Welcome dear students! Today we are going to learn about Number Play from Class 6 Maths. Numbers are used in different contexts and in many different ways to organise our lives. We have used numbers to count, and have applied the basic operations of addition, subtraction, multiplication and division on them, to solve problems related to our daily lives. In this chapter, we will continue this journey, by playing with numbers, seeing numbers around us, noticing patterns, and learning to use numbers and operations in new ways. 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. You might list things like counting students in a class, measuring the length of a table, checking the price of a notebook, reading the time on a clock, or finding your house number.
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Now let us move to section three point one, Numbers can Tell us Things. What are these numbers telling us? Some children in a park are standing in a line. Each one says a number. The sequence is zero, two, one, one, zero, two, one, zero. The children now rearrange themselves, and again each one says a number based on the arrangement. The new sequence is one, zero, two, zero, one, two, one, zero. Did you figure out what these numbers represent? Here is the hint. Could their heights be playing a role? A child says one if there is only one taller child standing next to them. A child says two if both the children standing next to them are taller. A child says zero, if neither of the children standing next to them are taller. That is each person says the number of taller neighbours they have. Let us try answering the questions below and share your reasoning. Question one: Can the children rearrange themselves so that the children standing at the ends say two? The answer is no. A child at the end of the line has only one neighbour. Therefore, they can have at most one taller neighbour. They can never say two. Question two: Can we arrange the children in a line so that all would say only zeros? Yes, we can. If we arrange the children in strictly decreasing order of height from left to right, every child will have shorter neighbours, so everyone will say zero. Question three: Can two children standing next to each other say the same number? Yes, they can. For example, two children in the middle of a line could both have two taller neighbours if the tallest children are on both sides of them. Question four: There are five children in a group, all of different heights. Can they stand such that four of them say one and the last one says zero? Why or why not? Yes, it is possible. Arrange them in a zigzag height pattern. The tallest child will say zero, and the others can each have exactly one taller neighbour. Question five: For this group of five children, is the sequence one, one, one, one, one possible? No, it is not possible. The tallest child will always have zero taller neighbours, so they must say zero. Therefore, we cannot have five ones. Question six: Is the sequence zero, one, two, one, zero possible? Why or why not? Yes, it is possible. The middle child says two, meaning both neighbours are taller. The children on the ends say zero, meaning their only neighbour is shorter. The ones next to the ends say one. This matches a mountain shape: short, medium, tall, medium, short. Question seven: How would you rearrange the five children so that the maximum number of children say two? The maximum number who can say two is two. Place the two tallest children in the middle, flanked by shorter children. The tallest child can never say two, but the second and third tallest can each have two taller neighbours if arranged carefully.
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Let us move to section three point two, Supercells. Observe the numbers written in the table. Why are some numbers coloured? Discuss. The table has two rows. First row: 43, 79, 75, 63, 10, 29, 28, 34. Second row: 200, 577, 626, 345, 790, 694, 109, 198. A cell is coloured if the number in it is larger than its adjacent cells. The number 626 is coloured as it is larger than 577 and 345, whereas 200 is not coloured as it is smaller than 577. The number 198 is coloured as it has only one adjacent cell with 109 in it, and 198 is larger than 109. Now let us do the Figure it Out questions. Question one: Colour or mark the supercells in the table below. The table has numbers: 6828, 670, 9435, 3780, 3708, 7308, 8000, 5583, 52. The supercells are 9435, 7308, and 8000. These are larger than their immediate left and right neighbours. Question two: Fill the table below with only four digit numbers such that the supercells are exactly the coloured cells. The given table has 5346, blank, blank, 1258, blank, blank, blank, 9635, blank. We need to place numbers so that only the specified cells are supercells. We can fill the blanks with smaller numbers like 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000 in a pattern that ensures only the marked ones are peaks. For example, fill as 5346, 4000, 3000, 1258, 2000, 3000, 4000, 9635, 8000. Question three: Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions. We can alternate high and low numbers to create peaks. For example, 900, 100, 800, 200, 700, 300, 600, 400, 500. This gives four supercells. Question four: Out of the nine numbers, how many supercells are there in the table above? Four. Question five: 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. For n cells in a row, the maximum number of supercells is the floor of n divided by two. The method is to arrange numbers in a zigzag pattern: high, low, high, low, and so on. Question six: Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not? No, it is not possible. In any finite list of distinct numbers, there must be at least one maximum value. That maximum value will be greater than its neighbours, so it will always be a supercell. Question seven: 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? Yes, the largest number will always be a supercell because it is greater than all its neighbours. The smallest number can never be a supercell because it is smaller than all its neighbours. Question eight: Fill a table such that the cell having the second largest number is not a supercell. Place the largest number next to the second largest. For example, 900, 800, 100. The 800 is not a supercell because 900 is larger. Question nine: 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? Yes. Arrange as smallest, second smallest, largest, third smallest. For example, 10, 20, 100, 90, 80. The 20 is the second smallest. Its neighbour is 10, which is smaller, so it is a supercell at the end. The second largest is 90, next to 100, so it is not a supercell. Question ten: Make other variations of this puzzle and challenge your classmates. You can try changing the rule to include diagonal neighbours or using three by three grids.
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Let us do the supercells activity with more rows. Here the neighbouring cells are those that are immediately to the left, right, top and bottom. The rule remains the same: a cell becomes a supercell if the number in it is greater than all the numbers in its neighbouring cells. In Table one, 8632 is greater than all its neighbours 4580, 8280, 4795 and 1944. Table one shows a four by four grid. Row one: 2430, 7500, 7350, 9870. Row two: 3115, 4795, 9124, 9230. Row three: 4580, 8632, 8280, 3446. Row four: 5785, 1944, 5805, 6034. Complete Table two with five digit numbers whose digits are one, zero, six, three, and nine in some order. Only a coloured cell should have a number greater than all its neighbours. Table two has blanks and some given numbers. The given numbers are 96,301, 36,109, 13,609, 60,319, 19,306, 60,193, 10,963. We need to fill the blanks with permutations of one, zero, six, three, nine. The biggest number in the table is 96,301. The smallest even number in the table is 10,963. The smallest number greater than 50,000 in the table is 60,193. Once you have filled the table above, put commas appropriately after the thousands digit.
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Now let us move to section three point three, Patterns of Numbers on the Number Line. We are quite familiar with number lines now. Let us 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. On a number line from 1000 to 10,000, we place them in order: 1050, 1500, 2180, 2754, 3050, 3600, 5030, 5300, 8400, 9590, 9950. Figure it Out: Identify the numbers marked on the number lines below, and label the remaining positions. Part a shows a number line from 2010 to 2020. The marks are at 2010, 2012, 2014, 2016, 2018, 2020. Part b shows 9996 to 9997. The marks are at 9996, 9996.2, 9996.4, 9996.6, 9996.8, 9997. Part c shows 15,077 to 15,083. The marks are 15,077, 15,078, 15,079, 15,080, 15,081, 15,082, 15,083. Part d shows 86,705 to 87,705. The marks are at intervals of 200. Put a circle around the smallest number and a box around the largest number in each of the sequences above. For a, circle 2010, box 2020. For b, circle 9996, box 9997. For c, circle 15,077, box 15,083. For d, circle 86,705, box 87,705.
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Next, we will learn about section three point four, Playing with Digits. We start writing numbers from 1, 2, 3 and so on. There are nine one digit numbers. Find out how many numbers have two digits, three digits, four digits, and five digits. Two digit numbers are from 10 to 99. That is 90 numbers. Three digit numbers are from 100 to 999. That is 900 numbers. Four digit numbers are from 1000 to 9999. That is 9000 numbers. Five digit numbers are from 10000 to 99999. That is 90000 numbers. Digit sums of numbers: Komal observes that when she adds up digits of certain numbers the sum is the same. For example, adding the digits of the number 68 will be same as adding the digits of 176 or 545. 6+8=14. 1+7+6=14. 5+4+5=14. Figure it Out question one: Digit sum 14. Part a: Write other numbers whose digits add up to 14. Examples: 59, 68, 77, 86, 95, 149, 158. Part b: What is the smallest number whose digit sum is 14? The smallest number is 59. Part c: What is the largest five digit whose digit sum is 14? The largest five digit number is 95000. Part d: How big a number can you form having the digit sum of fourteen? Can you make an even bigger number? You can make a number with fourteen ones: 111,111,111,111,111. You can make an even bigger number by adding zeros, like 1 followed by many zeros and then a one, but the digit sum must stay fourteen. Actually, to make the number bigger, you add more digits that sum to fourteen, like one followed by many zeros and ending with digits that sum to fourteen. The number can be arbitrarily large by adding zeros between digits. Question two: Find out the digit sums of all the numbers from 40 to 70. Share your observations. The digit sums cycle. Forty is 4, forty one is 5, up to forty nine is 13. Fifty is 5, up to fifty nine is 14. Sixty is 6, up to sixty nine is 15. Seventy is 7. The pattern increases by one each step, then drops by nine when the tens digit increases. Question three: Calculate the digit sums of three digit numbers whose digits are consecutive. For example, 345. 3+4+5=12. 234 sums to 9. 456 sums to 15. The pattern is multiples of three: 9, 12, 15, 18, 21. Yes, this pattern continues because consecutive digits differ by one, so their sum is always three times the middle digit.
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Digit Detectives: After writing numbers from one to one hundred, Dinesh wondered how many times he would have written the digit seven. Among the numbers one to one hundred, how many times will the digit seven occur? It occurs in 7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 87, 97. Counting them: seven appears once in each unit place from seven to ninety seven, that is ten times. It appears ten times in the tens place from seventy to seventy nine. But seventy seven has two sevens. So total is twenty times. Among the numbers one to one thousand, how many times will the digit seven occur? In each hundred block, seven appears twenty times. There are ten hundred blocks from one to one thousand. So twenty times ten equals two hundred times.
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Now let us explore section three point five, Pretty Palindromic Patterns. What pattern do you see in these numbers: 66, 848, 575, 797, 1111? These numbers read the same from left to right and from right to left. Try and see. Such numbers are called palindromes or palindromic numbers. All palindromes using 1, 2, 3: The numbers 121, 313, 222 are some examples of palindromes using the digits one, two, three. Write all possible three digit palindromes using these digits. They are 101, 111, 121, 131, 202, 212, 222, 232, 303, 313, 323, 333. Reverse and add palindromes: Now, look at these additions. Try to figure out what is happening. Steps to follow: Start with a two digit number. Add this number to its reverse. Stop if you get a palindrome or else repeat the steps of reversing the digits and adding. Try the same procedure for some other numbers, and perform the same steps. Stop if you get a palindrome. The examples show 34+43=77. 29+92=121. 48+84=132. Then 132+231=363. 76+67=143. Then 143+341=484. Explore: Will reversing and adding numbers repeatedly, starting with a two digit number, always give a palindrome? Yes, for two digit numbers, it always reaches a palindrome quickly. For three digit numbers, it is unknown. It is suspected that starting with 196 never yields a palindrome. Puzzle time: Write the number in words. I am a five 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? The units digit must be odd. Possible units digits are 1, 3, 5, 7, 9. If u is 1, t is 2, h is 4. The number is 42124. It is a palindrome, odd, and matches the clues. So the number is forty two thousand one hundred twenty four.
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Let us move to section three point six, The Magic Number of Kaprekar. D R Kaprekar was a mathematics teacher in a government school in Devlali, Maharashtra. He liked playing with numbers very much and found many beautiful patterns in numbers that were previously unknown. In 1949, he discovered a fascinating and magical phenomenon when playing with four digit numbers. Follow these steps and experience the magic for yourselves. Pick any four digit number having at least two different digits, say 6382. Take a four digit number. Make the largest number from these digits. Call it A. Make the smallest number from these digits. Call it B. Subtract B from A. Call it C. C = A - B. What happens if we continue doing this? A = 8632. B = 2368. C = 8632-2368 = 6264. Next, A = 6642. B = 2466. C = 6642-2466 = 4176. Next, A = 7641. B = 1467. C = 7641-1467 = 6174. Next, A = 7641. B = 1467. C = 6174. It repeats. Explore: Take different four 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 three digit numbers. What number will start repeating? For three digit numbers, the process always reaches 495.
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Now let us explore section three point seven, Clock and Calendar Numbers. On the usual twelve hour clock, there are timings with different patterns. For example, 4:44, 10:10, 12:21. Try and find out all possible times on a twelve hour clock of each of these types. Times with all same digits: 1:11, 2:22, 3:33, 4:44, 5:55, 10:01, 11:11, 12:21. Times with repeating pattern like ab:ba: 10:01, 11:11, 12:21, 13:31, 14:41, 15:51, 20:02, 21:12, 22:22, 23:32, 24:42, 25:52, 30:03, 31:13, 32:23, 33:33, 34:43, 35:53, 40:04, 41:14, 42:24, 43:34, 44:44, 45:54, 50:05, 51:15, 52:25, 53:35, 54:45, 55:55. Manish has his birthday on 20/12/2012 where the digits two, zero, one, and two repeat in that order. Find some other dates of this form from the past. Examples: 02/12/2012, 12/02/2012, 20/12/2012. His sister, Meghana, has her birthday on 11/02/2011 where the digits read the same from left to right and from right to left. Find all possible dates of this form from the past. Examples: 01/10/2010, 10/01/2010, 11/02/2011, 20/02/2002, 02/20/2020, 22/02/2022. Jeevan was looking at this year's calendar. He started wondering, why should we change the calendar every year? Can we not reuse a calendar? What do you think? Calendars repeat when the days of the week align with the dates. A common year has 365 days, which is fifty two weeks plus one day. So the calendar shifts by one day each year. A leap year has 366 days, shifting by two days. Calendars repeat every six or eleven years, or twenty eight years for leap years. Figure it Out question one: Pratibha uses the digits four, seven, three and two, and makes the smallest and largest four 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 four digits to make part a: the difference between the largest and smallest numbers greater than 5085. Use digits 9, 8, 1, 0. Largest 9810, smallest 1089. Difference is 8721, which is greater. Part b: difference less than 5085. Use digits 5, 4, 3, 2. Largest 5432, smallest 2345. Difference 3087, which is less. Part c: sum greater than 9779. Use digits 9, 8, 7, 6. Largest 9876, smallest 6789. Sum 16665, which is greater. Part d: sum less than 9779. Use digits 4, 3, 2, 1. Largest 4321, smallest 1234. Sum 5555, which is less. Question two: What is the sum of the smallest and largest five digit palindrome? What is their difference? Smallest five digit palindrome is 10001. Largest is 99999. Sum is 110000. Difference is 89998. Question three: The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that? Next is 10:10, which is nine minutes later. The one after that is 11:11, which is sixty one minutes after 10:10. Question four: How many rounds does the number 5683 take to reach the Kaprekar constant? Start with 5683. Largest 8653, smallest 3568. Difference 5085. Next A 8550, B 558. Difference 7992. Next A 9972, B 2799. Difference 7173. Next A 7731, B 1377. Difference 6354. Next A 6543, B 3456. Difference 3087. Next A 8730, B 0378. Difference 8352. Next A 8532, B 2358. Difference 6174. It takes eight rounds.
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Next is section three point eight, Mental Math. Observe the figure below. What can you say about the numbers and the lines drawn? The figure shows a middle column with numbers 400, 1500, 13000, 25000, 31000, 38800, 60000, 63000. Numbers in the middle column are added in different ways to get the numbers on the sides. 1500+1500+400=3400. The numbers in the middle can be used as many times as needed to get the desired sum. Draw arrows from the middle to the numbers on the sides to obtain the desired sums. Two examples are given. It is simpler to do it mentally. 38800 = 25000 + 400×2 + 13000. 3400 = 1500 + 1500 + 400. Can we make 1000 using the numbers in the middle? Why not? The smallest number available is 400. We cannot combine them to get exactly 1000 because the available numbers are multiples of 100, but we lack a 600. What about 14000, 15000 and 16000? Yes, it is possible. 14000 = 13000 + 1500 - 500 is not allowed. Using only addition, 15000 = 1500 × 10. 16000 = 13000 + 1500 + 1500. What thousands cannot be made? Thousands that are not combinations of 400, 1500, and 13000. Adding and Subtracting: Here, using the numbers in the boxes, we are allowed to use both addition and subtraction to get the required number. An example is shown. 40000, 7000, 300, 1500, 12000, 800. 39800 = 40000 - 800 + 300 + 300. Now solve for 45000, 5900, 17500, 21400. 45000 = 40000 + 7000 - 1500 - 500? We don't have 500. Let's use: 45000 = 40000 + 7000 - 1500 - 500 is not possible. Actually, 45000 = 40000 + 12000 - 7000. 5900 = 7000 - 1500 + 400? Not available. Let's use: 5900 = 7000 - 1500 + 400 is not in the box. We can do: 5900 = 7000 - 1500 + 300 + 100? Not available. The key is to combine the given numbers flexibly. 17500 = 12000 + 7000 - 1500. 21400 = 12000 + 7000 + 1500 + 800 + 100? We can approximate. The exercise encourages exploring combinations. Digits and Operations: An example of adding two five digit numbers to get another five digit number is 12350 + 24545 = 36895. An example of subtracting two five digit numbers to get another five digit number is 48952 - 24547 = 24405. Figure it Out question one: Write an example for each scenario. Five digit plus five digit to give a five digit sum more than 90250: 45000 + 46000 = 91000. Five digit plus three digit to give a six digit sum: 99900 + 100 = 100000. Four digit plus four digit to give a six digit sum: Not possible. Maximum is 9999 + 9999 = 19998. Five digit plus five digit to give a six digit sum: 50000 + 50000 = 100000. Five digit plus five digit to give 18500: Not possible. Minimum sum is 10000 + 10000 = 20000. Five digit minus five digit to give a difference less than 56503: 60000 - 50000 = 10000. Five digit minus three digit to give a four digit difference: 10000 - 1000 = 9000. Five digit minus four digit to give a four digit difference: 20000 - 10000 = 10000. Five digit minus five digit to give a three digit difference: 10000 - 9900 = 100. Five digit minus five digit to give 91500: 100000 - 8500 is invalid. Use 92000 - 500 = 91500. Question two: Always, Sometimes, Never? Part a: five digit plus five digit gives five digit. Sometimes true. True if sum < 100000. Part b: four digit plus two digit gives four digit. Sometimes true. True if sum < 10000. Part c: four digit plus two digit gives six digit. Never true. Max is 9999+99=10098. Part d: five digit minus five digit gives five digit. Sometimes true. Part e: five digit minus two digit gives three digit. Never true. Min 10000-99=9901.
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Now let us explore section three point nine, Playing with Number Patterns. Here are some numbers arranged in some patterns. Find out the sum of the numbers in each of the below figures. Should we add them one by one or can we use a quicker way? Part a shows a pattern with five rows. Row one has four numbers of 40. Row two has five numbers of 50. Row three has four numbers of 40. Row four has five numbers of 50. Row five has four numbers of 40. Total sum: three rows of 40×4 = 480. Two rows of 50×5 = 500. Total is 980. Part b shows a grid pattern with dots representing numbers. Count the dots and multiply by their values. Part c shows a table of 32s and 64s. There are four rows of eight 32s. That is 32×32 = 1024. Then there are rows with 64s. There are five rows of three 64s. That is fifteen 64s. 15×64 = 960. Total is 1984. Part d shows a dice pattern. Sum the dots on each face. Part e shows a hexagonal diagram with numbers 15, 25, 35 arranged in a pattern. Sum them by grouping. Part f shows a circular target diagram with concentric rings labeled center 1000, then 500, then 250, then 125. Sum is 1000+500+250+125 = 1875.
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Next is section three point ten, An Unsolved Mystery, the Collatz Conjecture. Look at the sequences below. The same rule is applied in all the sequences. Sequence a: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. Sequence b: 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Sequence c: 21, 64, 32, 16, 8, 4, 2, 1. Sequence d: 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Do you see how these sequences were formed? The rule is: one starts with any number. If the number is even, take half of it. If the number is odd, multiply it by 3 and add 1. Repeat. Notice that all four sequences above eventually reached the number 1. In 1937, the German mathematician, Lothar Collatz conjectured that the sequence will always reach 1, regardless of the whole number you start with. Even today, despite many mathematicians working on it, it remains an unsolved problem as to whether Collatz's conjecture is true. Collatz's conjecture is one of the most famous unsolved problems in mathematics. Make some more Collatz sequences like those above, starting with your favourite whole numbers. Do you always reach 1? Yes, for all tested numbers, you reach 1. Do you believe the conjecture of Collatz that all such sequences will eventually reach 1? Why or why not? I believe it because it has been tested for billions of numbers, but mathematically it is not yet proven.
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Now let us move to section three point eleven, Simple Estimation. At times, we may not know or need an exact count of things and an estimate is sufficient for the purpose at hand. For example, your school headmaster might know the exact number of students enrolled in your school, but you may only know an estimated count. How many students are in your school? About 150? 400? A thousand? Paromita's class section has 32 children. The other two sections of her class have 29 and 35 children. So, she estimated the number of children in her class to be about 100. Along with Class six, her school also has Classes seven to ten and each class has three sections each. She assumed a similar number in each class and estimated the number of students in her school to be around 500. Figure it Out: We shall do some simple estimates. It is a fun exercise, and you may find it amusing to know the various numbers around us. Remember, we are not interested in the exact numbers for the following questions. Share your methods of estimation with the class. Question one: Steps you would take to walk from the place you are sitting to the classroom door. Estimate about ten to twenty steps. Across the school ground from start to end. Estimate about fifty to one hundred steps. From your classroom door to the school gate. Estimate about one hundred to two hundred steps. From your school to your home. Estimate about five hundred to one thousand steps depending on distance. Question two: Number of times you blink your eyes or number of breaths you take. In a minute, about fifteen to twenty breaths. In an hour, about nine hundred to one thousand two hundred breaths. In a day, about twenty one thousand to twenty eight thousand breaths. Question three: Name some objects around you that are a few thousand in number. Examples: leaves on a tree, pages in a library, grains of sand in a handful. More than ten thousand in number. Examples: stars visible in the sky, books in a large library, people in a city. Estimate the answer. Try to guess within thirty seconds. Check your guess with your friends. Question one: Number of words in your maths textbook. More than 5000 or less than 5000? Less than 5000. Question two: Number of students in your school who travel to school by bus. More than 200 or less than 200? Depends on the school, but typically less than 200 for a small school, or more for a large one. Question three: Roshan wants to buy milk and three types of fruit to make fruit custard for five people. He estimates the cost to be ₹100. Do you agree with him? Why or why not? No, I do not agree. Milk costs about ₹30, and three fruits cost at least ₹20 each, totaling ₹90. It might be close, but likely more than ₹100. Question four: Estimate the distance between Gandhinagar in Gujarat to Kohima in Nagaland. Looking at the map of India, it spans almost the entire width of the country. Estimate about 2000 to 2500 kilometers. Question five: Sheetal is in Grade six and says she has spent around 13,000 hours in school till date. Do you agree with her? Why or why not? A school day is about six hours. One hundred eighty school days a year. 6×180 = 1080 hours per year. For six years, it is about 6480 hours. So 13000 is too high. Question six: Earlier, people used to walk long distances. Suppose you walk at your normal pace. Approximately, how long would it take you to go from your current location to one of your favourite places nearby? About thirty minutes. To any neighbouring state's capital city? About two to three days. From the southernmost point in India to the northernmost point in India? About thirty to forty days. Question seven: Make some estimation questions and challenge your classmates. You can ask how many tiles are on the school floor, or how many pages are in the library.
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Finally, let us explore section three point twelve, Games and Winning Strategies. Numbers can also be used to play games and develop winning strategies. Here is a famous game called twenty one. Rules for Game one: The first player says one, two or three. Then the two players take turns adding one, two, or three to the previous number said. The first player to reach twenty one wins. Play this game several times with your classmate. Are you starting to see the winning strategy? Which player can always win if they play correctly? What is the pattern of numbers that the winning player should say? The winning strategy is to say numbers that leave a remainder of one when divided by four. The winning player should say 1, 5, 9, 13, 17, 21. The second player can win if the first player does not start with one. Actually, if the first player starts with one, they control the game. The pattern increases by four each time. There are many variations of this game. Rules for Game two: The first player says a number between one and ten. Then the two players take turns adding a number between one and ten to the previous number said. The first player to reach ninety nine wins. Play this game several times with your classmate. See if you can figure out the corresponding winning strategy in this case. Which player can always win? What is the pattern of numbers that the winning player should say this time? The winning strategy is to reach numbers that leave a remainder of one when divided by eleven. The winning player should say 1, 12, 23, 34, 45, 56, 67, 78, 89, 99. The first player can always win by starting with one. Make your own variations of this game. Decide how much one can add at each turn, and what number is the winning number. Then play your game several times, and figure out the winning strategy and which player can always win. Figure it Out question one: There is only one supercell in this grid. If you exchange two digits of one of the numbers, there will be four supercells. Figure out which digits to swap. The grid is 16200, 39344, 29765, 23609, 62871, 45306, 19381, 50319, 38408. Swap the six and two in 16200 to make 26100. This changes the peaks and creates new supercells. Try This question two: How many rounds does your year of birth take to reach the Kaprekar constant? Apply the Kaprekar steps to your birth year digits. Question three: We are the group of five digit numbers between 35000 and 75000 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 50000? Largest is 73999. Smallest is 35111. Closest to 50000 is 49999. Question four: 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. Estimate about one hundred twenty days. Exact count depends on your calendar. Question five: Estimate the number of liters a mug, a bucket and an overhead tank can hold. Mug holds about 250 milliliters. Bucket holds about 15 liters. Overhead tank holds about 5000 liters. Question six: Write one five digit number and two three digit numbers such that their sum is 18670. Example: 18000 + 350 + 320 = 18670. Question seven: 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. Choose 300. Create a pattern of ten rows of 30. Question eight: Recall the sequence of Powers of two from Chapter one, Table one. Why is the Collatz conjecture correct for all the starting numbers in this sequence? Because powers of two are even. Dividing by two repeatedly will always reach one directly. Question nine: Check if the Collatz Conjecture holds for the starting number 100. 100 is even, half is 50. 50 is even, half is 25. 25 is odd, 3×25+1=76. 76 is even, half is 38. 38 is even, half is 19. 19 is odd, 3×19+1=58. 58 is even, half is 29. 29 is odd, 3×29+1=88. 88 is even, half is 44. 44 is even, half is 22. 22 is even, half is 11. 11 is odd, 3×11+1=34. 34 is even, half is 17. 17 is odd, 3×17+1=52. 52 is even, half is 26. 26 is even, half is 13. 13 is odd, 3×13+1=40. 40 is even, half is 20. 20 is even, half is 10. 10 is even, half is 5. 5 is odd, 3×5+1=16. 16 is even, half is 8. 8 is even, half is 4. 4 is even, half is 2. 2 is even, half is 1. Yes, it holds. Question ten: Starting with zero, players alternate adding numbers between one and three. The first person to reach twenty two wins. What is the winning strategy now? The target is twenty two. The winning numbers leave a remainder of two when divided by four. The winning player should say 2, 6, 10, 14, 18, 22. The first player should start with two.
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Let us review the Summary. Numbers can be used for many different purposes including, to convey information, make and discover patterns, estimate magnitudes, pose and solve puzzles, and play and win games. Thinking about and formulating set procedures to use numbers for these purposes is a useful skill and capacity, called computational thinking. Many problems about numbers can be very easy to pose, but very difficult to solve. Indeed, numerous such problems are still unsolved, for example, Collatz's Conjecture.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]