Class 6 Maths Chapter 3 Solutions: Number Play

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Solution sheet 1

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NCERT Class 6 Maths Chapter 3, question 1 — Source question

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.

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Five different possible situations in which numbers are used - 1. Time 2. Calendar 3. Counting objects/Marks 4. Measurement of height & weight 5. Money There could many more. Section 3.1 Page No. 55

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Section 3.1 · Page No. 55

NCERT Class 6 Maths Chapter 3, question 2 — Source question

What do you think these numbers mean?

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Refer page 56. Page No. 56

Source worked solution for Class 6 Chapter 3, question 2, part 1 — Source worked answer · part 1

Refer page 56. · Page No. 56

NCERT Class 6 Maths Chapter 3, question 3 — Source question

1. Can the children rearrange themselves so that the children standing at the ends say ‘2’?

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No; There will be no one standing on the other side of the child standing at the end.

Source worked solution for Class 6 Chapter 3, question 3, part 1 — Source worked answer · part 1

Refer page 56. · Page No. 56

NCERT Class 6 Maths Chapter 3, question 4 — Source question

2. Can we arrange the children in a line so that all would say only 0s?

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Yes; All the children in the line should be of same height.

Source worked solution for Class 6 Chapter 3, question 4, part 1 — Source worked answer · part 1

Refer page 56. · Page No. 56

NCERT Class 6 Maths Chapter 3, question 5 — Source question

3. Can two children standing next to each other say the same number?

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Yes; Refer picture on page 55.

Source worked solution for Class 6 Chapter 3, question 5, part 1 — Source worked answer · part 1

Page No. 56 · Ans. Yes; Refer picture on page 55.

NCERT Class 6 Maths Chapter 3, question 6 — Source question

4. 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?

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Yes, they can, if they are standing in ascending order of height. [1]

Source worked solution for Class 6 Chapter 3, question 6, part 1 — Source worked answer · part 1

Source worked solution for Class 6 Chapter 3, question 6, part 2 — Source worked answer · part 2

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Solution sheet 2

Ans. Yes; Refer picture on page 55.

NCERT Class 6 Maths Chapter 3, question 7 — Source question

5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?

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No; the tallest child at the end cannot say1.

Source worked solution for Class 6 Chapter 3, question 7, part 1 — Source worked answer · part 1

Ans. Yes; Refer picture on page 55.

NCERT Class 6 Maths Chapter 3, question 8 — Source question

6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?

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Yes, it is possible.

Source worked solution for Class 6 Chapter 3, question 8, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 3, question 9 — Source question

7. How would you rearrange the five children so that the maximum number of children say ‘2’?

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At the most only 2 children can say 2 as given is the following arrangement. Section 3.2 Page No. 57 Figure it out

Source worked solution for Class 6 Chapter 3, question 9, part 1 — Source worked answer · part 1

Page No. 57 · Figure it out

NCERT Class 6 Maths Chapter 3, question 10 — Source question

1. Colour or mark the supercells in the table below. 6828 670 9435 3780 3708 7308 8000 5583 52

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6828 670 9435 3780 3708 7308 8000 5583 52 [2]

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Solution sheet 3

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NCERT Class 6 Maths Chapter 3, question 11 — Source question

2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells. 5346 1258 9635

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One of the ways could be-5346; 9636.Try more 5346 5347 1000 1258 1100 1200 1300 9635 9636

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NCERT Class 6 Maths Chapter 3, question 12 — Source question

3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.

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110 100 150 130 280 200 230 210 270

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NCERT Class 6 Maths Chapter 3, question 13 — Source question

4. Out of the 9 numbers, how many supercells are there in the table above? ___________

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5

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NCERT Class 6 Maths Chapter 3, question 14 — Source question

5 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.

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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.

Source worked solution for Class 6 Chapter 3, question 14, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 3, question 15 — Source question

6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?

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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.

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NCERT Class 6 Maths Chapter 3, question 16 — Source question

7. 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?

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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.

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NCERT Class 6 Maths Chapter 3, question 17 — Source question

8. Fill a table such that the cell having the second largest number is not a supercell.

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One of the ways could be- 1 2 3 4 5 6 7 9 8 [3]

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Solution sheet 4

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NCERT Class 6 Maths Chapter 3, question 18 — Source question

9. Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?

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One of the ways is- 2 1 3 4 5 6 7 9 8 Second smallest number a super cell Second largest number 8 is not a supercell.

Source worked solution for Class 6 Chapter 3, question 18, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 3, question 19 — Source question

10. Make other variations of this puzzle and challenge your classmates.

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Some of these could be- Can you fill the table with 9 cells such that there are more than 5 super cells? Can you fill the table with 9 cells such that there are exactly 4 super cells? Page No. 58

Source worked solution for Class 6 Chapter 3, question 19, part 1 — Source worked answer · part 1

Page No. 58

NCERT Class 6 Maths Chapter 3, question 20 — Source question

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.

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Table 2 (One of the ways) – 96,310 96,301 36,109 39,160 96,103 13,609 60,319 19,306 13,906 10,396 60,193 60,931 10,369 10,963 10,936 69,031

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NCERT Class 6 Maths Chapter 3, question 21 — Source question

The biggest number in the table is ____________.

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The biggest number in the table is 96,310

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NCERT Class 6 Maths Chapter 3, question 22 — Source question

The smallest even number in the table is ____________.

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The smallest even number in the table is 10,396

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NCERT Class 6 Maths Chapter 3, question 23 — Source question

The smallest number greater than 50,000 in the table is ____________.

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The smallest number greater than 50,000 in the table is 60,193. Section 3.3 Page no.59

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Section 3.3 · Page no.59

NCERT Class 6 Maths Chapter 3, question 24 — Source question

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.

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2180 1050 8400 9950 3600 5030 2754 1000 9000 8000 2000 3000 4000 5000 6000 10000 7000 1500 5300 9590 [4]

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Solution sheet 5

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NCERT Class 6 Maths Chapter 3, question 25 — Source question

Identify the numbers marked on the number lines below, and label the remaining

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positions. (a). 2025 2030 1995 2000 2005 2010 2015 2035 1990 2020 (b). 10001 10000 9998 10002 9995 9996 9997 9993 9999 9994 (c). 15085 15077 15084 15078 15079 15080 15081 15082 15086 15083 (d). 91705 83705 90705 84705 85705 86705 87705 88705 92705 89705 Put a circle around the smallest number and a box around the largest number in each of the sequences above. Section 3.4 Page no. 60

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Section 3.4 · Page no. 60

NCERT Class 6 Maths Chapter 3, question 26 — Source question

Find out how many numbers have two digits, three digits, four digits, and five digits: 1 Digit numbers 2 Digit numbers 3 Digit numbers 4 Digit numbers 5 Digit numbers 9

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1 Digit numbers 2 Digit numbers 3 Digit numbers 4 Digit numbers 5 Digit numbers 9 90 900 9,000 90,000 [5] Figure it out

Source worked solution for Class 6 Chapter 3, question 26, part 1 — Source worked answer · part 1

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Solution sheet 6

Figure it out

NCERT Class 6 Maths Chapter 3, question 27 — Source question

1. 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 14? Can you make an even bigger number?

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a. Some such numbers are:248, 653, 356, 815, 833, 12335, 23351. b. 59 c. 95000 d. 95, 9005, 900005, 90000005, 9000000005, 90000000000005 …

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NCERT Class 6 Maths Chapter 3, question 28 — Source question

3. Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?

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123 → 1+2+3 = 6 234 → 2+3+4 = 9 345 → 3+4+5 = 12 456 → 4+5+6 = 15 567 → 5+6+7 = 18 678 → 6+7+8 = 21 789 → 7+8+9 = 24 • Yes, there is a pattern, all the sums are multiples of 3. • No. Page no. 61

Source worked solution for Class 6 Chapter 3, question 28, part 1 — Source worked answer · part 1

Page no. 61

NCERT Class 6 Maths Chapter 3, question 29 — Source question

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?

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• 20 times. • 300 times. [6] Section 3.5 Page no. 61

Source worked solution for Class 6 Chapter 3, question 29, part 1 — Source worked answer · part 1

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Solution sheet 7

Section 3.5 · Page no. 61

NCERT Class 6 Maths Chapter 3, question 30 — Source question

Write all possible 3-digit palindromes using these digits.

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Palindromes: 111, 121, 131 222, 212, 232 313, 323, 333 Explore Page no. 62

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Page no. 61 · Page no. 62

NCERT Class 6 Maths Chapter 3, question 31 — Source question

Will reversing and adding numbers repeatedly, starting with a 2-digit number,

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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

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NCERT Class 6 Maths Chapter 3, question 32 — Source question

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? _________________

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tth th h t u 1 2 4 2 1 Twelve thousand four hundred twenty one. Section 3.6 Page no. 63

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Section 3.6 · Page no. 63

NCERT Class 6 Maths Chapter 3, question 33 — Source question

Carry out these same steps with a few 3-digit numbers. What number will start repeating?

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Take a 3- Digit number say, 321. 321 -123 198 981 -189 792 972 -279 693 963 -369 594 954 -459 495 954 -459 495 The number 495 starts repeating. Try for other 3-digit numbers. [7] Section 3.7 Page no. 64

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Section 3.7 · Page no. 64

NCERT Class 6 Maths Chapter 3, question 34 — Source question

Try and find out all possible times on a 12-hour clock of each of these types.

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4:44 2:22 3:33 10:10 11:11 12:12 09:09 12:21 05:50 10:01 Think of some more!

Source worked solution for Class 6 Chapter 3, question 34, part 1 — Source worked answer · part 1

Think of some more!

NCERT Class 6 Maths Chapter 3, question 35 — Source question

Find some other dates of this form from the past.

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20/04/2004, 20/06/2006, Try for yourself!

Source worked solution for Class 6 Chapter 3, question 35, part 1 — Source worked answer · part 1

Think of some more!

NCERT Class 6 Maths Chapter 3, question 36 — Source question

Find all possible dates of this form from the past.

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01/02/2001, 02/02/2002, Think of some more!

Source worked solution for Class 6 Chapter 3, question 36, part 1 — Source worked answer · part 1

Think of some more!

NCERT Class 6 Maths Chapter 3, question 37 — Source question

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?

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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

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Page no. 64 · Figure it out

NCERT Class 6 Maths Chapter 3, question 38 — Source question

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 nu...

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Some of the possibilities are– a. 7431 – 1347 = 6084 b. 7433 – 3347 = 4086 c. 7433 + 3347 = 10780 d. 7431 + 1347 = 8778

Source worked solution for Class 6 Chapter 3, question 38, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 3, question 39 — Source question

2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference?

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Smallest 5 digit palindrome = 10001 largest 5 digit palindrome = 99999 Sum = 10001 + 99999 = 110,000 Difference = 99999 – 10001 = 89,998 [8]

Source worked solution for Class 6 Chapter 3, question 39, part 1 — Source worked answer · part 1

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Solution sheet 9

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NCERT Class 6 Maths Chapter 3, question 40 — Source question

3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?

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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.

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NCERT Class 6 Maths Chapter 3, question 41 — Source question

4. How many rounds does the number 5683 take to reach the Kaprekar constant?

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5683 8653 -3568 5085 8550 -5058 3492 9432 -2349 7083 8730 -3078 5652 6552 -2556 3996 9963 -3699 6264 6642 -2466 4176 7641 -1467 6174 1 2 3 4 5 6 7 8 It will take 8 rounds to reach the Kaprekar constant. Page no. 66 Section 3.8

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Page no. 66 · Section 3.8

NCERT Class 6 Maths Chapter 3, question 42 — Source question

Can we make 1,000 using the numbers in the middle? Why not? What about 14,000, 15,000 and 16,000? Yes, it is possible. Explore how. What thousands cannot be made?

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No; the only number which is smaller than 1000 is 400 and 1000 is not a multiple of 400. 14000 = 1500 × 8 + 400 × 5 = 12000 + 2000 = 14000 15000 = 13000 + 400 × 5 = 13000 + 2000 = 15000 16000 = 1500 × 8 + 400 × 10 = 12000 + 4000 = 16000 Only one thousand (1000) cannot be made. Figure it out

Source worked solution for Class 6 Chapter 3, question 42, part 1 — Source worked answer · part 1

Figure it out

NCERT Class 6 Maths Chapter 3, question 43 — Source question

1. Write an example for each of the below scenarios whenever possible. Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates.

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• 5 digit + 5 digit > 90,250 45,000 + 45,400 = 90,400 > 90,250 • 5 digit + 3 digit = 6 digit sum 99,999 + 999 = 100,998 • 4 digit + 4 digit = 6 digit sum Not possible as even the sum of the greatest 4 digit numbers will not give a six digit sum. (9999 + 9999 = 19,998) [9] • 5 digit + 5 digit = 6 digit sum 60,000 + 40,000 = 1,00,000 • 5 digit + 5 digit = 18,500 Not possible as smallest 5-digit number is 10,000. If both the numbers are 10,000 then the sum is 20,000, which is more than 18,500. • 5 digit – 5 digit < 56,503 80,000 – 50,000 < 56,503 < 56,503 • 5 digit – 3 digit = 4 digit difference 10,000 – 999 = 9001 • 5 digit – 4 digit = 4 digit difference 12,000 – 2,500 = 9,500 • 5 digit – 5 digit = 3 digit difference 50,999 – 50,000 = 999 • 5 digit – 5 digit = 91,500 Not possible as the difference of the greatest and the smallest 5 digit numbers, the maximum difference, can be 99,999 – 10,000 = 89,999 Some examples of other such questions are- 1. 5 digit + 5 digit = 7 digit sum 2. 4 digit + 4 digit = 2900 More such examples can be made.

Source worked solution for Class 6 Chapter 3, question 43, part 1 — Source worked answer · part 1

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Solution sheet 10

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NCERT Class 6 Maths Chapter 3, question 44 — Source question

2. Always, Sometimes, Never? Below are some statements. Think, explore and find out if each of the statement is ‘Always true’, ‘Only sometimes true’ or ‘Never true’. Why do you think so? Write your reasoning; discuss this with the class. a. 5-digit number + 5-digit number gives a 5-digit number b. 4-digit number + 2-digit number gives a 4-digit number c. 4-digit number + 2-digit number gives a 6-digit number d. 5-digit number – 5-digit number gives a 5-digit number e. 5-digit number – 2-digit number gives a 3-di...

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a. Only sometimes true. 20,000 + 80,000 = 1,00,000 not a 5 digit number b. Only sometimes true 9,999 + 99 = 10,098 not a 4 digit number c. Never true 9,999 + 99 = 10,098 On adding the greatest 4 digit and the greatest 2 digit numbers, we can reach only 5 digit number 10,098. So there is no possibility of getting a 6 digit number. d. Only sometimes true Ex. 12,000 – 10,000 = 2,000 5 digit – 5 digit = 4 digit e. Never true Ex. 10,000 – 99 = 9901 [10] Even if the greatest 2 digit numbers is subtracted from the smallest 5 digit number, 4 digit number will be obtained. Page no. 69 Section 3.10

Source worked solution for Class 6 Chapter 3, question 44, part 1 — Source worked answer · part 1

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Page no. 69 · Section 3.10

NCERT Class 6 Maths Chapter 3, question 45 — Source question

Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1? Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not?

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a) 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 b) 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 Yes we always reach 1. The even numbers are halved and when we have an odd number we convert it into an even number by multiplying by 3 and adding 1 so that eventually it can be halved again. The smallest even number is 2 so we will reach 1 for sure.

Source worked solution for Class 6 Chapter 3, question 45, part 1 — Source worked answer · part 1

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NCERT Class 6 Maths Chapter 3, question 46 — Source question

3. Name some objects around you that are: a. a few thousand in number b. more than ten thousand in number

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A few thousands: car numbers, 4 digit pin More than ten thousands: Salary, Mobile numbers.

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NCERT Class 6 Maths Chapter 3, question 47 — Source question

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 with him? Why or why not?

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Yes, it is possible with less quantity of serving and with purchase of 1-1-1 fruit of each type. However, it is not possible with costly fruits and more quantity of serving.

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NCERT Class 6 Maths Chapter 3, question 48 — Source question

4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland).

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2500 kilometer

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NCERT Class 6 Maths Chapter 3, question 49 — Source question

5. Sheetal is in Grade 6 and says she has spent around 13,000 hours in school till date. Do you agree with her? Why or why not?

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No, I do not agree with her. There are 6 school hours in a day and around 200 working days in a year. 𝟏𝟑,𝟎𝟎𝟎 𝟔×𝟐𝟎𝟎 = 10.8 years (Nursery, KG, 1,2,3,4,5,6) 8 years She is in school for 8 years, 13,000 hours is way too high.

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NCERT Class 6 Maths Chapter 3, question 50 — Source question

7. Make some estimation questions and challenge your classmates!

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Some such are • How many students are there in your school? • How many hours does a person sleep in his lifetime on an average? (More such questions can be made) [11] Section 3.12 Page No. 72 Figure it Out

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Page No. 72 · Figure it Out

NCERT Class 6 Maths Chapter 3, question 51 — Source question

1. There is only one supercell (number greater than all its neighbours) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap. 16,200 39,344 29,765 23,609 62,871 45,306 19,381 50,319 38,408

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If I exchange the digits 1 and 6 in the number 62,871 then there will be 4 Super cells. 16,200 39,344 29,765 23,609 12,876 45,306 19,381 50,319 38,408

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NCERT Class 6 Maths Chapter 3, question 52 — Source question

2. How many rounds does your year of birth take to reach the Kaprekar constant?

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Suppose the birth year is 1980, then, - 9810 -1089 8721 8721 -1278 7443 7443 -3447 3996 9963 -3699 6264 6642 -2466 4176 7641 -1467 6174 It takes 6 rounds. (Try now for your year of birth.)

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NCERT Class 6 Maths Chapter 3, question 53 — Source question

3. 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 number in our group? Who among us is the closest to 50,000?

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73,951 With repeating digit With non repeating digit Largest number → 73,999 Smallest number → 35,111 35,179 With repeating digit With non repeating digit Closest to 50000 51,111 51,379

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NCERT Class 6 Maths Chapter 3, question 54 — Source question

6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.

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18000 + 300 + 370 = 18670. Try for more. [12]

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NCERT Class 6 Maths Chapter 3, question 55 — Source question

7. 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.

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Number Chosen: 250 25 25 50 50 50 25 25 Or 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

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NCERT Class 6 Maths Chapter 3, question 56 — Source question

8. 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?

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when we divide 28 = 2x2x2x2x2x2x2x2 by 2 it become 27 and every time you divide by 2 the same will continue happening, until you are left with 2 which when divided by 2 will leave 1.

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NCERT Class 6 Maths Chapter 3, question 57 — Source question

9. Check if the Collatz Conjecture holds for the starting number 100.

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100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

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NCERT Class 6 Maths Chapter 3, question 58 — Source question

10 Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?

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Winning strategy is to be the first player. [13]

Source worked solution for Class 6 Chapter 3, question 58, part 1 — Source worked answer · part 1

Match, attempt, then check

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.

What do you think these numbers mean?

1. Can the children rearrange themselves so that the children standing at the ends say ‘2’?

2. Can we arrange the children in a line so that all would say only 0s?

3. Can two children standing next to each other say the same number?

4. 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?

5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?

6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?

7. How would you rearrange the five children so that the maximum number of children say ‘2’?

1. Colour or mark the supercells in the table below. 6828 670 9435 3780 3708 7308 8000 5583 52

2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells. 5346 1258 9635

3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.

4. Out of the 9 numbers, how many supercells are there in the table above? ___________

5 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.

6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?

7. 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?

8. Fill a table such that the cell having the second largest number is not a supercell.

9. Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?

10. Make other variations of this puzzle and challenge your classmates.

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 ____________.

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.

Identify the numbers marked on the number lines below, and label the remaining

Find out how many numbers have two digits, three digits, four digits, and five digits: 1 Digit numbers 2 Digit numbers 3 Digit numbers 4 Digit numbers 5 Digit numbers 9

1. 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 14? Can you make an even bigger number?

3. Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?

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?

Write all possible 3-digit palindromes using these digits.

Will reversing and adding numbers repeatedly, starting with a 2-digit number,

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? _________________

Carry out these same steps with a few 3-digit numbers. What number will start repeating?

Try and find out all possible times on a 12-hour clock of each of these types.

Find some other dates of this form from the past.

Find all possible dates of this form from the past.

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?

2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference?

3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?

4. How many rounds does the number 5683 take to reach the Kaprekar constant?

Can we make 1,000 using the numbers in the middle? Why not? What about 14,000, 15,000 and 16,000? Yes, it is possible. Explore how. What thousands cannot be made?

1. Write an example for each of the below scenarios whenever possible. Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates.

Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1? Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not?

3. Name some objects around you that are: a. a few thousand in number b. more than ten thousand in number

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 with him? Why or why not?

4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland).

5. Sheetal is in Grade 6 and says she has spent around 13,000 hours in school till date. Do you agree with her? Why or why not?

7. Make some estimation questions and challenge your classmates!

1. There is only one supercell (number greater than all its neighbours) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap. 16,200 39,344 29,765 23,609 62,871 45,306 19,381 50,319 38,408

2. How many rounds does your year of birth take to reach the Kaprekar constant?

3. 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 number in our group? Who among us is the closest to 50,000?

6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.

7. 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.

8. 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?

9. Check if the Collatz Conjecture holds for the starting number 100.

10 Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?