in solving problems can be used to solve puzzles and develop winning strategies in playing games.  Students may expect to discover and develop problem solving methods through working with puzzles, games and non-traditional problems in such areas as number theory, geometry, probability, logic, and statistics.  Non-mathematical problems explored in the class may come from varied disciplines.  Emphasis in this class will be on developing methods for finding solutions and discovering and proving why some shortcut methods work rather than simply finding answers or learning “tricks.”  Students will also be given the opportunity to appreciate the elegance and beauty often found in these solutions and the paths leading to them.

II. Instructor

Don Arni

  • B.S. Physics, University of Missouri-Columbia, 1973
  • M.Ed., Curriculum and Instruction, University of Missouri-Columbia
  • Mathematics and physics instructor
  • Glasgow R-II High School
  • Glasgow, Missouri   65254
  • darni@glasgow.k12.mo.us
III. Rationale for inclusion in a program for gifted students

Puzzles, games, and problem solving serve as both a vehicle and an end in this course.  As a vehicle, puzzles, games, and problem solving will be used to introduce students to a broad range of mathematical topics, many of which are not customarily included in traditional high school courses, and, as an end, students will be more experienced in solving problems with a wider variety of methods.

IV. Major topics covered
  • Defining problems, puzzles, and games
  • Polya’s strategies for problem solving
  • Classic problems in history—solved and unsolved
  • Sequences and series
  • Limits and derivatives
  • Logarithms and analytical geometry
  • Binary numbers and other base systems
  • Patterns
  • Pascal’s triangle
  • Combinations, permutations, and probability
  • Fibonacci numbers
  • Golden ratio
  • Networks
  • Inductive and deductive reasoning
  • Logic and logic games
  • Lateral thinking puzzles
  • Word play and word puzzles
  • Manipulative puzzles and topology
  • Empirical problem solving
  • Modeling and simulation
  • Number theory
  • Game theory
  • Mathematics and problem solving contests
  • Graphing and solving systems of equations
  • Trigonometry
  • Strategy games
  • Cooperative problem solving—team approach
  • Geometric constructions
  • Physics-related applications
V. Prerequisite knowledge:

Students should have successfully completed introductory courses in Algebra I and geometry and possess a desire to explore a variety of problem solving methods while investigating a broad range of mathematical and non-mathematical topics.  Students should also be prepared to be persistent in meeting the challenge of a mixture of puzzles and games.

VI. Learning objectives

The student will

  • develop working definitions for problems, puzzles, and games
  • identify which of Polya’s problem solving strategies they use when solving a problem
  • restate some of the classic solved and unsolved problems in history
  • define sequences recursively and by a general term formula
  • use the method of finite differences to find general term formula
  • identify arithmetic and geometric sequences
  • find sums of finite and infinite series
  • find limits of sequences and functions
  • use derivatives to solve maximum and minimum problems
  • solve problems by using logarithmic and exponential equations
  • use binary numbers and other base systems to solve puzzles, develop game strategies, and solve problems
  • use patterns in Pascal’s triangle to solve problems
  • calculate combinations, permutations, and probability
  • list the sequence of Fibonacci numbers and give examples of their occurrences in nature
  • calculate the golden ratio and cite its significance in mathematics, art, and architecture
  • solve networking problems of various types
  • distinguish between inductive and deductive reasoning and explain the importance of each
  • develop strategies to solve logic puzzles
  • use Boolean algebra to solve logic problems
  • solve lateral thinking puzzles
  • solve and design word puzzles
  • construct all 12 pentominoes
  • use pentominoes and tangrams to replicate challenge patterns
  • use topology to solve puzzles and problems
  • solve manipulative puzzles
  • use the empirical method to solve problems
  • use simulations and models to solve problems
  • use geometric constructions to solve problems
  • use number theory concepts to solve problems
  • use game theory to make decisions
  • compete in mathematical and problem solving contests
  • solve systems of equations
  • sketch curves of quadratic, cubic, and higher degree functions
  • use trigonometric functions to solve right triangle problems
  • use the law of sines and the law of cosines to solve problems
  • develop winning strategies for strategy games
  • solve problems cooperatively, using a team approach
  • solve problems using basic physics laws and principles
VII. Primary source material
  • Books
    • Art and Techniques of Simulation.  Quantitative Literacy Series.  Dale Seymour Publications.  1987.
    • Bergamini, David.  Mathematics.  Time-Life Books, Inc.  1980.
    • Britton and Seymour.  Introduction to Tesselations.  Dale Seymour Publications.  1989.
    • Camilli, Thomas.  A Case of Red Herrings Book B1.  Critical Thinking Press and Software.  1992.
    • ______.  A Case of Red Herrings Book B2.  Critical Thinking Press and Software.  1993.
    • Carlson, Roger.  “Random Digits and Some of Their Uses.”  Statistics by Example: Weighing Chances.  Addison-Wesley Publishing Company.  1973.
    • Conrad and Flegler.  The 1st High School Math League Problem Book.  Math League Press.  1989.
    • ______.  Math Contests! Volume 2.  Math League Press.  1992.
    • Coxford, Arthur F. and Joseph N. Payne.  Advanced Mathematics: A Preparation for Calculus.  Harcourt, Brace, Jovanovich.  1978.
    • Eves, Howard.  In Mathematical Circles—Series.  Prindle Weber and Schmidt, Inc.
    • Exploring Probability.  Quantitative Literacy Series.  Dale Seymour Publications.  1987.
    • Fixx, James F. Games for the Superintelligent.  Popular Library.  1972.
    • Gardner, Martin.  Aha! Gotcha: Paradoxes to Puzzle and Delight.  W. H. Freeman and Company.  1982.
    • ______.  Aha! Insight.  W. H. Freeman and Company.  1978.
    • ______.  Mathematical Carnival.  Vintage Books.  1975.
    • ______.  Mathematics Magic and Mystery.  Dover Publications, Inc.  1956.
    • Garland, Trudi Hammel.  Fascinating Fibonaccis, Mystery and Magic in Numbers.  Dale Seymour Publications.  1987.
    • Green and Hamberg.  Pascal’s Triangle.  Dale Seymour Publications.  1986.
    • Loyd, Sam.  Mathematical Puzzles of Sam Loyd.  Dover Publications.  1959
    • ______.  More Mathematical Puzzles of Sam Loyd.  Dover Publications.  1960.
    • Mottershead, Lorraine.  Metamorphosis, A Source Book of Mathematical Discovery.  Dale Seymour Publications.  1977.
    • Pappas, Theoni.  The Joy of Mathematics.  Wide World Publishing.  1989.
    • Polya, G.  How to Solve It.  Princeton University Press.  1973.
    • Problem Solving in School Mathematics—NCTM 1980 Yearbook.  National Council of Teachers of Mathematics.  1980.
    • Runion, Garth E.  The Golden Section.  Dale Seymour Publications.  1990.
    • Seymour and Shedd.  Finite Differences.  Dale Seymour Publications.  1973.
    • Sloane, Paul.  Lateral Thinking Puzzlers.  Sterling Publishing Co., Inc.  1992.
    • Sloane and MacHale.  Challenging Lateral Thinking Puzzles.  Sterling Publishing Co., Inc.  1993.
    • ______.  Great Lateral Thinking Puzzles.  Sterling Publishing Co., Inc.  1994.
    • A Sourcebook of Applications of School Mathematics.  National Council of Teachers of Mathematics.  1980.
    • Wujec, Tom.  Pumping Ions.  Doubleday.  1988.
  • Journals Games Magazine
  • The Mathematics Teacher
  • Quantum
  • Student Math Notes, MCTM Bulletin
VIII. Supplementary source material
  • AHSME Contests
  • AIME Contests
  • Pentominoes
  • Tangrams
  • Manipulative and Strategy Puzzles such as
  • Rubik’s Cubes
  • Tavern Puzzles—horseshoe and other interlocking chain type puzzles
  • Wooden Barrel and other 3D interlocking puzzles
  • Twisted Nail type puzzles, etc.
  • Strategy games such as Mastermind and other logic and strategy games will be used.