To develop students’ logical reasoning and problem-solving skills through solving the Binairo puzzle using binary patterns.
Purpose:
To introduce students to binary logic puzzles and help them understand how patterns, reasoning, and deduction are used to complete grids correctly.
Learning Outcomes:
By the end of this activity, students will be able to:
Understand the rules and structure of a Binairo puzzle.
Apply logical reasoning to fill the grid correctly.
Recognize and extend binary patterns (0 and 1).
Improve concentration, patience, and analytical thinking.
Connect the puzzle to real-life applications such as computer binary systems.
Skills Developed:
Logical reasoning, pattern recognition, problem-solving, analytical thinking, and decision-making.
Activity Rules / Guidelines:
Students followed the standard Binairo rules:
Each row and column must contain equal numbers of 0s and 1s.
No more than two same numbers can be adjacent (no three consecutive 0s or 1s).
No two rows or columns can be identical.
All cells must be filled using logic, not guessing.
Procedure:
Students were given the Binairo puzzle from Class 8 NCERT Ganita Prakash – Puzzle Time (Page 178).
The teacher explained the rules and demonstrated one example.
Students worked individually and in pairs to complete the puzzle.
They used logical reasoning and elimination methods to fill the grid.
The final solution was discussed and verified collectively.
Students reflected on strategies used and challenges faced.
Teacher’s Observations:
Students showed great interest in solving the Binairo puzzle. Initially, some students tried guessing, but gradually they understood the importance of logical deduction. Many students became more confident as they progressed and successfully completed the puzzle.
The activity enhanced their patience, concentration, and reasoning skills. Students were excited to learn that computers also use binary numbers (0 and 1).
Student’s Feedback / Reflections:
I really enjoyed solving the Binairo puzzle. At first, it was confusing, but later I understood the pattern and rules. It made me think carefully before filling each box. It was fun and challenging. I learned how logic is important in solving puzzles.
I thank the PM SHRI Scheme for giving me this opportunity to learn mathematics in an interesting and enjoyable way.
— By __________________
Photo Caption:
๐ธ Students actively solving the Binairo Puzzle from Ganita Prakash – Puzzle Time, enhancing logical and binary thinking skills during Math Circle Activity.
Present puzzles like:
“Four friends sit in a row. A is left of B, C is not near D. Find all possible seatings.”
Students use reasoning tables or sketches to test conditions.
Discuss possible arrangements and verify logic collectively.
Extend to circular arrangements or larger groups for advanced learners.
Teacher’s Observations:
Students collaborated enthusiastically, testing multiple arrangements logically. The activity encouraged clear communication and stepwise deduction.
Student’s Reflections:
I enjoyed figuring out who sits where — it felt like solving a mystery with logic! It made me realize how reasoning and order are connected. – By ____________
Photo Caption: Students solving seating arrangement challenges — sharpening logical and sequencing skills!
Objective:
To strengthen understanding of number properties and divisibility rules through a clue-based reasoning game.
Purpose:
To encourage logical deduction and use of mathematical properties to identify hidden numbers using given clues.
Learning Outcomes:
Apply divisibility rules for 2, 3, 5, 9, etc.
Strengthen reasoning through elimination and deduction.
Recognize relationships between digits and number characteristics.
Develop and solve peer-created riddles.
Skills Developed:
Analytical thinking, divisibility, logical deduction, creative problem design.
Procedure:
Present riddles such as:
“I’m a two-digit number. The sum of my digits is 9. I’m divisible by 3. Who am I?”
Students reason step-by-step to narrow down possible numbers.
Once familiar, students create their own number riddles for peers to solve.
Discuss different solving strategies (digit sum, factors, multiples).
Teacher’s Observations:
Students were highly engaged in reasoning out the correct answers. They creatively formed riddles that used multiple number properties together.
Student’s Reflections:
I felt like a detective solving puzzles! It helped me recall divisibility rules and think logically. Making my own riddle was the best part! – By ____________
Photo Caption: Students acting as Number Detectives — solving and creating riddles using math clues!
Present a story-based puzzle: e.g., “The thief’s locker number is a multiple of 3, sum of digits = 9, between 20–40.”
Guide students to test numbers logically to find the correct answer.
Invite them to create their own detective clues for peers.
Teacher’s Observations:
Students loved the detective theme and applied math concepts intuitively. Several created unique clues and exchanged puzzles enthusiastically.
Student’s Reflections:
It was exciting to be a math detective! I used reasoning to find the right number and felt like solving a real mystery. – By ____________
Photo Caption: Students acting as math detectives, solving numeric mysteries with logic and teamwork!
Students solve “across” and “down” clues like a crossword but with number-based answers.
Discuss strategies for using intersecting clues to confirm results.
Encourage teamwork for checking accuracy.
Teacher’s Observations:
Students collaborated effectively and enjoyed the puzzle-solving challenge. It boosted both confidence and fluency.
Student’s Reflections:
This was like solving a crossword, but with numbers! I learned to connect operations and think carefully before writing each answer. – By ____________
Photo Caption: Students working on Crossnumber puzzles – blending words and numbers creatively!
Present riddles like: “Think of a number, double it, add 6, halve it, subtract your original number.”
Guide students to express steps algebraically and simplify.
Encourage them to design their own “magic” riddles.
Teacher’s Observations:
Students were amazed to find that the result remains constant. They began forming their own riddles and reasoning algebraically.
Student’s Reflections:
I enjoyed creating my own math riddle! It made algebra feel like a mystery to solve. I understood how variables can represent any number. – By ____________
Photo Caption: Students exploring algebraic magic through creative number riddles!
Objective:
To help students identify and understand prime numbers through an engaging and competitive number hunt.
Purpose:
To reinforce the concept of prime numbers, promote quick number recognition, and spark curiosity about patterns and gaps between primes.
Learning Outcomes:
Identify prime and composite numbers up to 100.
Understand prime gaps and their irregular distribution.
Develop number sense and analytical reasoning.
Recognize the importance of primes in number theory and cryptography.
Skills Developed:
Prime recognition, mental computation, logical analysis, pattern discovery.
Procedure:
Students write numbers from 1 to 100 in a grid format.
They circle all the prime numbers as quickly as possible.
After completion, compare times and discuss patterns such as consecutive primes and prime gaps.
Reflect on why certain numbers (like even numbers > 2) are never prime.
Teacher’s Observations:
Students were enthusiastic and competitive, enhancing engagement. They quickly recognized prime patterns and discussed irregular prime gaps. Some connected the concept to divisibility rules.
Student’s Reflections:
This was a fun and fast-paced activity! I learned how prime numbers become rarer as numbers increase. Competing made it exciting, and I noticed cool patterns between primes. – By ____________
Photo Caption: Students participating in the Prime Number Race — circling primes up to 100 with focus and fun!
complete solutions forNCERT Class 9 Maths Ganita Manjari Chapter 3: The World of Numbers, covering all exercise sets, in-text questions, examples, and end-of-chapter exercises with full explanations.
NCERT Class 9 Maths Ganita Manjari Chapter 3 Solutions
Chapter Overview: The World of Numbers
This chapter traces the historical evolution of numbers—from ancient counting methods (like the Ishango Bone and finger counting) to the formal classification of numbers into Natural, Whole, Integer, Rational, Irrational, and Real numbers. It highlights India’s contribution through Brahmagupta (who formalised zero and negative numbers) and introduces concepts like density of rational numbers, proof by contradiction for irrationals, and decimal expansions.
Exercise Set 3.1 Solutions
1. A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?
Solution:
Given: 2 bags = 15 ingots
So, 1 bag = 15/2 ingots
For 12 bags = 12 × (15/2) = 6 × 15 = 90 ingots
Explanation: This is a unitary method problem. The key is finding the value of one unit (1 bag) and then multiplying by the required quantity.
2. Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.
Solution:
These numbers are prime numbers (numbers with exactly two factors: 1 and the number itself).
Next three primes after 19: 23, 29, 31
Explanation: The Ishango bone (c. 20,000 BCE) is one of the oldest known mathematical artefacts. Its columns contain prime numbers, suggesting early understanding of primality.
3. We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
Solution: Natural numbers are NOT closed under subtraction.
Examples:
5 − 3 = 2 (Natural number) ✓
3 − 5 = −2 (Not a natural number) ✗
Explanation: Closure means performing an operation on two numbers from a set always gives a result inside the same set. Since subtraction can yield negative numbers (which aren't natural numbers), the set fails closure.
4. Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?
Solution:
Fingers used for counting (excluding thumb) = 4 fingers
Joints per finger = 3
Total count on one hand = 4 × 3 = 12
Relation: Counting up to 12 on one hand naturally led to base-12 (duodecimal) systems, which explain why we have terms like "dozen" (12) and "gross" (144 = 12²).
Exercise Set 3.2 Solutions (Integers & Brahmagupta's Laws)
1. The temperature in Ladakh is recorded as 4°C at noon. By midnight, it drops by 15°C. What is the midnight temperature?
Solution:
Initial temperature = 4°C
Drop = 15°C (subtract)
Midnight temperature = 4 − 15 = −11°C
2. A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Calculate his final financial standing using integers.
Solution:
Debt = −850 (negative)
Profit = +1200 (positive)
Loss = −450 (negative)
Equation: (−850) + 1200 + (−450)
Step 1: −850 + 1200 = 350
Step 2: 350 − 450 = −100
Final standing: −₹100 (a loss of ₹100)
Explanation: This uses Brahmagupta's concept of representing fortunes as positive and debts as negative numbers.
3. Calculate the following using Brahmagupta's laws (Debt = Negative, Fortune = Positive):
Expression
Law Applied
Answer
(i) (−12) × 5
Negative × Positive = Negative
−60
(ii) (−8) × (−7)
Negative × Negative = Positive
56
(iii) 0 − (−14)
Zero minus debt = Fortune
14
(iv) (−20) ÷ 4
Negative ÷ Positive = Negative
−5
4. Explain, using a real-world example of debt, why subtracting a negative number equals adding a positive number.
Solution:
Suppose you have ₹10 and you owe ₹5 (i.e., you have −5 debt)
The expression 10 − (−5) means: "remove a debt of ₹5"
If the debt is cancelled/removed, your effective money increases by ₹5
So, 10 − (−5) = 10 + 5 = 15
Thus: Subtracting a negative = Adding a positive.
Exercise Set 3.3 Solutions (Rational Numbers)
1. Prove that the following rational numbers are equal:
Pair
Simplification
Conclusion
(i) 2/3 and 4/6
4/6 = (4÷2)/(6÷2) = 2/3
Equal
(ii) 5/4 and 10/8
10/8 = 5/4
Equal
(iii) −3/5 and −6/10
−6/10 = −3/5
Equal
(iv) 9/3 and 3
9/3 = 3
Equal
2. Find the sum:
Expression
LCM
Solution
(i) 2/5 + 3/10
10
4/10 + 3/10 = 7/10
(ii) 7/12 + 5/8
24
14/24 + 15/24 = 29/24
(iii) −4/7 + 3/14
14
−8/14 + 3/14 = −5/14
3. Find the difference:
Expression
LCM
Solution
(i) 5/6 − 1/4
12
10/12 − 3/12 = 7/12
(ii) 11/8 − 3/4
8
11/8 − 6/8 = 5/8
Explanation: For addition/subtraction of rational numbers, always find the LCM of denominators, convert to equivalent fractions, then operate on numerators.
In-Text Questions (Think & Reflect)
Q: Are all integers rational numbers? Justify.
Solution: Yes, all integers are rational numbers.
Justification: An integer n can be written as 1n, which is of the form qp with q=0. For example:
5 = 5/1
−3 = −3/1
0 = 0/1
Q: Between any two rational numbers, how many rational numbers exist?
Solution: Infinitely many rational numbers exist between any two distinct rational numbers.
Example: Between 1/2 = 0.5 and 3/4 = 0.75:
5/8 = 0.625
11/16 = 0.6875
9/16 = 0.5625 And countless more. This property is called density of rational numbers.
Q: Is √2 rational or irrational? Prove.
Solution: √2 is irrational.
Proof by contradiction:
Assume √2 is rational = p/q in simplest form (p, q integers, q ≠ 0, no common factors)
Squaring: 2 = p²/q² → p² = 2q²
So p² is even ⇒ p is even ⇒ p = 2k
Substitute: (2k)² = 2q² → 4k² = 2q² → 2k² = q²
So q² is even ⇒ q is even
This means p and q are both even ⇒ they have common factor 2
This contradicts our assumption that p/q was in simplest form
Therefore, √2 cannot be rational. Hence, it's irrational.
Q: Classify the decimal expansions:
0.375
0.333...
0.1010010001...
Solution:
Decimal
Type
Reason
0.375
Terminating
Ends after finite digits = 375/1000
0.333...
Non-terminating repeating
1/3 in decimal form
0.1010010001...
Non-terminating non-repeating
No pattern repetition → Irrational
Solved Examples from Chapter
Example: Locate √2 on the number line.
Solution:
Draw a number line. Mark O (0) and A (1).
At A, draw perpendicular AB of length 1 unit.
Join OB. By Pythagoras: OB = √(1² + 1²) = √2
Using compass, draw an arc with centre O and radius OB to intersect the number line at point P.
OP = √2. Point P represents √2.
Example: Simplify 232+52
Solution:
32+52=82
282=8
End of Chapter Exercises
1. Short Answer Type
Q1: Write three rational numbers between 2 and 3.
Solution: 2.1, 2.5, 2.75 (or as fractions: 21/10, 5/2, 11/4)
Q2: Identify as rational or irrational: (i) √9 = 3 → Rational (ii) 2√3 → Irrational (√3 is irrational) (iii) 0.121221222... → Irrational (non-terminating non-repeating)
Q3: Express 0.999... in p/q form.
Solution: Let x = 0.999... 10x = 9.999... Subtract: 10x − x = 9.999... − 0.999... → 9x = 9 → x = 1
Thus, 0.999... = 1 (This shows two different decimal representations for the same number!)
2. Long Answer Type
Q4: Prove that 3 is irrational.
Solution:
Assume √3 = p/q (simplest form)
Squaring: 3 = p²/q² → p² = 3q²
So p² is divisible by 3 ⇒ p is divisible by 3 ⇒ p = 3k
Substituting: (3k)² = 3q² → 9k² = 3q² → 3k² = q²
So q² is divisible by 3 ⇒ q is divisible by 3
Both p and q divisible by 3 ⇒ contradiction to simplest form assumption
Hence, √3 is irrational.
Q5: Represent 5 on the number line.
Solution:
Mark O (0) and A (2) on number line
At A, draw perpendicular AB = 1 unit
Join OB. OB = √(2² + 1²) = √4 + 1 = √5
Draw arc with centre O and radius OB to intersect number line at P
OP = √5
3. Multiple Choice Questions (MCQs)
Q1: Which of the following is an irrational number?
