Subject Enrichment Activity – Mathematics (Class 8) proportional reasoning

 

SUBJECT ENRICHMENT ACTIVITY – MATHEMATICS (CLASS 8)

(Ganita Prakash – Page 178)

Title of the Activity:

🧩 Binaro – A Logic Puzzle Using Proportional Reasoning

Binairo, also known as Takuzu, is a logic puzzle with simple rules. Binairo is generally played on a square grid with no particular size. Some cells start out filled with two symbols: here horizontal and vertical lines. The rest of the cells are empty. The task is to fill cells in such a way that: 1. Each row and each column must contain an equal number of horizontal and vertical lines. 2. More than two horizontal or vertical lines can’t be adjacent. 3. Each row is unique. Each column is unique. 

Puzzle: 

Solution 

 Solve the following Binairo puzzles: 

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Topic:

Proportional Reasoning & Logical Thinking (Patterns, Equality, Constraints)

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Aim:

To develop logical reasoning, proportional thinking, and pattern recognition skills by solving a Binaro puzzle using given mathematical rules.

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Learning Outcomes:

Students will be able to:

• Apply proportional reasoning (equal numbers in rows/columns)

• Identify and extend patterns logically

• Follow constraints systematically

• Improve problem-solving and critical thinking skills

• Work collaboratively and explain reasoning

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Materials Required:

• Ganita Prakash textbook

• Pencil and eraser

• Worksheet / notebook

• Ruler

• Coloured pencils (optional, for marking patterns)

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Introduction (Concept Overview):

Binaro (also called Takuzu) is a logic-based puzzle played on a square grid.
The grid is filled using two symbols only:

• Horizontal line (—)

• Vertical line (|)

The challenge is to fill the grid by following specific rules, ensuring balance, uniqueness, and logical consistency.

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Rules of the Binaro Puzzle:

1. Equal Proportion Rule:
   Each row and each column must contain an equal number of horizontal and vertical lines.

2. Adjacency Rule:
   No more than two identical lines (— or |) can be placed adjacent to each other horizontally or vertically.

3. Uniqueness Rule:

   • No two rows can be exactly the same.

   • No two columns can be exactly the same.

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Procedure:

1. Observe the partially filled grid carefully.

2. Count the number of cells in each row and column.

3. Since the grid is even-sized, divide the total number of cells by 2 to know how many of each symbol are needed.

4. Use the adjacency rule to avoid placing three identical symbols in a row or column.

5. Check row and column uniqueness after placing each symbol.

6. Continue until the grid is completely filled correctly.

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Observation Table:

Observation	Mathematical Reasoning
Equal symbols in each row	Proportional reasoning
No three identical symbols together	Pattern recognition
Rows are unique	Logical differentiation
Columns are unique	Combinatorial reasoning

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Solved Answers / Solutions (Explanation-Based):

Puzzle 1 (Left Grid): 

• Each row has 6 cells → 3 horizontal and 3 vertical lines.

• Missing symbols filled by checking adjacency.

• Final grid satisfies all three rules.

✔ Solution Valid:

• Equal count ✔

• No three adjacent ✔

• Rows & columns unique ✔

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Puzzle 2 (Middle Grid):

• Several rows already had two consecutive symbols.

• Third symbol forced to be opposite to avoid violation.

• Final solution balanced proportionally.

✔ Solution Valid

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Puzzle 3 (Right Grid):

• Columns used first to determine missing symbols.

• Uniqueness rule helped avoid repeating patterns.

• Final grid completed logically.

✔ Solution Valid

(Teacher Note: The solutions shown in the textbook image are correct and follow all three rules.)

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Result / Conclusion:

The Binaro puzzle can be solved successfully by applying proportional reasoning, logical constraints, and systematic checking. Each correct solution maintains balance, avoids repetition, and respects adjacency rules.

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Reflection (Student Responses – Sample):

• “I learned how proportions can be applied even without numbers.”

• “The uniqueness rule made the puzzle challenging but interesting.”

• “Logical elimination helped me decide which symbol to place.”

• “This activity improved my patience and thinking skills.”

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Higher Order Thinking Skills (HOTS):

Q1. Why must the grid size be even in a Binaro puzzle?

Answer:
Because each row and column must have equal numbers of two symbols. An odd number cannot be divided equally.

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Q2. What would happen if we allowed three symbols instead of two?

Answer:
The puzzle would become more complex. Proportional reasoning would involve ratios like 1:1:1, and uniqueness rules would be harder to maintain.

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Q3. Can Binaro be related to real-life situations?

Answer:
Yes. Examples include:

• Balanced team formation

• Timetable scheduling

• Computer binary systems (0 and 1)

• Resource allocation problems

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Q4. How is Binaro connected to proportional reasoning?

Answer:
Each row and column maintains a fixed ratio (1:1) between two symbols, which is the core idea of proportional reasoning.

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Extension Activity:

• Design your own 4×4 or 6×6 Binaro puzzle.

• Exchange puzzles with classmates and solve them.

• Try replacing symbols with colours or shapes.

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Assessment :

• ✔ Logical reasoning

• ✔ Accuracy of solution

• ✔ Explanation of steps

• ✔ Participation and collaboration

TEACHER ANSWER KEY

🔹 Explanation of Logic

• Each grid has an even number of cells, so symbols are split equally.

• If two identical symbols appear consecutively, the third must be opposite.

• If a row is close to completion, missing symbols are deduced proportionally.

• Uniqueness rule ensures only one valid solution.

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🔹 Final Solutions (as per textbook)

Puzzle 1 – Completed Grid

✔ Equal number of horizontal and vertical lines in every row and column
✔ All rows and columns unique
✔ No adjacency violations

(Matches the “Solution” grid shown in the textbook)

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Puzzle 2 – Completed Grid

✔ Proportional distribution maintained
✔ Logical placement using elimination method

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Puzzle 3 – Completed Grid

✔ All constraints satisfied
✔ No repeated rows or columns

Note for Teachers:
Students’ answers should exactly match the textbook solution grid. Any deviation violates at least one rule.

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🔟 Assessment Criteria (Teacher Use)

Criteria	Marks
Correct application of rules	4
Logical reasoning steps	3
Accuracy of final solution	2
Neatness & presentation	1
Total	10 Marks

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1️⃣1️⃣ Conclusion

This activity effectively integrates proportional reasoning with logical deduction, encouraging students to think mathematically beyond calculations. It strengthens analytical skills essential for higher mathematics.