Chapter 3: Number Play –QUESTION BANK ( Study Material cum) Worksheet
Chapter Subsections Covered
3.1 Numbers can Tell us Things 3.2 Supercells
3.3 Patterns of Numbers on the Number Line 3.4 Playing with Digits
3.5 Pretty Palindromic Patterns 3.6 The Magic Number of Kaprekar
3.7 Clock and Calendar Numbers 3.8 Mental Math
3.9 Playing with Number Patterns 3.10 Collatz Conjecture
3.11 Simple Estimation 3.12 Games and Winning Strategies
1. Multiple Choice Questions (MCQs)
Which of the following is a palindromic number?
a) 121 b) 132 c) 145 d) 234 (Recognising palindromes)
The Kaprekar constant is:
a) 6174 b) 1729 c) 1089 d) 1001 (Number pattern recognition
The sum of digits of 1089 is:
a) 18 b) 10 c) 9 d) 19 (Playing with digits)
In Kaprekar’s process, repeatedly subtracting the largest and smallest 4-digit numbers formed by digits leads to:
a) 1729 b) 6174 c) 1000 d) 9999 (Kaprekar’s constant)
The Collatz Conjecture states that any positive integer will eventually reach:
a) 0 b) 1 c) 2 d) 4 (Understanding conjectures)
Which of the following is not a palindrome?
a) 2002 b) 2332 c) 3223 d) 1234 (Palindrome test)
Which number is a palindrome?
a) 1001 b) 1111 c) 1221 d) All of these (Palindromes)
adding the digits of the number 68 will be same as adding the digits of _______
a) 108 b) 176 c) 729 d) 181 (digit sum)
1. In the 'taller neighbours' game, a child says '2'. What does this mean?
a) Both neighbours are shorter. b) Both neighbours are taller. c) One neighbour is taller and one is shorter. d) The child is the tallest. (Competency: Problem Solving & Reasoning)
A number in a grid is colored as a 'supercell' if:
a) It is an even number. b) It is larger than all its adjacent cells. c) It is a palindrome. d) Its digit sum is 10. (Competency: Analytical Thinking)
The Kaprekar constant for 4-digit numbers is:
a) 6174 b) 495 c) 1089 d) 9999 (Competency: Knowledge & Recall)
Which of these is a palindromic number?
a) 1234 b) 2002 c) 2012 d) 1001 (Competency: Pattern Recognition)
According to the Collatz Conjecture, if you start with an even number, what is the next step?
a) Multiply by 3 and add 1. b) Find its half. c) Reverse its digits. d) Subtract 1. (Competency: Understanding Concepts)
The smallest number whose digit sum is 14 is:
a) 59 b) 77 c) 149 d) 68 (Competency: Logical Reasoning)
In the 'Game of 21', if you want to force a win, you must aim to say the number:
a) 20 b) 17 c) 13 d) 1 (Competency: Strategic Thinking)
Estimate the number of breaths a person takes in one hour.
a) About 600 b) About 1000 c) About 900 d) About 1500 (Competency: Estimation & Application)
The digit sum of the number 5683 is:
a) 21 b) 22 c) 20 d) 19 (Competency: Numerical Calculation)
Which of these sequences will eventually reach 1 according to the Collatz Conjecture?
a) 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 b) 4, 2, 1
c) 6, 3, 10, 5, 16, 8, 4, 2, 1 d) All of the above (Competency: Critical Thinking)
A number on a clock that reads the same forwards and backwards is called a:
a) Kaprekar number b) Supercell c) Palindromic time d) Magic number (Competency: Knowledge & Recall)
To get the maximum number of supercells in a grid, you should place the largest numbers:
a) In the corners b) In the center c) Next to each other d) In cells with the fewest neighbors
(Competency: Analytical Thinking & Strategy)
The difference between the largest and smallest 4-digit numbers formed using 4, 7, 3, 2 is:
a) 5085 b) 5086 c) 5084 d) 5087 (Competency: Numerical Calculation)
How many 2-digit numbers are there?
a) 99 b) 89 c) 90 d) 100 (Competency: Knowledge & Recall)
In the mental math section, which number from the middle column is used multiple times to make 3,400?
a) 25,000 b) 400 c) 1,500 d) 13,000 (Competency: Mental Calculation & Application)
The largest 5-digit number with a digit sum of 14 is:
a) 95000 b) 94100 c) 93200 d) 90050 (Competency: Logical Reasoning & Problem Solving)
A calendar can be reused after how many years?
a) 5 years b) 6 years c) 11 years d) It depends on the leap year cycle. (Competency: Real-life Application & Reasoning)
The sum of the smallest and largest 5-digit palindromes is:
a) 109989 b) 100000 c) 199998 d) 109998 (Competency: Numerical Calculation & Analysis)
In the 'Game of 99', the winning strategy involves controlling multiples of:
a) 9 b) 10 c) 11 d) 12 (Competency: Strategic Thinking)
The number of times the digit '7' appears from 1 to 100 is:
a) 10 b) 11 c) 19 d) 20 (Competency: Systematic Counting & Reasoning)
2. Assertion and Reasoning Questions
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice.
a) Both A and R are true and R is the correct explanation of A.
b) Both A and R are true but R is NOT the correct explanation of A.
c) A is true but R is false.
d) A is false but R is true.
1. Assertion (A): The number 6174 is known as the Kaprekar constant.
Reason (R): For any 4-digit number with at least two different digits, the process of arranging digits in descending and ascending order and subtracting will always eventually reach 6174.
(Competency: Conceptual Understanding & Reasoning)
2. Assertion (A): The sequence 0, 1, 2, 1, 0 is possible for five children of different heights standing in a line.
Reason (R): The child saying '2' must have two taller neighbors, which is only possible if they are the shortest and in the middle.
(Competency: Logical Reasoning & Problem Solving)
3. Assertion (A): The cell with the smallest number in a grid can never be a supercell.
Reason (R): A supercell must be larger than all its adjacent cells.
(Competency: Analytical Thinking)
4. Assertion (A): The Collatz Conjecture has been proven true for all numbers.
Reason (R): Every sequence starting with a whole number will eventually reach the cycle 4, 2, 1.
(Competency: Knowledge & Critical Thinking)
5. Assertion (A): The sum of a 5-digit number and a 3-digit number can never be a 6-digit number.
Reason (R): The maximum sum of a 5-digit number (99,999) and a 3-digit number (999) is 100,998, which is a 6-digit number. (Competency: Numerical Analysis & Reasoning)
6.A: 121 is a palindrome.
R: A number that reads the same backward and forward is a palindrome. (Palindrome recognition)
(Competency: Conceptual Understanding)
7. A: Collatz sequence starting at 6 ends at 1.
R: The sequence goes 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. (Collatz conjecture)
(Competency: Strategic Thinking)
8. A: Kaprekar’s constant 6174 is reached in at most 7 steps for any 4-digit number (not all digits same).
R: The process converges by repeated subtraction. (Kaprekar’s process)
9. Assertion (A): The digit sum of 999 is 27.
Reason (R): The digit sum is the sum of all digits of a number.
(Competency: Numerical Calculation)
10. Assertion (A): The date 02/02/2020 is a palindromic date.
Reason (R): It reads the same forwards and backwards when written in DD/MM/YYYY format.
(Competency: Pattern Recognition & Real-life Connection)
3. True or False – justify your reason
All palindromes are divisible by 11. (False) (Competency:Palindrome test)
1089 is obtained in a digit reversal trick. (True) (Competency:Playing with digits)
Kaprekar’s constant is 1729. (False) (Competency:Kaprekar’s process)
6174 is known as Kaprekar’s number. (True) (Competency:Number property)
Collatz Conjecture ends at 1. (True) (Competency:Conjecture understanding)
Every palindrome is a square number. (False) (Competency:Palindrome recognition)
121 is both a palindrome and a perfect square. (True) (Competency:Pattern recognition)
The number of 3-digit numbers is greater than the number of 2-digit numbers. (True) (Competency: Conceptual Understanding)
The sum of two 5-digit numbers can be a 4-digit number. (False)(Competency: Numerical Reasoning)
The number 495 is the Kaprekar constant for 3-digit numbers. (True) (Competency: Knowledge & Recall)
4. Short Answer I (2 Marks) – 15 Questions
Write any three palindromic numbers between 100 and 200. (Competency:Palindrome identification)
State Kaprekar’s constant. (Competency:Kaprekar’s process)
Find the sum of digits of 1234. (Competency:Playing with digits)
Estimate the number of liters a mug, a bucket and an overhead tank can hold.
Write one digit number and two digit numbers such that their sum is 18,670.
Colour or mark the supercells in the table below
Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.
State Collatz Conjecture in your own words. (Understanding conjectures)
Write a palindrome less than 100. (Palindrome recognition)
Check if the Collatz Conjecture holds for the starting number 100
Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?
place the following numbers in their appropriate positions on the number line: 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 positions. Put a circle around the smallest number and a box around the largest number in each of the sequences above
Fill the table below such that we get as many supercells as possible USe numbers between 100 and 1000 without repetitions. OUt of the numbers, how many supercells are there in the table above?
5. Short Answer II (3 Marks) – 10 Questions
Show that 121 is both a palindrome and a square. (Application of patterns)
Complete Table 2 with digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘ , and ‘ in some order 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 5,000 in the table is___________
Show with an example how Kaprekar’s process works. (Kaprekar’s process)
Verify Collatz sequence starting with 6. (Collatz conjecture)
There is only one supercell (number greater than all its neighbours) in this gridIf you exchange two digits of one of the numbers, there will be 4 supercells Figure out which digits to swap
Find out how many numbers have two digits, three digits, four digits, and five digits
Estimate the product of 498 × 52 by rounding. (Estimation)
Complete the collatz conjecture. a) 12, 6, _________________________
b) 17, 52 ,________________ c) 21, 64,_______________, d ) 22, 11, ___________________
How many rounds does the number 5683 take to reach the Kaprekar constant?
Draw angles with the following degree measures a)140°b) 82°
The time now is 10:01 How many minutes until the clock shows the next palindromic time? What about the one after that?
What is the sum of the smallest and largest digit palindrome? What is their difference?
Write an example for each of the below scenarios whenever possible
5 digit + 5 digit to give a digit sum more than 90250
5 digit - 5 digit to give a difference less than 56503
5 digit + 3 digit to give a 6-digit sum
4-digit + 4-digit to give a 6-digit sum
5 digit – 4 digit to give a 4-digit difference
5 digit +5 digit to give a 6-digit sum
5 digit + 5 digit to give 18500
5 digit – 3 digit to give a 4-digit difference
5 digit – 5 digit to give a 3-digit difference
5digit − 5 digit to give 91500
Write always, sometimes, Never?
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 – digit number gives a 5 digit number
e) 5 digit number – 2-digit number gives a 3 digit number
6. Long Answer (5 Marks) – 10 Questions
Explain Kaprekar’s process in detail with an example starting with 3524. (Kaprekar’s constant)
How many rounds does your year of birth take to reach the Kaprekar constant?
Explain Collatz Conjecture using number 11. (Conjecture analysis)
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 mak e
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.
Explain the importance of estimation in real-life with two examples. (Estimation application)
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 digit 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?
7. Case-Based Questions (5 Sets, 4 MCQs each)
Case 1 – Kaprekar’s Process
D.R. Kaprekar, an Indian mathematician, discovered a fascinating number trick in 1949. If we take any 4-digit number (with at least two different digits), arrange its digits in descending and ascending order, and subtract, the process eventually leads to a special number – 6174, known as the Kaprekar constant. For example, starting with 3524, the largest number is 5432 and the smallest is 2345. Their difference is 3087. Repeating this step again and again finally brings us to 6174. This shows how numbers can settle into surprising and beautiful patterns. A student chooses number 3524 and applies Kaprekar’s process.
1.1) What is the largest number formed from 3524?
A) 5432 B) 4523 C) 5324 D) 5243
1.2) What is the smallest number formed?
A) 2543 B) 2345 C) 2435 D) 2534
1.3) What is their difference?
A) 2187 B) 4087 C) 3087 D) 6187
1.4) What constant is obtained after repeating the process?
A) 495 B) 6174 C) 9999 D) 2025
Case 2 – Collatz Conjecture
The Collatz Conjecture is a simple yet mysterious idea in mathematics. It says: start with any whole number – if it’s even, divide it by 2; if it’s odd, multiply it by 3 and add 1. Repeat the process. For example, starting with 10, we get 10 → 5 → 16 → 8 → 4 → 2 → 1. No matter where we begin, the sequence seems to always reach 1. This unsolved puzzle, posed by Lothar Collatz in 1937, continues to amaze mathematicians because although it looks simple, nobody has proved it true for all numbers. Starting with 10, the Collatz process is applied.
2.1) What is the next number after dividing 10 by 2?
A) 2 B) 3 C) 5 D) 8
2.2) What happens after reaching 5?
A) Stop at 5 B) Multiply by 3 and add 1 → 16 C) Divide by 2 → 2.5 D) Start again from 10
2.3) Does the sequence reach 1?
A) No B) Yes C) Only for even numbers D) Only for prime numbers
2.4) What does the Collatz conjecture state?
A) Every number eventually reaches 0 B) Every number eventually reaches 1
C) Only even numbers reach 1 D) Numbers cycle forever (CBQ, Collatz conjecture)
Case 4 – Estimation in Daily Life
Estimation helps us handle numbers in everyday life without always calculating exact values. For instance, a shopkeeper with 198 pencils, 102 pens, and 298 erasers can round them as 200, 100, and 300, giving an estimated total of about 600 items. The actual total is 598, very close to the estimate. Estimation saves time, allows quick decisions, and is especially useful in shopping, planning, or when exact figures are unnecessary. A shopkeeper has 198 pencils, 102 pens, and 298 erasers.
4.1 ) Estimate total items by rounding.
A) 500 B) 550 C) 600 D) 650
4.2) What is the actual total?
A) 598 B) 600 C) 602 D) 608
4.3) What is the difference between estimated and actual?
A) 0 B) 2 C) 5 D) 10
4.4) Why is estimation useful here?
A) It gives exact answers B) It saves time and helps in quick decisions
C) It avoids subtraction D) It always matches the total (CBQ, Estimation)
Case 5 – Number Games
Numbers can be turned into fun puzzles and games. One such game is reversing digits of 3-digit numbers and performing operations. For example, reversing 231 gives 132. Subtracting the smaller from the larger (231 – 132 = 99), and then adding the digits of the result (9 + 9 = 18) often shows interesting patterns. Playing with numbers in this way helps us notice hidden properties, predict results, and enjoy mathematics as a game of discovery. A game involves reversing digits of 3-digit numbers.
5.1) What is the reverse of 231?
A) 123 B) 132 C) 321 D) 213
5.2) Subtract smaller from larger (231 – 132).
A) 199 B) 109 C) 99 D) 111
5.3) Add digits of the result (99).
A) 9 B) 18 C) 27 D) 99
5.4) What pattern do you observe?
A) Always prime numbers B) Always multiples of 9
C) Always even numbers D) Always ends in 0 (CBQ, Number patterns)
