On December 19, 2025, the LUMS Math Circle hosted an engaging session titled “Think, Choose, Win: The Math Behind Strategy.” The session was jointly led by Dr. Lyyla Khalid and Dr. Waqas Ali Azhar; focused on introducing students to mathematical thinking through logical puzzles, strategic games, and fundamental ideas from game theory. 

The session aimed to demonstrate how mathematics can be used to analyze decisions, predict outcomes, and develop winning strategies in competitive and cooperative situations. 

Part I — Logical Reasoning and Strategy 

Dr. Waqas began the session with a warm-up puzzle known as The Age Riddle: 

Sana is twice as old as Fatima. Eight years ago, Sana’s age was three times Fatima’s age. How old are Sana and Fatima now? 

Students initially attempted to solve the problem through logical guesses. Dr. Waqas then demonstrated how to model the situation using variables and algebraic equations, emphasizing that mathematical modeling leads to a unique and reliable solution. This activity introduced students to the idea that structured reasoning is more powerful than intuition alone. 

He then presented a more complex age-based strategy problem involving a competition between boys and girls, where participants were added over several days to ensure that their team always won based on total age. Students observed how each side strategically introduced older members to change the outcome. 

Through this example, Dr. Waqas highlighted a central concept of game theory: 

players adapt their strategies based on the opponent’s previous moves.

The Nim Game 

Dr. Waqas next introduced the classical game of Nim, in which players take turns removing stones from piles, with the rule that stones can only be removed from one pile at a time. The player who removes the final stone wins. 

Using an example with piles of 3, 4, and 5 stones, two students played the game in front of the audience. The class analyzed which player had a winning strategy and how careful planning could guarantee victory. 

He also demonstrated a related line-removal game, where players could remove at most three lines per turn. Dr. Waqas showed that the winning strategy is to always leave the opponent with M + 1 objects (where M is the maximum number removable per turn). This illustrated the importance of pattern recognition and forward planning. 

Before concluding his segment, Dr. Waqas posed a challenge problem: 

How many different games of Nim can be formed using only five, six, and seven stones? 

Students were encouraged to explore this question further as homework.

Part II — Introduction to Game Theory (Dr. Lyyla Khalid) 

Dr. Lyyla Khalid began her part with a humorous anecdote about two students who lied about having a flat tire to avoid an exam. When asked the next day, “Which tire was it?”, students were asked how they would answer. This story served as a bridge into the central idea of strategic thinking: choosing an action based on what others might do or expect. 

She formally introduced game theory as a branch of mathematics and economics that studies situations in which outcomes depend on the choices of multiple decision-makers. Students learned that not all decisions are strategic, but strategic interactions arise when “my outcome depends on your choice too.” 

Through everyday examples—such as Rock–Paper–Scissors, competing for a swing, or sharing chores—students identified which situations qualify as strategic interactions. 

Core Concepts of Game Theory 

Dr. Lyyla explained the main building blocks of a game: 

• Players 

• Strategies 

• Payoffs 

• Information 

• Timing 

She introduced the idea of Nash Equilibrium as a situation in which no player benefits from changing their decision unilaterally — also called a “no-regrets strategy profile.” 

This concept was explored using several interactive examples: 

• Guessing Game 

Students analyzed a game where players pick numbers between 1 and 10, and the winner is closest to two-thirds of the average. Through reasoning, they discovered how strategic thinking leads toward equilibrium, which is 1 in this game. 

• Lost Luggage Game 

Two players secretly report suitcase values. The airline pays based on the lower claim and gives a bonus to the lower bidder. Students reasoned through incentives and identified the Nash equilibrium, which is again 1 in this game. 

• Prisoner’s Dilemma 

Using the famous two-prisoner scenario, students learned about dominant strategies and why rational players may choose outcomes that are not socially optimal.

Decision Trees and Backward Induction 

Dr. Lyyla also introduced backward induction, a method of solving multi-stage games by analyzing the final step first and reasoning backward. This method was demonstrated using: 

• The Pizza Slice Game 

• The Grab-or-Share Game 

• The Ice Cream Vendor Location Game 

These examples showed how rational players anticipate future consequences and adjust their actions accordingly.

Real-World Applications 

To conclude, Dr. Lyyla highlighted how game theory applies to real-world scenarios such as: 

• Competition between companies (e.g., Pepsi vs. Coca-Cola) 

• Pricing strategies in restaurants 

• Oil production decisions by OPEC 

• National defense and budget decisions 

Students were able to see that game theory is not merely theoretical, but a powerful tool for understanding economic, political, and social behavior. 

Conclusion 

The session successfully combined logical puzzles, competitive games, and strategic reasoning to introduce students to the mathematics of decision-making. Through active participation and guided discussion, students learned how mathematical tools can be used to analyze strategy, predict outcomes, and understand real-life conflicts and cooperation. 

The session concluded with lively discussion and problem-solving, leaving participants with a deeper appreciation for the role of mathematics in strategic thinking. Certificates were distributed as souvenirs in recognition of the students’ efforts.

Acknowledgment 

Heartfelt thanks to Dr. Lyyla Khalid and Dr. Waqas Ali Azhar for guiding students from playful puzzles to sophisticated ideas in game theory. Their ability to connect everyday situations with deep mathematical reasoning made the session both educational and enjoyable. 

Special appreciation is extended to Ms. Noreen Sohail, Ms. Shazia Zafar, Mr. Qamar Abbas, and Mr. Javaid Qayoom for their excellent organizational support, which ensured the smooth execution of the session. 

Here are some highlights from the event: