Class 6 Maths Chapter 3 Number Play NCERT Solutions (Step-by-Step Answers, Activities & Complete Guide | Ganita Prakash)
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................................................................................................................... Introduction
Numbers are used everywhere in daily life. We use them for:
counting objects
measuring quantities
telling time
identifying patterns
solving puzzles
In Chapter 3 – Number Play, students explore numbers in creative ways. The chapter focuses on patterns, logical thinking, and interesting mathematical puzzles.
Instead of just performing calculations, students learn how numbers behave in different situations.
Where Do We Use Numbers?
Examples of real-life use of numbers:
Time on a clock
Calendar dates
Marks in exams
Height and weight measurements
Money transactions
Numbers help us organise our daily life.
Q. Think about various situations where we use numbers. List five different situations in
which numbers are used. See what your classmates have listed, share, and discuss.
Ans. Five different possible situations in which numbers are used -
1. Time
2. Calendar
3. Counting objects/Marks
4. Measurement of height & weight
5. Money
Section 3.1
Page No. 56
Q1. Can the children rearrange themselves so that the children standing at the ends say
‘2’?
Ans. No; There will be no one standing on the other side of the child standing at the end.
Q2. Can we arrange the children in a line so that all would say only 0s?
Ans. Yes; All the children in the line should be of same height.
Q3. Can two children standing next to each other say the same number?
Ans. Yes; Refer picture on page 55.
Q4. There are 5 children in a group, all of different heights. Can they stand such that four
of them say ‘1’ and the last one says ‘0’? Why or why not?
Ans. Yes, they can, if they are standing in ascending order of height.
Q5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible? Ans. No; the tallest child at the end cannot say1.Q6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
Ans. Yes, it is possible. Q7. How would you rearrange the five children so that the maximum number of children
say ‘2’?
Ans. At the most only 2 children can say 2 as given is the following arrangement. Section 3.2
Page No. 57
Figure it out
Q1. Colour or mark the supercells in the table below.
Q5 Find out how many supercells are possible for different numbers of cells.
Do you notice any pattern? What is the method to fill a given table to get the
maximum number of supercells? Explore and share your strategy.
Ans. For even number of cells say,2,4,6,… the number of supercells would be respectively,
2/2 =1,4/2 =2,6/2=3,…
For odd number of cells , say 1,3,5,7,… the number of supercells would be respectively
(1+1)/2= 1, (3+1)/2 = 2, (5+1)/2= 3,(7+1)/2 = 4,…
To get the maximum number ofsupercells, we have to start by filling the first cell as
super cell & then fill alternately.
Q6. Can you fill a supercell table without repeating numbers such that there are no
supercells? Why or why not?
Ans. No; the cell which is filled by the greatest number among the given numbers chosen, will
become super cell irrespective of its position in the table.
Q7. Will the cell having the largest number in a table always be a supercell? Can the cell
having the smallest number in a table be a supercell? Why or why not?
Ans. Yes, the largest number in a table will always be a supercell.
No, the smallest number in a table can never be a supercell as the number in all the
adjacent cells will be greater than it.
Q8. Fill a table such that the cell having the second largest number is not a supercell.
Ans. One of the ways could be- Page No. 58
Q. Complete Table 2 with 5-digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘3’, and ‘9’ in
some order. Only a coloured cell should have a number greater than all its
neighbours.
Ans. Table 2 (One of the ways) – Q. The biggest number in the table is ____________.
Ans. The biggest number in the table is 96,310
Q. The smallest even number in the table is ____________.
Ans. The smallest even number in the table is 10,396
Q. The smallest number greater than 50,000 in the table is ____________.
Ans. The smallest number greater than 50,000 in the table is 60,193.
Section 3.3
Page no.59
Q. We are quite familiar with number lines now. Let’s see if we can place some numbers
in their appropriate positions on the number line. Here are the numbers: 2180, 2754,
1500, 3600, 9950, 9590, 1050, 3050, 5030, 5300 and 8400.
Page no. 61
Q. Among the numbers 1–100, how many times will the digit ‘7’ occur? Among the
numbers 1–1000, how many times will the digit ‘7’ occur?
Ans.
• 20 times.
• 300 times.
Section 3.5
Page no. 61
Q. Write all possible 3-digit palindromes using these digits.
Ans. Palindromes: 111, 121, 131
222, 212, 232
313, 323, 333
Explore
Page no. 62
Q. Will reversing and adding numbers repeatedly, starting with a 2-digit number,
always give a palindrome? Explore and find out.*
Some of these are-
12
+21
.........
33
...........
47
+74
........
121
........
Try more
Yes, it will always give a palindrome.
Puzzle time
Q. I am a 5-digit palindrome.
I am an odd number.
My ‘t’ digit is double of my ‘u’ digit.
My ‘h’ digit is double of my ‘t’ digit.
Who am I? _______________Section 3.7
Page no. 64
Q. Try and find out all possible times on a 12-hour clock of each of these types.
Ans. 4:44 2:22 3:33
10:10 11:11 12:12 09:09
12:21 05:50 10:01 Think of some more!
Q. Find some other dates of this form from the past.
Ans. 20/04/2004, 20/06/2006, Try for yourself!
Q. Find all possible dates of this form from the past.
Ans. 01/02/2001, 02/02/2002, Think of some more!
Q. Will any year’s calendar repeat again after some years? Will all dates and days in a
year match exactly with that of another year?
Ans. Yes,
The calendar repeats itself after 6 years if only one leap year is included in these 6 years.
If 2 leap years are included, then it will repeat after 5 years.
Page no. 64
Figure it out
Q.1. Pratibha uses the digits ‘4’, ‘7’, ‘3’ and ‘2’, and makes the smallest and largest 4
digit numbers with them: 2347 and 7432. The difference between these two
numbers is 7432 – 2347 = 5085. The sum of these two numbers is 9779. Choose 4
digits to make:
a. the difference between the largest and smallest numbers greater than 5085.
b. the difference between the largest and smallest numbers less than 5085.
c. the sum of the largest and smallest numbers greater than 9779.
d. the sum of the largest and smallest numbers less than 9779.
Ans. Some of the possibilities are–
a. 7431 – 1347 = 6084
b. 7433 – 3347 = 4086
c. 7433 + 3347 = 10780
d. 7431 + 1347 = 8778
Q.2. What is the sum of the smallest and largest 5-digit palindrome? What is their
difference?
Ans. Smallest 5 digit palindrome = 10001
largest 5 digit palindrome = 99999
Sum = 10001 + 99999 = 110,000
Difference = 99999 – 10001 = 89,998
Q.3. The time now is 10:01. How many minutes until the clock shows the next palindromic
time? What about the one after that?
Ans. Time Now → 10:01
Next palindrome time → 11:11
After 1 hr. 10 min = 70 min the clock will show next palindrome time.
Next palindrome time = 12:21 which will occur after 2 hr. 20 min = 140 min from
10:01.
Section 3.1 Numbers Can Tell Us Things
In this activity, children stand in a line and say numbers based on how many taller neighbours they have.
Rule:
A child says 1 if one neighbour is taller.
A child says 2 if both neighbours are taller.
A child says 0 if none are taller.
Question 1
Can the children at the ends say “2”?
Solution
No.
Reason:
Children at the ends have only one neighbour, not two.
So they cannot have two taller neighbours.
Question 2
Can all children say 0?
Solution
Yes.
If all children have equal height, then none of them will have taller neighbours.
Therefore every child says 0.
Question 3
Can two children standing together say the same number?
Solution
Yes.
Example:
Two children may both have exactly one taller neighbour, so both say 1.
Section 3.2 Supercells
A supercell is a cell that contains a number larger than its neighbouring cells.
Example:
In the table
200 577 626
The number 626 is a supercell because it is larger than its neighbours.
Question
Will the largest number in the table always be a supercell?
Solution
Yes.
The largest number must be greater than all neighbouring numbers.
Therefore it will always be a supercell.
Question
Can the smallest number be a supercell?
Solution
No.
A supercell must be larger than its neighbours.
Since the smallest number is smaller than other numbers, it cannot be a supercell.
Section 3.3 Patterns on the Number Line
Example numbers:
1050
1500
2180
2754
3600
5030
5300
8400
9590
9950
These numbers must be placed correctly between 1000 and 10,000 on the number line.
Learning Idea
Students understand:
number order
number magnitude
spacing between numbers
Digit Sum Activity
Example:
68 → 6 + 8 = 14
176 → 1 + 7 + 6 = 14
Both have digit sum 14.
Smallest number with digit sum 14
Digits must add to 14.
Example:
5 + 9 = 14
Smallest number = 59
Largest 5-digit number with digit sum 14
Largest arrangement:
95000
Because
9 + 5 = 14
Digit Detective Puzzle
How many times does digit 7 appear from 1 to 100?
Solution
Numbers containing 7:
7
17
27
37
47
57
67
77
87
97
70
71
72
73
74
75
76
78
79
Total occurrences = 20
Section 3.5 Palindromic Numbers
A palindrome reads the same forward and backward.
Examples:
66
575
848
1111
Example:
121 → same both directions.
3-digit palindromes using digits 1,2,3
111
121
131
212
222
232
313
323
333
Reverse and Add Method
Example
12 + 21 = 33
33 is a palindrome.
Sometimes we repeat the process until a palindrome appears.
Section 3.6 Kaprekar Constant
Steps:
1 Choose a 4-digit number (example 6382)
2 Arrange digits in descending order
3 Arrange digits in ascending order
4 Subtract
Example:
8632
2368
8632 − 2368 = 6264
Repeat process.
Eventually the number becomes
6174
This is called the Kaprekar Constant.
Section 3.7 Clock Number Patterns
Examples of interesting clock times:
4:44
11:11
12:21
10:10
These show patterns or palindromes.
Section 3.8 Mental Math
Students learn to combine numbers mentally to reach target values.
Example
1500 + 1500 + 400 = 3400
This improves mental calculation skills.
Section 3.9 Number Patterns
Students observe patterns in grids of numbers and find faster ways to calculate sums.
Instead of adding numbers one by one, students identify repeating structures.
Section 3.10 Collatz Conjecture
Rule:If number is even → divide by 2
If number is odd → multiply by 3 and add 1
Example starting from 12:
12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Mathematicians believe every number eventually reaches 1, but it has not been fully proven.
Section 3.11 Estimation
Estimation means guessing a reasonable value without exact calculation.
Examples:
students in school
steps to classroom
distance between cities
Estimation helps in real-life decision making.
Section 3.12 Games with Numbers
Example game: Reach 21
Rules:
Players add 1, 2, or 3 to the previous number.
Goal: reach 21.
Winning strategy:
Say numbers in pattern
4 → 8 → 12 → 16 → 20 → 21
Chapter Summary
In this chapter students learned:
number patterns
digit sums
palindromes
Kaprekar constant
estimation
mathematical puzzles
number games
These activities develop logical thinking and problem solving skills.
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Q1. In what different situations do we use numbers?
Numbers are used in time, calendars, counting objects, measuring height/weight, and money transactions.
Q2. What do the children’s numbers in Section 3.1 represent?
Each child says the number of taller neighbours they have: 0 (none taller), 1 (one taller), or 2 (both taller).
Q3. Can children at the ends of the line say ‘2’?
No, because children at the ends have only one neighbour, not two.
Q4. What is a supercell in Section 3.2?
A supercell is a number in a table that is greater than all its adjacent neighbours (top, bottom, left, right).
Q5. Will the largest number in a table always be a supercell?
Not always. It depends on whether it is greater than its immediate neighbours. If not, it won’t be a supercell.
Q6. How many 2-digit, 3-digit, 4-digit, and 5-digit numbers exist?
- 2-digit: 90 (10–99)
- 3-digit: 900 (100–999)
- 4-digit: 9000 (1000–9999)
- 5-digit: 90,000 (10,000–99,999)
Q7. What is a digit sum?
The digit sum is the sum of all digits in a number. Example: 176 → 1+7+6 = 14.
Q8. What are palindromic numbers?
Palindromic numbers read the same forwards and backwards. Example: 121, 575, 1111.
Q9. What is the reverse-and-add palindrome method?
Start with a number, add it to its reverse, and repeat until a palindrome is formed. Example: 34+43=77 (palindrome).
Q10. What is Kaprekar’s constant?
For any 4-digit number (with at least two different digits), repeatedly subtract smallest arrangement from largest arrangement of digits. You always reach 6174.
Q11. What is the Collatz Conjecture?
Start with any number: if even, divide by 2; if odd, multiply by 3 and add 1. The sequence is conjectured to always reach 1.
Q12. What is estimation in mathematics?
Estimation means approximating values instead of exact counts. Example: estimating school strength as ~500 students.
Q13. What is the game of 21?
Two players alternately add 1, 2, or 3 to the previous number. The player who reaches 21 wins. Strategy ensures one player can always win.
Q14. What is the variation of the 99 game?
Players alternately add numbers between 1 and 10. The player who reaches 99 wins. Strategy is similar to the 21 game.
Q15. What is computational thinking?
Computational thinking is the ability to use numbers and logical steps to solve problems, discover patterns, and create strategies.
