Mathematical Swindles Part 1 - Some Puzzles & Games

Logical reasoning, using the tools and techniques of mathematics, is the key to the development and better understanding of all the sciences. The purpose of this HiMAP Pull-Out Section is to involve the reader with a simple, yet powerful technique for mathematical reasoning. This technique will be used to determine the best strategy one should use to ensure winning in puzzles and games. Nim games are used to illustrate the process; however, the process has numerous applications in solving complex problems that arise in business, government, and industry. This lesson will increase your understanding of the power of the reasoning techniques described, rather than promote swindling!

Note: The information below was created with the assistance of AI.

Level of Mathematics
This lesson is suitable for:

Upper middle school and high school students (grades 8–12).

Especially well-matched with Algebra I, Discrete Mathematics, Math Enrichment, or Game Theory electives.

Can also be used at the introductory college level in courses on logic, critical thinking, or decision science.

The mathematics involves logic, pattern recognition, and introductory game theory rather than computation-heavy content.

Application Areas
This lesson bridges mathematical reasoning with strategic thinking in various domains:

Game Theory:
Modeling competitive interactions and decision-making.

Foundational to economics, politics, cybersecurity, and AI.

Recreational Mathematics:
Developing critical thinking through games and puzzles (Nim, tic-tac-toe, toe-tac-tic).

Understanding fairness, strategy, and optimal play.

Consumer Awareness & Logic:

Identifying "swindles" and deceptive scenarios.

Teaching students how to dissect logical misdirection in real-world contexts.

Computer Science & Artificial Intelligence:

Introduction to game trees and strategic modeling (referenced in relation to chess and backgammon AI).

Behavioral Strategy in Real-Life Contexts:

Decision-making under competitive or deceptive conditions.

Applications in sports strategy, political campaigning, and business negotiation.

Prerequisites
Students should have:

Familiarity with basic arithmetic and algebraic reasoning.

Ability to follow multi-step logical sequences.

Experience with simple combinatorial thinking (optional but helpful).

An openness to exploring abstract ideas through concrete examples (puzzles, games, etc.).

No advanced mathematics is required—emphasis is placed on reasoning, logic, and strategic foresight.

Subject Matter
Core Mathematical Concepts

Strategic Reasoning:
Determining and executing strategies to optimize outcomes.

Exploring guaranteed win or tie strategies in deterministic games.

Game Types and Analysis:
Strictly determined games: where an optimal strategy ensures a specific outcome.

Swindle games: situations with deceptive or hidden winning strategies.

Examples and Game Exploration:

Tic-tac-toe: Tie strategies and predictable outcomes.

Toe-tac-tic: A twist where forcing the opponent to win becomes the goal.

Nim: A classic impartial game where backward analysis leads to a guaranteed win for the first player if played correctly.

Game Theory Language and Structure:
Terms such as strategy, optimal play, strictly determined, and game tree.

Introduction to modeling games mathematically and analyzing possible outcomes.

Correlation to Mathematics Standards
Common Core State Standards – High School
Mathematical Practices

MP1: Make sense of problems and persevere in solving them.

MP2: Reason abstractly and quantitatively.

MP3: Construct viable arguments and critique the reasoning of others.

MP4: Model with mathematics.

MP7: Look for and make use of structure.

High School Algebra and Modeling

HSA-CED.A.1: Create equations to model real-world situations.

HSA-REI.D.11: Solve problems using logic and interpretation.

Discrete Mathematics and Logic (NCTM or Advanced Electives)

Algorithmic Thinking

Recursive and Backward Reasoning

Decision Trees and Strategy Analysis