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NCERT Solutions · Maths Ch 3
Number Play
45 questions · 4 exercises
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Ex 3.1
Ex 3.2
Ex 3.3
Ex 3.4
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 3.1 · 5 QUESTIONS
Numbers can Tell us Things
Q1
Some children in a line each say a number: a child says '1' if only one neighbour is taller, '2' if both neighbours are taller, '0' if neither neighbour is taller. (a) Can the children rearrange themselves so that the children standing at the ends say '2'? (b) Can we arrange the children in a line so that all would say only 0s?
(2 marks)
Q2
Can two children standing next to each other say the same number? (Refer to the picture of the children in the book.)
(2 marks)
Q3
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?
(3 marks)
Q4
For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible? Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
(3 marks)
Q5
How would you rearrange the five children so that the maximum number of children say '2'?
(2 marks)
2
EXERCISE 3.2 · 9 QUESTIONS
Supercells
Q6
Colour or mark the supercells in the table: 6828, 670, 9435, 3780, 3708, 7308, 8000, 5583, 52 (a cell is a supercell if its number is greater than all its adjacent cells).
(5 marks)
Q7
Fill the table with only 4-digit numbers such that the supercells are exactly the coloured cells. (A row of 9 cells, with 5346, 1258 and 9635 already placed.)
(2 marks)
Q8
Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions. Out of the 9 numbers, how many supercells are there in the table?
(3 marks)
Q9
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?
(2 marks)
Q10
Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?
(2 marks)
Q11
Will the cell having the largest number in a table always be a supercell? Can the cell having the smallest number be a supercell? Why or why not?
(2 marks)
Q12
Fill a table such that the cell having the second largest number is not a supercell. Then fill a table such that the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?
(2 marks)
Q13
Make other variations of the supercell puzzle and challenge your classmates.
(2 marks)
Q14
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. The biggest number in the table is ____. The smallest even number in the table is ____. The smallest number greater than 50,000 in the table is ____.
(3 marks)
3
EXERCISE 3.3 · 15 QUESTIONS
Number Lines, Digits, Palindromes, Kaprekar
Q15
Identify the numbers marked on the number lines below, and label the remaining positions. (a) 2010, 2020 (b) 9996, 9997 (c) 15,077, 15,078, 15,083 (d) 86,705, 87,705. Put a circle around the smallest number and a box around the largest number in each of the sequences.
(2 marks)
Q16
Find out how many numbers have two digits, three digits, four digits and five digits.
(2 marks)
Q17
Digit sum 14: (a) Write other numbers whose digits add up to 14. (b) What is the smallest number whose digit sum is 14? (c) What is the largest 5-digit number whose digit sum is 14? (d) How big a number can you form having the digit sum of 14? Can you make an even bigger number?
(3 marks)
Q18
Find out the digit sums of all the numbers from 40 to 70. Share your observations.
(2 marks)
Q19
Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?
(3 marks)
Q20
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?
(3 marks)
Q21
Write all possible 3-digit palindromes using the digits 1, 2, 3.
(2 marks)
Q22
Puzzle: 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?
(3 marks)
Q23
Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.
(3 marks)
Q24
Carry out the Kaprekar steps with a few 3-digit numbers (take the largest number A, the smallest number B from the digits, subtract C = A − B and repeat). What number will start repeating?
(3 marks)
Q25
Pratibha uses the digits 4, 7, 3 and 2, and makes the smallest and largest 4-digit numbers: 2347 and 7432. The difference is 7432 − 2347 = 5085 and the sum is 9779. Choose 4 digits to make: (a) the difference between the largest and smallest numbers greater than 5085 (b) the difference less than 5085 (c) the sum of the largest and smallest numbers greater than 9779 (d) the sum less than 9779.
(3 marks)
Q26
What is the sum of the smallest and largest 5-digit palindrome? What is their difference?
(2 marks)
Q27
The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?
(3 marks)
Q28
How many rounds does the number 5683 take to reach the Kaprekar constant?
(3 marks)
Q29
Mental math: Can we make 1,000 using the numbers in the middle column (400, 1,500, 13,000, 25,000, …)? Why not? What about 14,000, 15,000 and 16,000?
(3 marks)
4
EXERCISE 3.4 · 16 QUESTIONS
Mental Math, Estimation, End of Chapter
Q30
Write an example for each of the following whenever possible: (a) 5-digit + 5-digit > 90,250 (b) 5-digit + 3-digit = 6-digit sum (c) 4-digit + 4-digit = 6-digit sum (d) 5-digit + 5-digit = 6-digit sum (e) 5-digit + 5-digit = 18,500 (f) 5-digit − 5-digit (3 marks)
Q31
Always, Sometimes, Never? Decide for each: (a) 5-digit + 5-digit gives a 5-digit number (b) 4-digit + 2-digit gives a 4-digit number (c) 4-digit + 2-digit gives a 6-digit number (d) 5-digit − 5-digit gives a 5-digit number (e) 5-digit − 2-digit gives a 3-digit number
(2 marks)
Q32
Here are numbers arranged in patterns (a: a pattern of 4 rows of 40 and 5 rows of 50...). Find the sum of the numbers in each figure. Should we add them one by one or can we use a quicker way?
(3 marks)
Q33
Steps you would take to walk: (a) from where you are sitting to the classroom door (b) across the school ground (c) from your classroom door to the school gate (d) from your school to your home. Estimate the number of steps.
(2 marks)
Q34
Estimate the number of times you blink your eyes or breaths you take: (a) in a minute (b) in an hour (c) in a day.
(2 marks)
Q35
Estimate the answers: (1) Number of words in your maths textbook: more than 5000 or less than 5000? (2) Number of students in your school who travel by bus: more than 200 or less than 200? (3) Roshan wants to buy milk and 3 types of fruit to make fruit custard for 5 people. He estimates the cost to be ₹100. Do you agree? (4) Estimate the distance between Gandhinagar (Gujarat) and Kohima (Nagaland). (5) Sheetal is in Grade 6 and says she has spent around 13,000 hours in school till date. Do you agree?
(3 marks)
Q36
There is only one supercell in this grid (a number greater than all its neighbours). If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap. Grid: row 1: 16,200 39,344 29,765; row 2: 23,609 62,871 45,306; row 3: 19,381 50,319 38,408.
(3 marks)
Q37
How many rounds does your year of birth take to reach the Kaprekar constant?
(3 marks)
Q38
We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest? Who among us is the closest to 50,000?
(3 marks)
Q39
Estimate the number of holidays you get in a year including weekends, festivals and vacation. Then try to get an exact number and see how close your estimate is.
(2 marks)
Q40
Estimate the number of litres a mug, a bucket and an overhead tank can hold.
(2 marks)
Q41
Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.
(2 marks)
Q42
Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.
(3 marks)
Q43
Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence?
(2 marks)
Q44
Check if the Collatz conjecture holds for the starting number 100.
(5 marks)
Q45
Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?
(2 marks)