Class 6 › Mathematics › Ganita Prakash Number Play Class 6 Mathematics · Ganita Prakash · Chapter 3
Practice all → Summary Questions & answers Practise this chapter → This chapter explores numbers through games, puzzles and patterns, building computational thinking. It opens with a line of children each saying the number of taller neighbours they have, showing how numbers convey information. Supercells are cells whose number is larger than all neighbouring cells; students fill grids to maximise supercells. The chapter places large numbers on number lines and counts how many 1-, 2-, 3-, 4- and 5-digit numbers exist (\(9, 90, 900, 9000, 90000\)). It studies digit sums, palindromes (numbers that read the same both ways, such as 575 or 12321), and the reverse-and-add method. A highlight is the Kaprekar constant: repeatedly making the largest minus the smallest number from a 4-digit number always reaches 6174. Clock and calendar number patterns, mental-math methods, and number-pattern sums are covered. The famous unsolved Collatz conjecture is introduced \u2014 halving even numbers and applying \(3n+1\) to odd numbers always seems to reach 1. The chapter ends with simple estimation and winning strategies for number games like reaching 21.
Supercell
A cell whose number is greater than all of its neighbouring cells.
Palindrome
A number that reads the same from left to right and right to left, such as 66, 575 or 12321.
Kaprekar constant
The number 6174, reached by repeatedly subtracting the smallest from the largest arrangement of a 4-digit number\u2019s digits.
Digit sum
The number obtained by adding all the digits of a number, e.g. the digit sum of 68 is \(6+8=14\).
Collatz conjecture
The unproven claim that halving even numbers and using \(3n+1\) for odd numbers always reaches 1.
Estimation
Finding an approximate value when an exact count is not needed.
Q1 What is a supercell in a number grid? Answer Q2 What is a palindromic number? Give an example. Answer Q3 What is the Kaprekar constant? Answer Q4 How many 3-digit numbers and 5-digit numbers are there? Answer Q5 What is the smallest number whose digit sum is 14? Answer Q6 What rule defines a Collatz sequence? Answer Q7 What is the sum of the smallest and largest 5-digit palindromes? Answer Q8 In the game of reaching 21 by adding 1, 2 or 3, what is the winning strategy? Answer Q9 What is estimation and why is it useful? Answer Q10 Why is the Collatz conjecture famous? Answer Q1 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. Answer Q1 1. Can the children rearrange themselves so that the children standing at the ends say ‘2’? Answer Q2 2. Can we arrange the children in a line so that all would say only 0s? Answer Q3 3. Can two children standing next to each other say the same number? Answer Q4 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? Answer Q5 5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible? Answer Q6 6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not? Answer Q7 7. How would you rearrange the five children so that the maximum number of children say ‘2’? Answer Q1 1. Colour or mark the supercells in the table below. 6828 670 9435 3780 3708 7308 8000 5583 52 Answer Q2 2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells. 5346 1258 9635 Answer Q3 3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions. Answer Q4 4. Out of the 9 numbers, how many supercells are there in the table above? ___________ Answer Q5 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. Answer Q6 6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not? Answer Q7 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? Answer Q8 8. Fill a table such that the cell having the second largest number is not a supercell. Answer Q9 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? Answer Q10 10. Make other variations of this puzzle and challenge your classmates. Answer Q1 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. Answer Q2 The biggest number in the table is ____________. Answer Q3 The smallest even number in the table is ____________. Answer Q4 The smallest number greater than 50,000 in the table is ____________. Answer Q1 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 above. Answer Q1 Find out how many numbers have two digits, three digits, four digits, and five digits. 1-digit numbers From 1–9 2-digit numbers 3-digit numbers 4-digit numbers 5-digit numbers 9 Answer Q1 1. Digit sum 14 a. Write other numbers whose digits add up to 14. Answer Q2 1. Digit sum 14 b. What is the smallest number whose digit sum is 14? Answer Q3 1. Digit sum 14 c. What is the largest 5-digit whose digit sum is 14? Answer Q4 1. Digit sum 14 d. How big a number can you form having the digit sum of 14? Can you make an even bigger number? Answer Q5 2. Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class. Answer Q6 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? Answer Q1 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? Answer Q1 Write all possible 3-digit palindromes using these digits. The numbers 121, 313, 222 are some examples of palindromes using the digits ‘1’, ‘2’, ‘3’. Answer Q1 Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.* Answer Q1 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? _________________ tth th h t u Answer Q1 Take different 4-digit numbers and try carrying out these steps. Find out what happens. Check with your friends what they got. You will always reach the magic number ‘6174’! The number ‘6174’ is now called the ‘Kaprekar constant’. Carry out these same steps with a few 3-digit numbers. What number will start repeating? Answer Q1 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. Answer Q2 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: b. the difference between the largest and smallest numbers less than 5085. Answer Q3 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: c. the sum of the largest and smallest numbers greater than 9779. Answer Q4 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: d. the sum of the largest and smallest numbers less than 9779. Answer Q5 2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference? Answer Q6 3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that? Answer Q7 4. How many rounds does the number 5683 take to reach the Kaprekar constant? Answer Q1 1. Write an example for each of the below scenarios whenever possible. 5-digit + 5-digit to give a 5-digit sum more than 90,250 5-digit – 5-digit to give a difference less than 56,503 4-digit + 4-digit to give a 6-digit sum 5-digit − 4-digit to give a 4-digit difference 5-digit + 3-digit to give a 6-digit sum 5-digit – 3-digit to give a 4-digit difference 5-digit + 5-digit to give a 6-digit sum 5-digit − 5-digit to give a 3-digit difference 5-digit + 5-digit to give 18,500 5-digit − 5-digit to give 91,500 Answer Q2 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 and discuss this with the class. a. 5-digit number + 5-digit number gives a 5-digit number Answer Q3 2. Always, Sometimes, Never? b. 4-digit number + 2-digit number gives a 4-digit number Answer Q4 2. Always, Sometimes, Never? c. 4-digit number + 2-digit number gives a 6-digit number Answer Q5 2. Always, Sometimes, Never? d. 5-digit number – 5-digit number gives a 5-digit number Answer Q6 2. Always, Sometimes, Never? e. 5-digit number – 2-digit number gives a 3-digit number Answer Q1 1. Steps you would take to walk: a. From the place you are sitting to the classroom door b. Across the school ground from start to end c. From your classroom door to the school gate d. From your school to your home Answer Q2 2. Number of times you blink your eyes or number of breaths you take: a. In a minute b. In an hour c. In a day Answer Q3 3. Name some objects around you that are: a. a few thousand in number b. more than ten thousand in number Estimate the answer Try to guess within 30 seconds. Check your guess with your friends. 1. Number of words in your maths textbook: a. More than 5000 b. Less than 5000 Answer Q4 2. Number of students in your school who travel to school by bus: a. More than 200 b. Less than 200 Answer Q5 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? Answer Q6 4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland). Hint: Look at the map of India to locate these cities. Answer Q7 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? Answer Q8 6. Earlier, people used to walk long distances as they had no other means of transport. Suppose you walk at your normal pace. Approximately, how long would it take you to go from: a. Your current location to one of your favourite places nearby. b. Your current location to any neighbouring state’s capital city. c. The southernmost point in India to the northernmost point in India. Answer Q1 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 45,306 19,381 50,319 38,408 62,871 Answer Q2 2. How many rounds does your year of birth take to reach the Kaprekar constant? Answer Q3 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? Answer Q4 4. 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. Answer Q5 5. Estimate the number of liters a mug, a bucket and an overhead tank can hold. Answer Q6 6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670. Answer Q7 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. Answer Q8 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? Answer Q9 9. Check if the Collatz Conjecture holds for the starting number 100. Answer Q10 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? Answer
A cell in a grid is a supercell if its number is
A greater than all its neighbouring cells B equal to its neighbours C smaller than all its neighbours D an even number
Flashcards & interactive practice 
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