MATHEMATICS SUBJECT ENRICHMENT ACTIVITY
Class: VIII
Chapter: We Distribute, Yet Things Multiply
Activity Title: Coin Conjoin – Flipping Triangular Coin Patterns
🔷 Topic
Triangular Numbers, Patterns, and Optimization of Moves
Coin Conjoin Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside down by moving one coin at a time. How many moves are needed? What is the minimum number of moves? A triangle of 3 coins can be inverted (turned upside down) with a single move, and a triangle of 6 coins can be inverted by moving 2 coins. The 10-coin triangle can be flipped with just 3 moves; did you figure out how? Find out the minimum possible moves needed to flip the next bigger triangle having 15 coins. Try the same for bigger triangular numbers. Is there a simple way to calculate the minimum number of coin moves needed for any such triangular arrangement?
🎯 Aim of the Activity
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To explore triangular number arrangements using coins
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To develop logical reasoning and spatial visualization
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To find the minimum number of moves required to transform a pattern
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To identify patterns and general rules for larger numbers
🧰 Materials Required
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10, 15 or more identical coins
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Flat surface or chart paper
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Pencil and notebook
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Ruler (optional)
🧠 Prior Knowledge
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Counting numbers
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Understanding triangular arrangements
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Basic reasoning and pattern recognition
🧩 Activity Description (Given Situation)
Coins are arranged in the shape of a triangle.
The challenge is to turn the triangle upside down by moving one coin at a time, using the minimum number of moves.
🪜 Procedure / Steps
Step 1: Forming the Triangle
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Arrange 10 coins in a triangular shape (4 rows: 1, 2, 3, 4).
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Observe the orientation (point facing upward).
Step 2: Understanding Smaller Cases
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Try flipping:
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Triangle of 3 coins → 1 move
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Triangle of 6 coins → 2 moves
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Step 3: Flipping the 10-Coin Triangle
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Identify coins that remain in the same position in the inverted triangle.
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Move only the coins that do not overlap with the final shape.
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Count each coin shift as one move.
Step 4: Extending the Pattern
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Repeat the activity for:
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15 coins
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21 coins (optional)
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Record observations.
📊 Observation Table
| Number of Coins | Rows | Moves Needed | Coins That Stay | Coins Moved |
|---|---|---|---|---|
| 3 | 2 | 1 | 2 | 1 |
| 6 | 3 | 2 | 4 | 2 |
| 10 | 4 | 3 | 7 | 3 |
| 15 | 5 | 4 | 11 | 4 |
✅ Answers / Solutions
✔ Solution for 10 Coins
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Minimum number of moves required = 3
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Reason: 7 coins overlap in both triangles
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Only 3 coins need repositioning
✔ Solution for 15 Coins
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Minimum number of moves required = 4
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Coins form 5 rows
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Overlapping coins = 11
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Coins to move = 15 − 11 = 4
🔍 Key Pattern Observed
For a triangle with n rows:
🔑 Minimum number of moves = n − 1
| Rows (n) | Coins | Minimum Moves |
|---|---|---|
| 2 | 3 | 1 |
| 3 | 6 | 2 |
| 4 | 10 | 3 |
| 5 | 15 | 4 |
💭 Reflections (Student Response)
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I learned that not all coins need to be moved.
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Patterns help reduce effort and save time.
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Mathematics helps us find the best solution, not just any solution.
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Logical thinking is more important than trial and error.
🔥 Higher Order Thinking Skills (HOTS)
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Why do some coins remain fixed during inversion?
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Can you predict the answer without physically moving coins?
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How is this activity related to triangular numbers?
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What happens as the number of coins becomes very large?
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Can you design a similar puzzle with square numbers?
📌 Conclusion
The Coin Conjoin activity shows how simple distribution leads to multiplication of ideas.
By observing patterns, students discover a general mathematical rule that reduces effort and increases efficiency.
🧮 Mathematical Link
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Triangular Numbers:
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Optimization & reasoning
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Visual geometry
✨ Extension Activities
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Predict moves for 21 coins
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Draw before–after diagrams
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Convert the rule into a formula
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Create your own coin puzzle
