NCERT Solutions - Class 6 Mathematics
Chapter 3: Number Play
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:[cite: 1]
- Time[cite: 1]
- Calendar[cite: 1]
- Counting objects/Marks[cite: 1]
- Money[cite: 1]
- Measurement of height & weight[cite: 1]
Page No. 55 & 56
Section 3.1
Q. What do you think these numbers mean?
Ans. Refer page 56.[cite: 1]
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.[cite: 1]
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.[cite: 1]
Q3. Can two children standing next to each other say the same number?
Ans. Yes; Refer picture on page 55.[cite: 1]
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.[cite: 1]
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 say 1.[cite: 1]
Q6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
Ans. Yes, it is possible.[cite: 1]
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 in the arrangement.[cite: 1]
Page No. 57 - 58
Section 3.2 — Figure it Out
Q1. Colour or mark the supercells in the table below.
Ans. Grid values: 6828 | 670 | 9435 | 3780 | 3708 | 7308 | 8000 | 5583 | 52[cite: 1]
Q2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.
Ans. One of the ways could be:[cite: 1]
5346 | 5347 | 1000 | 1258 | 1100 | 1200 | 1300 | 9635 | 9636[cite: 1]
5346 | 5347 | 1000 | 1258 | 1100 | 1200 | 1300 | 9635 | 9636[cite: 1]
Q3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.
Ans. 110 | 100 | 150 | 130 | 280 | 200 | 230 | 210 | 270[cite: 1]
Q4. Out of the 9 numbers, how many supercells are there in the table above?
Ans. 5[cite: 1]
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...): Number of supercells = n / 2 (e.g., 2/2=1, 4/2=2, 6/2=3...).[cite: 1]
For odd number of cells (say 1, 3, 5, 7...): Number of supercells = (n + 1) / 2 (e.g., (1+1)/2=1, (3+1)/2=2, (5+1)/2=3...).[cite: 1]
To get the maximum number of supercells, start by filling the first cell as a supercell and then fill alternately.[cite: 1]
For even number of cells (say 2, 4, 6...): Number of supercells = n / 2 (e.g., 2/2=1, 4/2=2, 6/2=3...).[cite: 1]
For odd number of cells (say 1, 3, 5, 7...): Number of supercells = (n + 1) / 2 (e.g., (1+1)/2=1, (3+1)/2=2, (5+1)/2=3...).[cite: 1]
To get the maximum number of supercells, start by filling the first cell as a supercell and then fill alternately.[cite: 1]
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 a supercell irrespective of its position in the table.[cite: 1]
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.[cite: 1]
No, the smallest number in a table can never be a supercell as the numbers in all adjacent cells will be greater than it.[cite: 1]
Yes, the largest number in a table will always be a supercell.[cite: 1]
No, the smallest number in a table can never be a supercell as the numbers in all adjacent cells will be greater than it.[cite: 1]
Q8. Fill a table such that the cell having the second largest number is not a supercell.
Ans. One of the ways could be: 1 2 3 4 5 6 7 9 8[cite: 1]
Q9. 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?
Ans. One of the ways is: 2 1 3 4 5 6 7 9 8 (Where 2 is the second smallest and a supercell, while 8 is the second largest and not a supercell).[cite: 1]
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.
• The biggest number in the table is: 96,310[cite: 1]
• The smallest even number in the table is: 10,396[cite: 1]
• The smallest number greater than 50,000 in the table is: 60,193[cite: 1]
• The biggest number in the table is: 96,310[cite: 1]
• The smallest even number in the table is: 10,396[cite: 1]
• The smallest number greater than 50,000 in the table is: 60,193[cite: 1]
Page No. 59
Section 3.3 — Number Lines
Q. Identify the numbers marked on the number lines, label remaining positions, and put a circle around the smallest number and a box around the largest number in each sequence.
Ans.
- (a) Smallest: 1990 | Largest: 2035[cite: 1]
- (b) Smallest: 9993 | Largest: 10002[cite: 1]
- (c) Smallest: 15077 | Largest: 15086[cite: 1]
- (d) Smallest: 83705 | Largest: 92705[cite: 1]
Page No. 60 - 61
Section 3.4 — Digit Counts and Sums
Q. Find out how many numbers have 1-digit, 2-digits, 3-digits, 4-digits, and 5-digits.
Ans.
| 1-Digit | 2-Digit | 3-Digit | 4-Digit | 5-Digit |
|---|---|---|---|---|
| 9[cite: 1] | 90[cite: 1] | 900[cite: 1] | 9,000[cite: 1] | 90,000[cite: 1] |
Q. 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?
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?
Ans.
a. Some numbers: 248, 653, 356, 815, 833, 12335, 23351.[cite: 1]
b. Smallest number = 59[cite: 1]
c. Largest 5-digit number = 95000[cite: 1]
d. Examples: 95, 9005, 900005, 90000005, 9000000005, 90000000000005...[cite: 1]
a. Some numbers: 248, 653, 356, 815, 833, 12335, 23351.[cite: 1]
b. Smallest number = 59[cite: 1]
c. Largest 5-digit number = 95000[cite: 1]
d. Examples: 95, 9005, 900005, 90000005, 9000000005, 90000000000005...[cite: 1]
Q. Calculate the digit sums of 3-digit numbers whose digits are consecutive (e.g., 345). Do you see a pattern?
Ans.
123 → 1+2+3 = 6[cite: 1]
234 → 2+3+4 = 9[cite: 1]
345 → 3+4+5 = 12[cite: 1]
456 → 4+5+6 = 15[cite: 1]
567 → 5+6+7 = 18[cite: 1]
678 → 6+7+8 = 21[cite: 1]
789 → 7+8+9 = 24[cite: 1]
Pattern: Yes, all the digit sums are multiples of 3.[cite: 1]
123 → 1+2+3 = 6[cite: 1]
234 → 2+3+4 = 9[cite: 1]
345 → 3+4+5 = 12[cite: 1]
456 → 4+5+6 = 15[cite: 1]
567 → 5+6+7 = 18[cite: 1]
678 → 6+7+8 = 21[cite: 1]
789 → 7+8+9 = 24[cite: 1]
Pattern: Yes, all the digit sums are multiples of 3.[cite: 1]
Q. How many times will the digit '7' occur among 1-100 and 1-1000?
Ans.
• 1 to 100: 20 times[cite: 1]
• 1 to 1000: 300 times[cite: 1]
• 1 to 100: 20 times[cite: 1]
• 1 to 1000: 300 times[cite: 1]
Page No. 61 - 62
Section 3.5 — Palindromes
Q. 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?
Ans. 12,421 (Twelve thousand four hundred twenty one).[cite: 1]
Page No. 63
Section 3.6 — Kaprekar Routine (3-Digit)
Q. Carry out these same steps with a few 3-digit numbers. What number will start repeating?
Ans. Starting with 321:
321 - 123 = 198 → 981 - 189 = 792 → 972 - 279 = 693 → 963 - 369 = 594 → 954 - 459 = 495 → 954 - 459 = 495.[cite: 1]
The number 495 starts repeating.[cite: 1]
321 - 123 = 198 → 981 - 189 = 792 → 972 - 279 = 693 → 963 - 369 = 594 → 954 - 459 = 495 → 954 - 459 = 495.[cite: 1]
The number 495 starts repeating.[cite: 1]
Page No. 64 - 66
Section 3.7 — Figure it Out
Q1. Choose 4 digits (from 4, 7, 3, 2) to form largest and smallest numbers to satisfy conditions...
Ans.
a. Difference > 5085: 7431 - 1347 = 6084[cite: 1]
b. Difference < 5085: 7433 - 3347 = 4086[cite: 1]
c. Sum > 9779: 7433 + 3347 = 10780[cite: 1]
d. Sum < 9779: 7431 + 1347 = 8778[cite: 1]
a. Difference > 5085: 7431 - 1347 = 6084[cite: 1]
b. Difference < 5085: 7433 - 3347 = 4086[cite: 1]
c. Sum > 9779: 7433 + 3347 = 10780[cite: 1]
d. Sum < 9779: 7431 + 1347 = 8778[cite: 1]
Q2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference?
Ans.
Smallest 5-digit palindrome = 10001[cite: 1]
Largest 5-digit palindrome = 99999[cite: 1]
• Sum = 10001 + 99999 = 110,000[cite: 1]
• Difference = 99999 - 10001 = 89,998[cite: 1]
Smallest 5-digit palindrome = 10001[cite: 1]
Largest 5-digit palindrome = 99999[cite: 1]
• Sum = 10001 + 99999 = 110,000[cite: 1]
• Difference = 99999 - 10001 = 89,998[cite: 1]
Q3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?
Ans.
• Next palindrome time = 11:11 (Occurs after 1 hr 10 min = 70 min).[cite: 1]
• Following palindrome time = 12:21 (Occurs after 2 hr 20 min = 140 min).[cite: 1]
• Next palindrome time = 11:11 (Occurs after 1 hr 10 min = 70 min).[cite: 1]
• Following palindrome time = 12:21 (Occurs after 2 hr 20 min = 140 min).[cite: 1]
Q4. How many rounds does the number 5683 take to reach the Kaprekar constant?
Ans. It will take 8 rounds to reach 6174.[cite: 1]
Page No. 66 - 68
Section 3.8 — Figure it Out & True/False Statements
Q. Always, Sometimes, Never?
Ans.
a. 5-digit + 5-digit = 5-digit: Only sometimes true[cite: 1]
b. 4-digit + 2-digit = 4-digit: Only sometimes true[cite: 1]
c. 4-digit + 2-digit = 6-digit: Never true[cite: 1]
d. 5-digit - 5-digit = 5-digit: Only sometimes true[cite: 1]
e. 5-digit - 2-digit = 3-digit: Never true[cite: 1]
a. 5-digit + 5-digit = 5-digit: Only sometimes true[cite: 1]
b. 4-digit + 2-digit = 4-digit: Only sometimes true[cite: 1]
c. 4-digit + 2-digit = 6-digit: Never true[cite: 1]
d. 5-digit - 5-digit = 5-digit: Only sometimes true[cite: 1]
e. 5-digit - 2-digit = 3-digit: Never true[cite: 1]
Page No. 69
Section 3.10 — Collatz Sequence
Q. Make some Collatz sequences starting with your favourite whole numbers. Do you always reach 1?
Ans.
a) Starting with 28: 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1[cite: 1]
b) Starting with 19: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1[cite: 1]
Yes, we always reach 1.[cite: 1]
a) Starting with 28: 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1[cite: 1]
b) Starting with 19: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1[cite: 1]
Yes, we always reach 1.[cite: 1]
Page No. 72
Section 3.12 — Figure it Out (Miscellaneous)
Q1. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.
Ans. Swap digits 1 and 6 in the number 62,871 (making it 12,876).[cite: 1]
Q2. How many rounds does your year of birth take to reach the Kaprekar constant?
Ans. For example, for 1980, it takes 6 rounds.[cite: 1]
Q3. Group of 5-digit numbers between 35,000 and 75,000 with all odd digits:
Ans.
• Largest number: 73,999 (repeating) / 73,951 (non-repeating)[cite: 1]
• Smallest number: 35,111 (repeating) / 35,179 (non-repeating)[cite: 1]
• Closest to 50,000: 51,111 (repeating) / 51,379 (non-repeating)[cite: 1]
• Largest number: 73,999 (repeating) / 73,951 (non-repeating)[cite: 1]
• Smallest number: 35,111 (repeating) / 35,179 (non-repeating)[cite: 1]
• Closest to 50,000: 51,111 (repeating) / 51,379 (non-repeating)[cite: 1]
Q9. Check if the Collatz Conjecture holds for the starting number 100.
Ans. Sequence: 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.[cite: 1]
Q10. Game: Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy?
Ans. The winning strategy is to be the first player.[cite: 1]
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