Playing with Numbers — Questions and Answers
Welcome to HSLC Guru. This page gives complete, worked answers to every Work it Out exercise (3.1 to 3.9) of ASSEB Class 6 New Mathematics Chapter 3, Playing with Numbers (সংখ্যাৰ সৈতে খেলা), along with a set of extra practice questions.
Summary
Numbers are a part and parcel of our daily life — we use them for counting, marketing, telling time and playing games. This chapter explores numbers by playing with them: magic cells, placing numbers on the number line, playing with digits, digit sums, palindromic numbers, the Kaprekar constant, the number 1089, clocks and calendars, mental math, the Collatz conjecture, math magic and estimation.
Some striking results of the chapter are: there are 9 one-digit, 90 two-digit, 900 three-digit, 9000 four-digit and 90000 five-digit numbers. A palindromic number reads the same from left to right and right to left (such as 66, 282, 363). For any 4-digit number with at least two different digits, repeatedly forming the largest and smallest arrangements and subtracting always leads to 6174, the Kaprekar constant.
The German mathematician Lothar Collatz proposed a conjecture in 1937 — take any number, halve it if it is even, or triple it and add one if it is odd, and you always reach 1. The chapter also teaches converting between 12-hour and 24-hour clock timings, the British, American and ISO date formats, and how estimation helps us make reasonable guesses in everyday life.
Summary: This page gives complete, worked answers to every Work it Out exercise (3.1 to 3.9) of ASSEB Class 6 New Mathematics Chapter 3, Playing with Numbers (সংখ্যাৰ সৈতে খেলা), covering magic cells, numbers on the number line, playing with digits and digit sums, palindromic numbers, the Kaprekar constant 6174, the number 1089, the 12-hour and 24-hour clock and date formats, mental math, the Collatz conjecture, math magic and estimation, along with extra MCQs, fill in the blanks, true or false and short answer questions.
Textbook Questions and Answers
Let’s Play Ludo — Starter Questions
Ten children play Ludo, each throwing a dice once. The numbers they get are: 3, 2, 6, 1, 5, 4, 2, 4, 2, 1.
1. How many of them got even numbers?
Answer: The even numbers are 2, 6, 4, 2, 4, 2 — so 6 of them got even numbers.
2. How many of them got odd numbers?
Answer: The odd numbers are 3, 1, 5, 1 — so 4 of them got odd numbers.
3. Arrange the numbers in ascending order.
Answer: 1, 1, 2, 2, 2, 3, 4, 4, 5, 6.
4. How many consecutive pairs of numbers are there whose sum is even?
Answer: The consecutive-pair sums are 3+2=5, 2+6=8, 6+1=7, 1+5=6, 5+4=9, 4+2=6, 2+4=6, 4+2=6, 2+1=3. Those with an even sum are (2,6), (1,5), (4,2), (2,4), (4,2) — so 5 pairs.
5. How many consecutive pairs of numbers are there whose sum is odd?
Answer: Of the 9 consecutive pairs, the remaining 4 have an odd sum — (3,2), (6,1), (5,4), (2,1).
6. Which numbers have occurred the most and the least?
Answer: The number 2 occurred the most (3 times). The numbers 3, 5 and 6 each occurred the least — just once.
7. What is the difference between the largest and smallest numbers that they got?
Answer: Largest = 6, smallest = 1; difference = 6 − 1 = 5.
Magic Cells — The Rule
A cell is coloured if the number in it is smaller than both of its adjacent cells. An end cell (with only one adjacent cell) is coloured if it is smaller than that single neighbour. For example, in the table below 12 is coloured (smaller than 65 and 31) and 205 is coloured (its only neighbour is 582 and 205 < 582).
| Row | Numbers | |||||||
|---|---|---|---|---|---|---|---|---|
| Top row | 45 | 82 | 77 | 65 | 12 | 31 | 30 | 36 |
| Bottom row | 205 | 582 | 631 | 350 | 795 | 699 | 114 | 203 |
Applying the rule, the coloured cells (smaller than both neighbours) are 45, 12, 30 in the top row and 205, 350, 114 in the bottom row — 6 cells in all.
Work it Out 3.1
1. Colour the numbers in the following magic cells using the same pattern as discussed above: 9843, 794, 4732, 3971, 2587, 8753, 8947, 9107, 456, 997.
Answer: Only a number smaller than both its neighbours is coloured. Checking each — 794 (smaller than 9843 and 4732), 2587 (smaller than 3971 and 8753) and 456 (smaller than 9107 and 997). So the coloured cells are 794, 2587, 456 — 3 cells.
2. Fill up the following magic cells with only 4-digit numbers using the same pattern: 3986, ___, 6837, ___, 8105, ___, 7516.
Answer: If each blank is filled with a 4-digit number smaller than both its neighbours, the pattern holds. One possible answer is 3986, 1234, 6837, 2345, 8105, 3456, 7516. Here the coloured cells are 1234, 2345 and 3456 (each smaller than both neighbours). Many such fillings are possible.
3. Using the numbers between 1,000 and 10,000 without repeating, construct such magic cells in as many ways as possible. Out of the 9 cells, how many numbers are coloured?
Answer: Placing large and small numbers alternately gives more coloured cells. For example, a strip of nine distinct 4-digit numbers 5000, 1200, 8000, 2300, 9000, 3400, 7000, 1500, 6000 has 1200, 2300, 3400 and 1500 coloured (each smaller than both neighbours) — that is 4 of the 9 cells. The count depends on the arrangement; at most 4 of the 9 cells can be coloured.
4. Using the table in question 1, colour a cell if the number in it is larger than its adjacent cells.
Answer: Now the numbers larger than both neighbours are coloured — 9843 (end, larger than 794), 4732 (larger than 794 and 3971), 9107 (larger than 8947 and 456) and 997 (end, larger than 456). So the coloured cells are 9843, 4732, 9107, 997 — 4 cells.
Work it Out 3.2 (Number Line)
1. By observing the natural numbers on the following number lines, fill up the remaining blank positions.
Answer: Find the equal spacing from the two known marks and the number of gaps between them, then fill the blanks.
- (a) Between 4000 and 12000 each step is 2000: 4000, 6000, 8000, 10000, 12000.
- (b) 26945 and 26947 are consecutive natural numbers, so each step is 1: …, 26944, 26945, 26946, 26947, 26948, …
- (c) The spacing between 1743, 1837 and 1978 is equal, each step being 47: 1743, 1790, 1837, 1884, 1931, 1978.
Playing with Digits — Counting Numbers
Fill the following table using the pattern above. (For example, the 4-digit numbers run from 1000 to 9999, that is (9999 − 1000) + 1 = 9000 numbers.)
| 1-digit numbers (1–9) | 2-digit numbers (10–99) | 3-digit numbers (100–999) | 4-digit numbers (1000–9999) | 5-digit numbers (10000–99999) |
|---|---|---|---|---|
| 9 | 90 | 900 | 9000 | 90000 |
Work it Out 3.3 (Digit Sums)
1. Write five other numbers whose digit sum is 12.
Answer: 39, 48, 84, 156 and 903 (each has digit sum 12 — for example 1+5+6 = 12 and 9+0+3 = 12).
2. What is the largest two-digit number whose digit sum is 12?
Answer: Make the tens digit as large as possible; 9 + 3 = 12, so the answer is 93.
3. Fill in the blank.
- (i) The 3-digit number 1 6 __ has digit sum 13 → 1 + 6 + __ = 13, so the digit is 6; the number is 166.
- (ii) The 4-digit number 2 __ 7 5 has digit sum 17 → 2 + __ + 7 + 5 = 17, so the digit is 3; the number is 2375.
- (iii) The 5-digit number __ 9 4 6 5 has digit sum 30 → __ + 24 = 30, so the digit is 6; the number is 69465.
- (iv) The 6-digit number 5 __ 6 __ 3 8 has digit sum 31 → the two missing digits add up to 31 − 22 = 9. The pairs are (0,9), (1,8), …, (9,0) — so 10 such six-digit numbers can be formed (for example 546538).
4. Determine the digit sum of the 3-digit numbers whose digits are consecutive odd (for example, 135).
Answer: Such numbers are 135, 357, 579. Their digit sums are 1+3+5 = 9, 3+5+7 = 15 and 5+7+9 = 21.
5. Determine the digit sum of the 3-digit numbers whose digits are consecutive even.
Answer: Such numbers are 024, 246, 468. Their digit sums are 0+2+4 = 6, 2+4+6 = 12 and 4+6+8 = 18.
6. Did you see any pattern in the two problems above (questions 4 and 5)? Are the digit sums divisible by any specific number?
Answer: Yes. For three consecutive odd or even digits the digit sum is always three times the middle digit (for example 357 gives 3 × 5 = 15). So the digit sums are always divisible by 3.
7. Reverse the order of the digits in the numbers of questions 4 and 5 and find the digit sums.
Answer: Reversing the order does not change the digits, so the digit sum stays the same — for example 135 → 531 still sums to 9, and 246 → 642 still sums to 12.
Searching Digits in a Number
From 1 to 100 the digit 3 occurs 20 times. How many times does the digit 5 occur? The digit 8? And ‘0’? Is there any similarity in how the digits 1 to 9 occur?
Answer: From 1 to 100 — the digit 5 occurs 20 times (in the units place 5, 15, …, 95 = 10 times, and in the tens place 50–59 = 10 times); likewise the digit 8 occurs 20 times. The digit 0 occurs 11 times (10, 20, …, 90 = 9 times, plus two zeros in 100). The similarity: each of the digits 1 to 9 occurs equally often, 20 times (the digit 1 once more because of 100); only 0 occurs fewer times.
Operations of Numbers — Place Suitable Numbers
Place suitable numbers for each of the following. (Many answers are possible; one example is given for each.)
| Required | One example |
|---|---|
| 3-digit + 3-digit = 4-digit number | 500 + 700 = 1200 |
| 4-digit − 3-digit = 3-digit number | 1200 − 500 = 700 |
| 6-digit + 6-digit = 6-digit number | 500000 + 400000 = 900000 |
| 5-digit − 5-digit = 3-digit number | 10500 − 10200 = 300 |
| 6-digit − 5-digit = 437125 | 500000 − 62875 = 437125 |
| 4-digit + 5-digit = a number more than 47126 | 9000 + 40000 = 49000 |
Work it Out 3.4 (Palindromic Numbers)
A number that reads the same from left to right and right to left is a palindromic number (such as 363, 5335). Reversing a number and adding it repeatedly produces a palindromic number.
1. Find the palindromic number formed from 78, 489 and 3467.
- 78 → 78 + 87 = 165 → 165 + 561 = 726 → 726 + 627 = 1353 → 1353 + 3531 = 4884 (palindrome).
- 489 → 489 + 984 = 1473 → 1473 + 3741 = 5214 → 5214 + 4125 = 9339 (palindrome).
- 3467 → 3467 + 7643 = 11110 → 11110 + 1111 = 12221 (palindrome).
2. Solve the puzzle: I am a 5-digit palindrome. My unit digit is one less than its tens digit. The product of the unit and tens digits is the hundreds digit. The sum of all the digits is 16. Find me!
Answer: Let the palindrome be abcba, where the unit digit = a and the tens digit = b. Then a = b − 1, the hundreds digit c = a × b, and 2a + 2b + c = 16. Solving gives a = 2, b = 3, c = 2 × 3 = 6. So the number is 23632 (2+3+6+3+2 = 16, which checks out).
3. What is the sum of the smallest and largest 5-digit palindromic numbers? What is their difference?
Answer: The smallest 5-digit palindrome is 10001 and the largest is 99999. Sum = 10001 + 99999 = 110000; difference = 99999 − 10001 = 89998.
Work it Out 3.5 (Kaprekar Constant)
For any 4-digit number with at least two different digits, let B = the largest number formed by rearranging its digits, C = the smallest, and D = B − C. Repeating this process always leads to 6174, the Kaprekar constant.
1. Try the process for the 4-digit numbers 1435, 8751 and 8632.
- 8632 → 8632 − 2368 = 6264 → 6642 − 2466 = 4176 → 7641 − 1467 = 6174 (3 rounds).
- 8751 → 8751 − 1578 = 7173 → 7731 − 1377 = 6354 → 6543 − 3456 = 3087 → 8730 − 378 = 8352 → 8532 − 2358 = 6174 (5 rounds).
- 1435 → 5431 − 1345 = 4086 → 8640 − 468 = 8172 → 8721 − 1278 = 7443 → 7443 − 3447 = 3996 → 9963 − 3699 = 6264 → 4176 → 6174 (7 rounds).
2. Take some more 4-digit numbers and try to reach the Kaprekar constant.
Answer: Any such number gives the same result. For example, 3524 → 5432 − 2345 = 3087 → 8730 − 378 = 8352 → 8532 − 2358 = 6174. Every 4-digit number with at least two different digits eventually settles at 6174.
3. How many rounds does the number 9632 take to reach the Kaprekar constant?
Answer: 9632 → 7263 → 5265 → 3996 → 6264 → 4176 → 6174 — so it takes 6 rounds.
4. Try the same steps with some 3-digit numbers to get a Kaprekar constant.
Answer: For 3-digit numbers the same process always leads to 495. For example, 352 → 532 − 235 = 297 → 972 − 279 = 693 → 963 − 369 = 594 → 954 − 459 = 495. So the 3-digit Kaprekar constant is 495.
Work it Out 3.6 (The Number 1089)
For any 3-digit number whose first and third digits differ by at least 2, take B = the number with digits reversed, C = larger minus smaller, D = C reversed, and E = C + D. The result is always 1089. Such plays existed in the ancient Assamese system of arithmetic called ‘Kaithali Anka’, composed by Dandiram Dutta.
1. Try the process for the 3-digit numbers 281, 164 and 752.
- 164 → 461 − 164 = 297 → 297 + 792 = 1089.
- 752 → 752 − 257 = 495 → 495 + 594 = 1089.
- 281 → its first and third digits differ by only 1, so 281 − 182 = 99; treating this as the 3-digit 099, 099 + 990 = 1089.
Work it Out 3.7 (Clock and Calendar)
Subtracting 12:00 hrs from a 24-hour time gives the p.m. time on a 12-hour clock; up to 12:00 hrs no conversion is needed.
1. Convert these 24-hour timings to 12-hour timings: 5:10 hrs, 11:45 hrs, 15:40 hrs, 18:20 hrs, 17:45 hrs, 22:10 hrs.
| 24-hour | 12-hour |
|---|---|
| 5:10 hrs | 5:10 a.m. |
| 11:45 hrs | 11:45 a.m. |
| 15:40 hrs | 3:40 p.m. |
| 18:20 hrs | 6:20 p.m. |
| 17:45 hrs | 5:45 p.m. |
| 22:10 hrs | 10:10 p.m. |
2. Convert these 12-hour timings to 24-hour timings: 1:45 a.m., 4:45 p.m., 3:20 p.m., 9:35 p.m.
| 12-hour | 24-hour |
|---|---|
| 1:45 a.m. | 01:45 hrs |
| 4:45 p.m. | 16:45 hrs |
| 3:20 p.m. | 15:20 hrs |
| 9:35 p.m. | 21:35 hrs |
3. Determine some timings in the 12-hour and 24-hour clock that follow a palindrome pattern.
Answer: On a 12-hour clock — 1:01, 2:02, 3:03, 10:01, 11:11, 12:21, and so on (for example 12:21 reads as 1221 either way). On a 24-hour clock — 01:10, 02:20, 12:21, 13:31, 20:02, 21:12, 23:32, and so on.
4. Write the timings that are the smallest and largest palindromic numbers on a 24-hour clock.
Answer: The smallest palindromic time is 00:00 and the largest is 23:32.
5. Write a few timings that give a palindromic number when converted from 24-hour to 12-hour clock.
Answer: For example 13:31 hrs → 1:31 p.m. (131), 14:02 hrs → 2:02 p.m. (202) and 15:03 hrs → 3:03 p.m. (303) — after conversion the timings read as palindromic numbers.
Mental Math
Add or subtract the middle-column numbers (17000, 200, 4200, 50000, 2000) to obtain the remaining left- and right-column numbers. (For example, 6400 = 4200 + 2000 + 200; 47400 = 50000 − 2000 − 200 − 200 − 200.)
| Number | One possible way |
|---|---|
| 23400 | 17000 + 4200 + 2000 + 200 |
| 15000 | 17000 − 2000 |
| 8400 | 4200 + 4200 |
| 63200 | 50000 + 17000 − 4200 + 200 + 200 |
| 1400 | 2000 − 200 − 200 − 200 |
| 58800 | 50000 + 4200 + 4200 + 200 + 200 |
| 19200 | 17000 + 2000 + 200 |
| 56200 | 50000 + 4200 + 2000 |
Work it Out 3.8 (The Collatz Conjecture)
1. Draw a shape for the numbers 18, 25, 31, 42 and 100 based on the Collatz Conjecture.
Answer: The rule is — if the number is even, divide by 2; if it is odd, triple it and add 1; continue until you reach 1. The Collatz sequence for each number is —
- 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1.
- 18 → 9 → 28 → 14 → 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
- 25 → 76 → 38 → 19 → 58 → 29 → 88 → 44 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
- 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.
- 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → … (after many steps) … → 16 → 8 → 4 → 2 → 1 (106 steps in all).
Drawing each sequence as a branch, the branches all merge into the common tail … 16, 8, 4, 2, 1, forming a tree-like shape.
Math Magic
Magic 1: Think of a number from 0 to 10 → multiply by 2 → add 2 and multiply by 5 → subtract 1 → the unit digit is always 9. For example 8 → 8 × 2 = 16 → (16 + 2) × 5 = 90 → 90 − 1 = 89 → unit digit 9.
Magic 2: Think of a 3-digit number with no repeated digit → reverse its digits → subtract the smaller from the larger → add the digits of the result → you always get 18. For example 376 → 673 → 673 − 376 = 297 → 2 + 9 + 7 = 18.
Work it Out 3.9 (Estimation)
1. Try estimating the following. (Estimation is a reasonable guess, not an exact figure; answers vary by school and locality.)
- (a) Total number of students in your school — if about 40 per class and 12 classes, roughly 480–500.
- (b) Distance between school and home — say about 2 km.
- (c) Water you consume in a week — about 2–3 litres a day, so roughly 15–20 litres.
- (d) Total population of your village or locality — say about 2000.
- (e) Length of your school playground — say about 100 metres.
- (f) Seema estimates ₹200 for 1 kg apples and 2 kg oranges. If apples are ₹120/kg and oranges ₹60/kg, the cost is 120 + 120 = ₹240 — so ₹200 is probably too low; the estimate should be revised depending on the actual rates.
- (g) Roshan thinks he can run 1 km in 10 minutes, that is 6 km per hour — this is reasonable for a Class VI student, so I agree.
- (h) Today’s temperature and whether it is hotter than yesterday — estimate from the season and the day’s conditions.
- (i) Students present in school today — usually about 90% of the total, so about 450 out of 500.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The Kaprekar constant is — (a) 1089 (b) 6174 (c) 495 (d) 1000
Answer: (b) 6174
2. The 3-digit Kaprekar constant is — (a) 495 (b) 594 (c) 6174 (d) 297
Answer: (a) 495
3. Which of these is a palindromic number? (a) 123 (b) 231 (c) 363 (d) 450
Answer: (c) 363
4. How many 4-digit numbers are there in all? (a) 900 (b) 9000 (c) 90000 (d) 1000
Answer: (b) 9000
5. In the Collatz conjecture, what is done to an odd number? (a) divide by 2 (b) triple it and add 1 (c) subtract 1 (d) add 2
Answer: (b) triple it and add 1
6. The time 17:45 hrs (24-hour) on a 12-hour clock is — (a) 7:45 a.m. (b) 5:45 a.m. (c) 5:45 p.m. (d) 7:45 p.m.
Answer: (c) 5:45 p.m.
7. The smallest 5-digit palindromic number is — (a) 10000 (b) 10001 (c) 11111 (d) 12321
Answer: (b) 10001
8. The ancient Assamese system of arithmetic linked with the number 1089 game is — (a) Kaithali Anka (b) Decimal (c) Roman (d) Binary
Answer: (a) Kaithali Anka
9. How many times does the digit 5 occur from 1 to 100? (a) 10 (b) 11 (c) 20 (d) 21
Answer: (c) 20
10. The last number in a Collatz sequence is always — (a) 0 (b) 1 (c) 2 (d) 6174
Answer: (b) 1
Fill in the Blanks
- A number that reads the same from left and right is called a ______ number. (palindromic)
- There are ______ two-digit numbers in all. (90)
- The Kaprekar process on any 4-digit number (with at least two different digits) gives ______. (6174)
- Subtracting ______ from a 24-hour time gives the p.m. time. (12:00 hrs)
- The Collatz conjecture was proposed in ______ by Lothar Collatz. (1937)
True or False
- 363 is a palindromic number. — True
- There are 999 three-digit numbers in all. — False (there are 900)
- In a Collatz sequence an even number is divided by 2. — True
- The Kaprekar process on any 6-digit number gives 6174. — False (6174 is for 4-digit numbers)
- 16:10 hrs means 4:10 p.m. — True
Short Answer Questions
1. What is a palindromic number? Give two examples.
Answer: A number that reads the same from left to right and from right to left is a palindromic number. Examples are 363 and 5335.
2. Briefly state the process for reaching the Kaprekar constant.
Answer: Take a 4-digit number with at least two different digits, arrange its digits to form the largest number (B) and the smallest number (C), and find D = B − C. Repeating this always ends at 6174.
3. Write the Collatz sequence for 13.
Answer: 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
4. What is 22:00 hrs on a 12-hour clock?
Answer: 22:00 hrs − 12:00 hrs = 10:00 p.m.
5. What is estimation? Give an example.
Answer: Estimation is finding an approximate value by a reasonable guess instead of an exact calculation. For example, if a bus covers 120 km in 3 hours we take its speed as about 40 km/h and estimate the remaining travel time.
Key Terms
| Term | Meaning |
|---|---|
| Even number | A number divisible by 2 (2, 4, 6, …) |
| Odd number | A number not divisible by 2 (1, 3, 5, …) |
| Number line | An equally spaced line that shows the position of numbers |
| Digit | The symbols 0–9 that make up numbers |
| Digit sum | The value obtained by adding all the digits of a number |
| Palindromic number | A number that reads the same from both ends |
| Kaprekar constant | The magic result 6174 for 4-digit numbers |
| Collatz Conjecture | The rule: halve if even, else 3n+1, always reaching 1 |
| Estimation | Finding an approximate value by a reasonable guess |
| Kaithali Anka | The ancient Assam system of arithmetic and measurement |