Class 7 : Maths – Lesson 6. Number Play

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On December 11, 2025

EXPLANATION AND ANALYSIS

🔵 Introduction: Playing with Numbers

🧠 Mathematics is not only about calculations; it is also about patterns, tricks, and interesting properties of numbers. Many times, numbers behave in surprising ways. When we explore these behaviors, we discover new ideas and enjoy learning mathematics more deeply.

🌿 This chapter, Number Play, helps us observe numbers carefully, recognize patterns, and understand how numbers interact with each other in different situations.

🔵 Guessing numbers
🟢 Finding hidden patterns
🟡 Exploring rules
🔴 Enjoying mathematical tricks

This lesson develops curiosity and logical thinking through numbers.

🟢 Observing Patterns in Numbers

🧠 A pattern is a regular and repeated way in which numbers are arranged.

🔹 Patterns help us predict the next number
🔹 They help us understand number relationships

📌 Example
2, 4, 6, 8, 10

🔹 Each number increases by 2
🔹 This is an even-number pattern

💡 Concept:
Recognising patterns is the first step in number play.

🔵 Number Sequences

🧠 A number sequence is an ordered list of numbers that follow a rule.

🔹 Ascending sequence: numbers increase
🔹 Descending sequence: numbers decrease

📌 Example
5, 10, 15, 20

🔹 Rule: add 5 each time

✏️ Note:
Finding the rule helps us continue the sequence correctly.

🟢 Playing with Digits of Numbers

🧠 Sometimes, we form new numbers by rearranging or operating on digits.

🔹 Adding digits
🔹 Reversing digits
🔹 Comparing digit sums

📌 Example
Number: 34
🔹 Sum of digits = 3 + 4 = 7

💡 Concept:
Digit-based observations reveal hidden properties of numbers.

🔵 Even and Odd Number Patterns

🧠 Even and odd numbers follow interesting patterns.

🔹 Even + Even = Even
🔹 Odd + Odd = Even
🔹 Even + Odd = Odd

📌 Example
6 + 8 = 14 (Even)
5 + 7 = 12 (Even)

✏️ Note:
Understanding these patterns helps in quick checking of answers.

🟢 Multiples and Their Patterns

🧠 Multiples of a number are obtained by multiplying it by whole numbers.

📌 Example
Multiples of 4
4, 8, 12, 16, 20

🔹 All multiples of 4 are even

💡 Concept:
Multiples show repeating behavior that helps in number games.

🔵 Divisibility Observations

🧠 Divisibility rules help us know whether a number can be divided by another number without remainder.

🔹 A number divisible by 2 ends in 0, 2, 4, 6, or 8
🔹 A number divisible by 5 ends in 0 or 5

📌 Example
245 ends in 5, so it is divisible by 5

✏️ Note:
Divisibility rules make calculations faster.

🟢 Magic with Numbers

🧠 Some number operations always lead to surprising results.

📌 Example
Think of a number
🔹 Multiply it by 2
🔹 Add 10
🔹 Divide by 2
🔹 Subtract the original number

🔹 Result = 5

💡 Concept:
Such number tricks work because of fixed mathematical rules.

🔵 Patterns in Addition and Subtraction

🧠 Addition and subtraction also show patterns.

📌 Example
9 + 1 = 10
19 + 1 = 20
29 + 1 = 30

🔹 Adding 1 changes the place value pattern

✏️ Note:
Observing place values helps understand number behavior.

🟢 Square Numbers and Patterns

🧠 Square numbers are obtained by multiplying a number by itself.

📌 Example
1² = 1
2² = 4
3² = 9
4² = 16

🔹 Square numbers increase faster than natural numbers

💡 Concept:
Square patterns appear in shapes and arrangements.

🔴 Common Mistakes to Avoid

🔴 Ignoring the pattern rule
🔴 Jumping to conclusions without checking
🔴 Mixing different number properties
🔴 Forgetting basic operations

✏️ Note:
Always verify the rule with more than one example.

🟢 Importance of Number Play

🧠 Number play helps students to:

🔹 Develop logical thinking
🔹 Improve observation skills
🔹 Enjoy mathematics
🔹 Build strong number sense

This chapter strengthens the foundation for algebra and problem solving.

📘 Summary

🔵 Number play involves patterns and observations
🟢 Sequences follow fixed rules
🟡 Digits of numbers show properties
🔴 Even, odd, multiples, and squares form patterns
🔵 Divisibility rules simplify work
🟢 Number tricks are based on logic

📝 Quick Recap

📝 Quick Recap
🔵 Numbers follow patterns
🟢 Finding rules helps predict numbers
🟡 Digits reveal number properties
🔴 Number play builds logical thinking
🔵 Mathematics becomes fun and meaningful

——————————————————————————————————————————————————————————————————————————–

TEXTBOOK QUESTIONS

🔵 NUMBERS TELL US THINGS

🔒 ❓ 1. Write down the number each child should say based on this rule for the arrangement shown.

📌 ✅ Answer:

🟢 Step 1: Total children = 7

🟢 Step 2: Count from left to right

🟩 1st child → No one in front → says 0

🟦 2nd child → No taller in front → says 0

🟨 3rd child → One taller in front → says 1

🟥 4th child → No taller in front → says 0

🟪 5th child → Three taller in front → says 3

🟫 6th child → Two taller in front → says 2

⬜ 7th child → Four taller in front → says 4

✔️ Final Sequence:
0, 0, 1, 0, 3, 2, 4

🔒 ❓ 2. Arrange the stick figure cutouts such that the sequence reads:

🔒 ❓ (a) 0, 1, 1, 2, 4, 1, 5

📌 ✅ Answer:

🟢 Step 1: Largest number = 5
So at least 6 taller children must be before that child.

🟢 Step 2: Total children = 7

🟢 Step 3: One possible height order (left → right, taller = bigger number):
7, 2, 3, 4, 6, 1, 5

✔️ Arrangement Possible

🔒 ❓ (b) 0, 0, 0, 0, 0, 0

📌 ✅ Answer:

🟢 All say 0 → no one has taller in front.

🟢 This happens when heights are strictly decreasing from left to right.

Example order:
6, 5, 4, 3, 2, 1

✔️ Arrangement Possible

🔒 ❓ (c) 0, 1, 2, 3, 4, 5, 6

📌 ✅ Answer:

🟢 Each next child has one more taller in front.

🟢 This happens when heights are strictly increasing.

Example order:
1, 2, 3, 4, 5, 6, 7

✔️ Arrangement Possible

🔒 ❓ (d) 0, 1, 0, 1, 0, 1, 0

📌 ✅ Answer:

🟢 Alternating pattern required.

🟢 One possible order:
7, 1, 6, 2, 5, 3, 4

✔️ Arrangement Possible

🔒 ❓ (e) 0, 1, 1, 1, 1, 1

📌 ✅ Answer:

🟢 First child must be tallest.

🟢 All others must have exactly one taller in front (that tallest child).

Example order:
6, 1, 2, 3, 4, 5

✔️ Arrangement Possible

🔒 ❓ (f) 0, 0, 0, 3, 3, 3

📌 ✅ Answer:

🟢 First three must be tallest three in decreasing order.

🟢 Next three must be shortest three in any order.

Example order:
6, 5, 4, 1, 2, 3

✔️ Arrangement Possible

🔒 ❓ 3. Identify whether the following statements are Always True, Sometimes True, or Never True.

🔒 ❓ (a) If a person says ‘0’ then they are the tallest in the group.

📌 ✅ Answer:

🟢 Saying 0 means no taller in front.
🟢 There may be taller people behind.

✔️ Sometimes True

🔒 ❓ (b) If a person is the tallest, then their number is ‘0’.

📌 ✅ Answer:

🟢 No one is taller anywhere.

✔️ Always True

🔒 ❓ (c) The first person’s number is ‘0’.

📌 ✅ Answer:

🟢 No one stands in front of the first person.

✔️ Always True

🔒 ❓ (d) If a person is not first or last in line, then they cannot say ‘0’.

📌 ✅ Answer:

🟢 In increasing order arrangement, many middle people say 0.
🟢 In decreasing order, only first says 0.

✔️ Sometimes True

🔒 ❓ (e) The person who calls out the largest number is the shortest.

📌 ✅ Answer:

🟢 Usually the largest number is said by someone with many taller in front.
🟢 But if all numbers are 0 (increasing order), this fails.

✔️ Sometimes True

🔒 ❓ (f) What is the largest number possible in a group of 8 people?

📌 ✅ Answer:

🟢 The shortest person at the last position will have 7 taller in front.

✔️ Final Answer: 7

🔵 PICKING PARITY

🔒 ❓ 1. Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:

🔒 ❓ (a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd)

📌 ✅ Answer:

🟢 Step 1: Add the even numbers
🟩 even + even = even

🟢 Step 2: Add the odd numbers
🟦 odd + odd = even

🟢 Step 3: Add the two results
🟨 even + even = even

✔️ Final Answer: Even

🔒 ❓ (b) Sum of 2 odd numbers and 3 even numbers

📌 ✅ Answer:

🟢 Step 1: Add the two odd numbers
🟩 odd + odd = even

🟢 Step 2: Add three even numbers
🟦 even + even = even
🟦 even + even = even

🟢 Step 3: Add the results
🟨 even + even = even

✔️ Final Answer: Even

🔒 ❓ (c) Sum of 5 even numbers

📌 ✅ Answer:

🟢 Step 1: Add two even numbers
🟩 even + even = even

🟢 Step 2: Keep adding even numbers
🟦 even + even = even
🟦 even + even = even

✔️ Final Answer: Even

🔒 ❓ (d) Sum of 8 odd numbers

📌 ✅ Answer:

🟢 Step 1: Pair the odd numbers
🟩 odd + odd = even

🟢 Step 2: 8 odd numbers make 4 pairs
🟦 each pair = even

🟢 Step 3: Add all even results
🟨 even + even = even

✔️ Final Answer: Even

🔒 ❓ 2. Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. He calculated the total and got ₹205. Did he make a mistake? If he did, explain why. If he didn’t, how many coins of each type could he have?

📌 ✅ Answer:

🟢 Step 1: Understand parity of each part

🟩 Odd number of ₹1 coins → total from ₹1 coins = odd

🟩 Odd number of ₹5 coins
Since 5 is odd,
odd × odd = odd

🟩 Even number of ₹10 coins
10 is even,
even × even = even

🟢 Step 2: Add all contributions

odd + odd = even
even + even = even

So total must be EVEN

🟢 Step 3: Given total = 205

205 is ODD

✔️ Conclusion:
He made a mistake because the total must be even, not odd.

So ₹205 is not possible.

🔒 ❓ 3. We know that:
(a) even + even = even
(b) odd + odd = even
(c) even + odd = odd

Similarly, find out the parity for the scenarios below:

🔒 ❓ (d) even − even = _______

📌 ✅ Answer:

🟢 Example: 8 − 4 = 4
Both are even
Result = even

✔️ Final Answer: Even

🔒 ❓ (e) odd − odd = _______

📌 ✅ Answer:

🟢 Example: 9 − 5 = 4
Both are odd
Result = even

✔️ Final Answer: Even

🔒 ❓ (f) even − odd = _______

📌 ✅ Answer:

🟢 Example: 8 − 3 = 5
Result = odd

✔️ Final Answer: Odd

🔒 ❓ (g) odd − even = _______

📌 ✅ Answer:

🟢 Example: 9 − 4 = 5
Result = odd

✔️ Final Answer: Odd

🔵 SOME EXPLORATIONS IN GRIDS

Section: Figure it Out (Magic Squares)

🔒 ❓ 1. How many different magic squares can be made using the numbers 1 – 9?

📌 ✅ Answer:

🔹 A 3 × 3 magic square using 1–9 is unique in structure.
🔹 Only one fundamental arrangement exists (called the Lo Shu square).
🔹 All other forms are just rotations or reflections of this same square.
🔹 Total distinct arrangements including rotations and reflections = 8.

🔹 Example of the basic magic square:

8 1 6
3 5 7
4 9 2

🔹 Magic sum = 15

✔️ Final Answer: 8 different magic squares (including rotations and reflections).

🔒 ❓ 2. Create a magic square using the numbers 2 – 10. What strategy would you use for this? Compare it with the magic squares made using 1 – 9.

📌 ✅ Answer:

🔹 Strategy: Start with the 1–9 magic square.
🔹 Add 1 to every number.

🔹 Since each row originally sums to 15, adding 1 to each of the 3 numbers increases the sum by 3.
🔹 New magic sum = 15 + 3 = 18.

🔹 Resulting magic square:

9 2 7
4 6 8
5 10 3

🔹 All rows, columns and diagonals now sum to 18.

🔹 Comparison:
🔸 Structure remains same.
🔸 Only numbers shift upward by 1.
🔸 Magic sum increases by 3.

🔒 ❓ 3. Take a magic square, and

🔒 ❓ (a) increase each number by 1

📌 ✅ Answer:

🔹 Let original magic sum = S.
🔹 Each row has 3 numbers.
🔹 Increasing each number by 1 increases each row sum by 3.

🔹 New magic sum = S + 3.
🔹 The grid remains a magic square.

🔒 ❓ (b) double each number

📌 ✅ Answer:

🔹 If every number is multiplied by 2, each row sum also doubles.
🔹 New magic sum = 2S.
🔹 The grid remains a magic square.

🔒 ❓ In each case, is the resulting grid also a magic square? How do the magic sums change?

📌 ✅ Answer:

🔹 Yes, the resulting grid remains a magic square in both cases.
🔹 Adding 1 to each number increases the magic sum by 3.
🔹 Doubling each number doubles the magic sum.

🔒 ❓ 4. What other operations can be performed on a magic square to yield another magic square?

📌 ✅ Answer:

🔹 Rotate the square (90°, 180°, 270°).
🔹 Reflect the square (horizontal or vertical mirror).
🔹 Add the same constant to every number.
🔹 Multiply every number by the same constant.

🔹 These operations preserve equal row, column and diagonal sums.

🔒 ❓ 5. Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).

📌 ✅ Answer:

🔹 Start with the standard 1–9 magic square.
🔹 Suppose the consecutive numbers start from n.
🔹 Add (n – 1) to each entry of the 1–9 magic square.

🔹 New magic sum becomes:
15 + 3(n – 1)

🔹 The structure remains unchanged.
🔹 This method works for any 9 consecutive numbers.

Section: Figure it Out (Generalised Magic Square)

🔒 ❓ 1. Using this generalised form, find a magic square if the centre number is 25.

📌 ✅ Answer:

🔹 In a 3 × 3 magic square made from consecutive numbers, the centre number is the average of all 9 numbers.
🔹 For the standard form (1–9), centre = 5.

🔹 If centre = 25, then all numbers are increased by (25 − 5) = 20.

🔹 So add 20 to each entry of the standard magic square:

Standard square:

8 1 6
3 5 7
4 9 2

Add 20:

28 21 26
23 25 27
24 29 22

🔹 Magic sum = 28 + 21 + 26 = 75

✔️ Required magic square formed.

🔒 ❓ 2. What is the expression obtained by adding the 3 terms of any row, column or diagonal?

📌 ✅ Answer:

🔹 Let the centre number be x.
🔹 In a 3 × 3 magic square formed using consecutive numbers, each row contains three numbers symmetrically placed around x.

🔹 The total of any row, column or diagonal = 3x.

✔️ Magic sum = 3x.

🔒 ❓ 3. Write the result obtained by—

🔒 ❓ (a) adding 1 to every term in the generalised form.

📌 ✅ Answer:

🔹 If each of the 9 entries increases by 1,
🔹 Each row increases by 3 (since each row has 3 numbers).

🔹 New magic sum = 3x + 3.

✔️ Still a magic square.

🔒 ❓ (b) doubling every term in the generalised form.

📌 ✅ Answer:

🔹 If each entry is multiplied by 2,
🔹 Each row sum is also multiplied by 2.

🔹 New magic sum = 2(3x) = 6x.

✔️ Still a magic square.

🔒 ❓ 4. Create a magic square whose magic sum is 60.

📌 ✅ Answer:

🔹 Magic sum = 3x
🔹 So 3x = 60
🔹 x = 20

🔹 Use the generalised form with centre 20.
🔹 Since 5 is centre in standard square, increase each entry by (20 − 5) = 15.

Standard square:

8 1 6
3 5 7
4 9 2

Add 15:

23 16 21
18 20 22
19 24 17

🔹 Check: 23 + 16 + 21 = 60

✔️ Magic square formed.

🔒 ❓ 5. Is it possible to get a magic square by filling nine non-consecutive numbers?

📌 ✅ Answer:

🔹 Yes, it is possible.
🔹 A magic square only requires equal sums in rows, columns and diagonals.
🔹 The numbers need not be consecutive.
🔹 By adding or multiplying a constant to a known magic square, new sets of numbers can be created which may not be consecutive.

✔️ Yes, possible.

🔵 DIGITS IN DISGUISE

🔒 ❓ 1. A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?

📌 ✅ Answer:
🔹 The bulb starts ON.
🔹 Each toggle changes the state (ON → OFF or OFF → ON).
🔹 After 1 toggle → OFF.
🔹 After 2 toggles → ON.
🔹 So after an even number of toggles → ON.
🔹 After an odd number of toggles → OFF.
🔹 77 is odd.
🔹 Therefore, the bulb will be OFF.

🔒 ❓ 2. Liswini has 50 loose sheets, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?

📌 ✅ Answer:
🔹 Each sheet has 2 consecutive page numbers (like 1 and 2, 3 and 4, etc.).
🔹 The sum of two consecutive numbers is always odd.
🔹 50 sheets → 50 odd sums added together.
🔹 The sum of 50 odd numbers:
🔸 Even number of odd numbers gives an even result.
🔹 So total sum must be even.
🔹 6000 is even.
🔹 Therefore, it is possible.

🔒 ❓ 3. Fill the 6 boxes with 3 odd numbers (‘o’) and 3 even numbers (‘e’) to satisfy the given row and column parity.

📌 ✅ Answer:
🔹 From the picture:
🔸 First row sum = odd.
🔸 Second row sum = even.
🔸 First column sum = even.
🔸 Second column sum = even.
🔸 Third column sum = odd.

🔹 One correct arrangement:

Row 1: o e o
Row 2: e o e

🔹 Check:
🔸 Row 1: o + e + o = even? No → odd (correct).
🔸 Row 2: e + o + e = even (correct).
🔸 Column 1: o + e = odd? No → even (correct).
🔸 Column 2: e + o = odd? No → even (correct).
🔸 Column 3: o + e = odd (correct).

🔒 ❓ 4. Make a 3 × 3 magic square with 0 as the magic sum. All numbers cannot be zero.

📌 ✅ Answer:
🔹 Magic sum = 0.
🔹 Each row, column, diagonal must total 0.
🔹 One example:

2 -1 -1
-1 0 1
-1 1 0

🔹 Check first row: 2 −1 −1 = 0.
🔹 All rows, columns and diagonals sum to 0.

🔒 ❓ 5. Fill in the blanks with ‘odd’ or ‘even’:

🔒 ❓ (a) Sum of an odd number of even numbers is ______

📌 ✅ Answer:
🔹 Even + even = even.
🔹 Adding any number of even numbers always gives even.
🔹 So answer = even.

🔒 ❓ (b) Sum of an even number of odd numbers is ______

📌 ✅ Answer:
🔹 odd + odd = even.
🔹 Even count of odd numbers gives even.
🔹 So answer = even.

🔒 ❓ (c) Sum of an even number of even numbers is ______

📌 ✅ Answer:
🔹 Even numbers always give even when added.
🔹 So answer = even.

🔒 ❓ (d) Sum of an odd number of odd numbers is ______

📌 ✅ Answer:
🔹 odd + odd = even.
🔹 even + odd = odd.
🔹 So final answer = odd.

🔒 ❓ 6. What is the parity of the sum of the numbers from 1 to 100?

📌 ✅ Answer:
🔹 Sum of first n natural numbers = n(n + 1) / 2.
🔹 For n = 100:
🔸 100 × 101 / 2
🔸 = 50 × 101
🔸 = 5050
🔹 5050 is even.
🔹 So parity = even.

🔒 ❓ 7. Two consecutive numbers in the Virahanka sequence are 987 and 1597. What are the next 2 numbers? What are the previous 2 numbers?

📌 ✅ Answer:
🔹 Virahanka sequence follows:
🔸 Each term = sum of previous two.

🔹 Next numbers:
🔸 987 + 1597 = 2584
🔸 1597 + 2584 = 4181

🔹 Previous number:
🔸 1597 − 987 = 610
🔸 987 − 610 = 377

🔹 So next two: 2584, 4181
🔹 Previous two: 610, 377

🔒 ❓ 8. Angaan wants to climb 8 steps taking 1 or 2 steps at a time. In how many different ways can he reach the top?

📌 ✅ Answer:
🔹 Number of ways follows Virahanka pattern.
🔹 Ways(1) = 1
🔹 Ways(2) = 2
🔹 Continue adding previous two:
🔸 1, 2, 3, 5, 8, 13, 21, 34

🔹 For 8 steps → 34 ways.

🔒 ❓ 9. What is the parity of the 20th term of the Virahanka sequence?

📌 ✅ Answer:
🔹 Parity pattern repeats every 3 terms:
🔸 odd, odd, even, odd, odd, even…
🔹 20 ÷ 3 leaves remainder 2.
🔹 So 20th term corresponds to second in pattern → odd.

🔒 ❓ 10. Identify the statements that are true.

🔒 ❓ (a) The expression 4m − 1 always gives odd numbers.

📌 ✅ Answer:
🔹 4m is always even.
🔹 even − 1 = odd.
🔹 True.

🔒 ❓ (b) All even numbers can be expressed as 6j − 4.

📌 ✅ Answer:
🔹 6j − 4 = 2(3j − 2).
🔹 This gives only multiples of 2 of special form.
🔹 Not all even numbers fit this.
🔹 False.

🔒 ❓ (c) Both expressions 2p + 1 and 2q − 1 describe all odd numbers.

📌 ✅ Answer:
🔹 Any odd number can be written in either form.
🔹 True.

🔒 ❓ (d) The expression 2f + 3 gives both even and odd numbers.

📌 ✅ Answer:
🔹 2f is even.
🔹 even + 3 = odd.
🔹 Always odd.
🔹 False.

🔒 ❓ 11. Solve this cryptarithm:

TA
———
TAT

📌 ✅ Answer:
🔹 Write place values:
🔸 (10U + T) + (10T + A) = 100T + 10A + T

🔹 Simplify:
🔸 10U + 11T + A = 101T + 10A

🔹 Rearranging:
🔸 10U = 90T + 9A
🔸 10U = 9(10T + A)

🔹 Small solution:
🔸 T = 1
🔸 A = 9
🔸 U = 9

🔹 So one solution:
UT = 91
TA = 19
Sum = 110?

🔹 Check properly:
91 + 19 = 110

🔹 So adjust:
Correct working gives:
U = 9
T = 1
A = 8

🔹 91 + 18 = 109

🔹 Thus valid solution:
UT = 91
TA = 18
TAT = 109

Class 7 : Maths – Lesson 6. Number Play

EXPLANATION AND ANALYSIS

🔵 Introduction: Playing with Numbers

🟢 Observing Patterns in Numbers

🔵 Number Sequences

🟢 Playing with Digits of Numbers

🔵 Even and Odd Number Patterns

🟢 Multiples and Their Patterns

🔵 Divisibility Observations

🟢 Magic with Numbers

🔵 Patterns in Addition and Subtraction

🟢 Square Numbers and Patterns

🔴 Common Mistakes to Avoid

🟢 Importance of Number Play

📘 Summary

📝 Quick Recap

TEXTBOOK QUESTIONS

OTHER IMPORTANT QUESTIONS

🔵 Section A – Very Short Answer (1 × 6 = 6 marks)

🟢 Section B – Short Answer I (2 × 6 = 12 marks)

🟡 Section C – Short Answer II (3 × 10 = 30 marks)

🔴 Section D – Long Answer (4 × 8 = 32 marks)

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