MATH CIRCLE ACTIVITY 3 The Tower of Hanoi Puzzle

🧮 MATH CIRCLE · NO 3

🎯 ACTIVITY - 3 : THE TOWER OF HANOI

DATE: 25-07-2025 DAY: Friday

Recursive Thinking | Pattern Analysis | Strategic Puzzle Solving

🎯 AIM

Main Objective:
To enhance students’ comprehension of recursive thinking, logical analysis, and strategic problem-solving through the Tower of Hanoi challenge.

"MISTAKES are PROOF you are TRYING" — embrace the recursive journey!

Purpose Statement:
To investigate the underlying patterns of the Tower of Hanoi puzzle, recognize its recursive nature, and cultivate effective thinking strategies in problem-solving.

Fig: Starting configuration — goal to move all discs to third rod.

📚 LEARNING OUTCOMES

• ✅ Recognize recurring patterns in recursive tasks.
• ✅ Strengthen logical reasoning and forward-thinking skills.
• ✅ Comprehend the mathematical formula: Minimum Moves = 2ⁿ – 1 (n = number of discs).
• ✅ Connect the puzzle to real-life applications such as algorithm design and data handling in computer science.

Discs (n)	1	2	3	4	5	6
Min Moves	1	3	7	15	31	63

Pattern: 1, 3, 7, 15, 31... Rule: Multiply by 2 and add 1 → \(2^n - 1\)

⚙️ RULES OF THE PUZZLE

• Rule 1: Move only one disc at a time.
• Rule 2: A larger disc must never be placed over a smaller one.
• Rule 3: All discs must be transferred from the first rod to the third rod using the middle rod as a helper.

Visual: Three rods — start (left), auxiliary (middle), target (right).

📋 PROCEDURE (Step-by-Step)

1. Setup: Begin with all discs arranged on the starting rod (source) in decreasing size (largest at bottom).
2. Attempt: Students attempt to shift all discs to the target rod by strictly following the rules.
3. Observe patterns: As the number of discs increases, encourage students to notice recursive repetition and think ahead.
4. Discuss recursive formula: The minimum moves follow \(2^n - 1\). Guide students to derive the relation.
5. Reflection: Connect the puzzle to algorithm design like Tower of Hanoi recursion in programming.

Step 1: Initial arrangement

Step 2: Move via auxiliary rod

Final: All discs on target

👩‍🏫 TEACHER’S FEEDBACK & OBSERVATIONS

This hands-on activity successfully introduced the concept of recursion in a practical, interactive way.

Students remained engaged throughout, using critical thinking to plan each move.

It was especially rewarding to see them independently discover the formula \(2^n - 1\) for calculating minimum moves.

Overall, this activity served as a great platform to reinforce patience and strategic planning.

💡 Additional Teacher Insight: The puzzle naturally introduces stack data structure (LIFO) and is excellent for computational thinking. Pair programming with physical discs boosted collaboration.

Fig: Collaborative learning — students analyzing the recursive pattern.

🗣️ STUDENT’S FEEDBACK & REFLECTIONS

Ananya (Grade 7):
“I found this puzzle really interesting! It pushed me to think before each move.”

Rohan (Grade 8):
“Initially I struggled, but with more practice, I began to spot patterns that made solving easier.”

Diya (Grade 6):
“It was challenging, especially with more discs, but I enjoyed the logical thinking it required.”

🙏 "I am thankful to the PM SHRI Scheme for giving me this opportunity to learn in a fun and meaningful way."

— Students of Math Circle

---

Beyond the classroom: Students explored similar puzzles on Brilliant.org, Khan Academy, and Code.org, strengthening their recursive thinking.

Joyful learning & discovery

Teamwork & logical discussion

🔍 PATTERN SPOTLIGHT & CHALLENGES

Pattern Challenge

Sequence: 1, 3, 7, 15, 31, __?
What comes next?
✅ Answer: 63 (because 2⁶ - 1 = 63)

Quick Think: 31 is the nth term. Find n.
Answer: n = 5 (since 2⁵ - 1 = 31)

Educational Platforms

Explore more recursion puzzles on: Brilliant.org, GeeksforGeeks, Khan Academy, and MIT Scratch. These platforms offer interactive Hanoi simulators and algorithmic challenges.

Real-world connection: Disk backup rotation, recursive file system traversal, and AI pathfinding.

Mobile friendly | All images fast-loading inline SVG | PDF: Print > Save as PDF

Math Circle · Activity 3: Tower of Hanoi | Recursion & Strategic Thinking