Mathematical Swindles Part 2 - 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. In this HiMAP Pull-Out Section, we continue the investigation of game-playing strategies, which we had begun to do in the previous issue. Winning strategies for Nim games are discovered using a technique known as backward analysis of losing combinations. This technique also has numerous applications in solving complex problems in business, government, and industry. Most recently, it has even been employed to discover some rather startling results in chess endgame analysis. This article will increase one's understanding of the power of the logical reasoning techniques described, as well as one's ability to use them to determine whether certain mathematical puzzles and games are honest tests of skill and intelligence or merely swindles.

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

Level of Mathematics
This lesson is ideal for:

High school students in grades 9–12, particularly in:

Algebra I

Discrete Mathematics

Math Enrichment or Puzzle/Logic electives

Also applicable for introductory college-level logic, game theory, or decision science courses.

It builds on logical reasoning and pattern recognition rather than computation-heavy content.

Application Areas
This module has wide-ranging relevance across both academic and real-world contexts:

Game Theory & Strategy:
Developing and analyzing winning strategies for deterministic games.

Modeling adversarial decision-making.

Computer Science & Artificial Intelligence:
Understanding algorithms and retrograde analysis in games like chess and Nim.

Foundations for AI strategy planning in board games.

Business & Military Planning:
Applying backward reasoning to strategic planning, negotiation, and resource management.

Behavioral Decision-Making:
Simulating real-life decisions under constraints using simplified game models.

Prerequisites
Students should be comfortable with:

Basic logic and sequential reasoning

Understanding conditional strategies (e.g., "if I move here, then…")

Working with patterns and recursive reasoning

Simple arithmetic operations

No algebra or calculus is required, making it accessible to a broad range of learners focused on critical thinking.

Subject Matter
Key Concepts and Content Areas
Backward Analysis (Retrograde Reasoning)

Core method for determining optimal strategies in Nim and other strictly determined games.

Used in advanced applications like chess endgame analysis.

Nim Game Strategy (with 15–17 coins)

Identifying “losing combinations” and planning to leave these to the opponent.

Exploring variations based on who takes the last coin.

Take-a-Line Game

A spatial-logic game involving grid-based strategies and adjacency constraints.

Combines spatial awareness with backward strategy logic.

Endgame Analysis in Chess

Overview of Kenneth Thompson’s retrograde algorithm for chess.

Highlights practical AI use of backward logic to outplay even grandmasters.

Generalization to Real-World Strategy

Applying backward reasoning to business, military, and policy decisions.

Identifying worst-case scenarios and working backward to avoid them.

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 (e.g., through game scenarios).

MP7: Look for and make use of structure.

Algebra and Discrete Math Connections

HSA-CED.A.1–3: Create equations or logical models that describe real-world situations.

HSA-REI.D.10–12: Solve systems using reasoning, including backwards approaches.

NCTM Discrete Math Topics

Decision trees

Recursion and iterative strategies

Strategic games and modeling