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Number Play — Class 6 Mathematics

Ganita Prakash · Chapter 3

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Chapter summary

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 (). 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 to odd numbers always seems to reach 1. The chapter ends with simple estimation and winning strategies for number games like reaching 21.

Numbers can Tell us ThingsSupercellsPatterns on the Number LinePlaying with DigitsPalindromes and Kaprekar ConstantEstimation and Winning Strategies

Key ideas & terms

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 .
Collatz conjecture: The unproven claim that halving even numbers and using for odd numbers always reaches 1.
Estimation: Finding an approximate value when an exact count is not needed.

Extra questions & answers

What is a supercell in a number grid?

A cell is a supercell if the number written in it is greater than all the numbers in its neighbouring cells.

What is a palindromic number? Give an example.

A number that reads the same forwards and backwards, such as 575, 1111 or 12321.

What is the Kaprekar constant?

It is 6174; any 4-digit number with at least two different digits reaches 6174 by repeatedly subtracting the smallest arrangement from the largest.

How many 3-digit numbers and 5-digit numbers are there?

There are 900 three-digit numbers and 90000 five-digit numbers.

What is the smallest number whose digit sum is 14?

The number 59, since and no smaller number has digit sum 14.

What rule defines a Collatz sequence?

Start with any number; if it is even take half, if it is odd multiply by 3 and add 1; repeat. The sequence always seems to reach 1.

What is the sum of the smallest and largest 5-digit palindromes?

The smallest is 10001 and the largest is 99999, so their sum is 110000.

In the game of reaching 21 by adding 1, 2 or 3, what is the winning strategy?

Aim to say the numbers ; reaching each of these multiples-of-4-plus-1 positions forces a win.

What is estimation and why is it useful?

Estimation gives an approximate value when an exact count is not needed, helping us judge magnitudes quickly.

Why is the Collatz conjecture famous?

Because despite many mathematicians trying, no one has proved that every starting number reaches 1, so it remains an unsolved problem.

Think about various situations where we use numbers.

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.

Numbers are used in many ways. Five examples are: telling time on a clock, dates on a calendar, counting students or objects, measuring height, weight or distance, and money, prices and phone numbers. Other uses include roll numbers, page numbers and scores in games.

Try answering the questions below and share your reasoning.

1. Can the children rearrange themselves so that the children standing at the ends say ‘2’?

No. A child at an end has only one neighbour, so the maximum number they can say is . They can never say because they do not have two neighbours.

2. Can we arrange the children in a line so that all would say only 0s?

Yes, if all children have the same height. Then no child has a taller neighbour, so everyone says . More generally any arrangement where no child is shorter than its neighbours also works, but equal heights is the simplest.

3. Can two children standing next to each other say the same number?

Yes. For example in the picture on page 55 two adjacent children both say or both say . This can happen depending on the relative heights.

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?

Yes. If they stand in strictly increasing order of height (shortest to tallest) from one end to the other, then the tallest child at the end has no taller neighbour and says , and each of the other four has exactly one taller neighbour (the next taller child) and says .

5. For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?

No. With all heights different, the tallest child must exist. If the tallest child is in the middle it would have taller neighbours and say . If the tallest child is at an end it also says . So at least one must appear, so five s is impossible.

6. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?

Yes, it is possible. An example height order from left to right is medium, tall, tallest, tall, medium where valleys say or and the peak says . One arrangement like in rank order can produce . The ends say because only one neighbour each and the middle child is taller than both neighbours.

7. How would you rearrange the five children so that the maximum number of children say ‘2’?

At most two children can say in a line of five distinct heights. To achieve this put the two tallest children not at the ends and separate them by a shorter child, for example order by height rank or tall-short-tall pattern such that two interior short children each have two taller neighbours. One maximum arrangement is shortest in the middle with taller children on both sides.

Figure it Out

1. Colour or mark the supercells in the table below. 6828 670 9435 3780 3708 7308 8000 5583 52

A supercell is larger than both adjacent cells. Here so is a supercell, and so is a supercell, and so is a supercell, and has only one neighbour but so it is not. Thus supercells are , and .

2. Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells. 5346 1258 9635

We need numbers larger than neighbours only on the coloured positions. One example row of nine numbers is where , and are coloured and are greater than their neighbours, while other cells are smaller than an adjacent coloured cell.

3. Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.

To maximise supercells colour alternate cells. For cells the maximum is supercells. One way is to place numbers in decreasing size on alternate positions, e.g. where positions are supercells. No two supercells can be adjacent.

4. Out of the 9 numbers, how many supercells are there in the table above? ___________

With the filling described above there are supercells, which is the maximum for cells.

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.

For cells, maximum supercells is . For even , gives supercells. For odd , gives supercells. Strategy: put the largest numbers on every alternate cell starting with the first cell, and fill the cells in between with the smallest numbers.

6. Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?

No. The greatest number among all cells must be larger than its neighbours, wherever it is placed. Therefore at least one supercell always exists if numbers are distinct.

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?

Yes, the largest number is always a supercell because it is larger than both neighbours. No, the smallest number can never be a supercell because at least one neighbour is larger.

8. Fill a table such that the cell having the second largest number is not a supercell.

Put the largest and second largest next to each other, so the second largest has the largest as a neighbour and fails. Example for cells: where (second largest 's neighbour) is largest, making second largest not a supercell.

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?

Yes, it is possible by placing the second smallest between two even smaller numbers while keeping second largest adjacent to the largest. Example: order in terms of size rank where rank is second smallest and rank is second largest adjacent to rank .

10. Make other variations of this puzzle and challenge your classmates.

Examples: Can you get exactly supercells with cells? Can you get more than supercells? Students can create their own tables and ask friends to find supercells.

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.

One valid completion is: Row1: , , , Row2: , , , Row3: , , , Row4: , , , where only green cells are greater than top/bottom/left/right neighbours.

The biggest number in the table is ____________.

The biggest number in the table is .

The smallest even number in the table is ____________.

The smallest even number in the table is .

The smallest number greater than 50,000 in the table is ____________.

The smallest number greater than that can be formed is .

Figure it Out

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.

a) The line is counting by : marks are - smallest circled, largest boxed. b) Counting by : . c) Counting by : to . d) Counting by : . In each, the first tick is smallest and last tick is largest.

Find out how many numbers have two digits, three digits, four digits, and five digits.

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

1-digit: ( to ). 2-digit: ( to ). 3-digit: ( to ). 4-digit: ( to ). 5-digit: ( to ).

Figure it Out

1. Digit sum 14 a. Write other numbers whose digits add up to 14.

Examples: , , , , , , , all have digit sum , etc.

1. Digit sum 14 b. What is the smallest number whose digit sum is 14?

The smallest number with digit sum is (since and any one-digit number max is , so two digits are needed and is smaller than , etc.). If allowed leading, is smallest with no zero in front, but overall smallest positive integer is .

1. Digit sum 14 c. What is the largest 5-digit whose digit sum is 14?

The largest 5-digit number with sum is because and putting largest digits at the left gives maximum value.

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?

You can make arbitrarily large numbers by adding zeros in the middle, e.g. , , , . Since you can insert any number of zeros between and without changing sum, there is no largest number.

2. Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.

Digit sums: , , , , , , , , , , , etc. Pattern increases by each step until , then drops to at and rises again.

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?

, , , , , , . All sums are multiples of and increase by each time. Pattern stops at because is not three consecutive increasing digits in the usual sense.

Digit Detectives

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?

From to , digit appears times ( with counting twice, and ). From to , it appears times ( times in each position: units, tens, hundreds).

Write all possible 3-digit palindromes using these digits.

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’.

All possibilities with digits and same outer digits: .

Explore

Will reversing and adding numbers repeatedly, starting with a 2-digit number, always give a palindrome? Explore and find out.*

Yes, for any 2-digit starting number the reverse-and-add process always reaches a palindrome quickly. Example (palindrome in one step), (palindrome in one step), , (palindrome in two steps). Experiments show all -digit numbers reach a palindrome.

Puzzle time

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

Let the number be . is odd (units). and . For single digits, gives , so number is . would give , but then number would be even, not allowed. So answer is .

Explore

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?

For -digit numbers with at least two different digits, repeated Kaprekar routine always reaches . For -digit numbers the analogous constant is . Example: and then repeats.

Figure it Out

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.

Example: choose digits gives largest and smallest , difference .

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.

Example: digits gives largest and smallest , difference .

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.

Example: same digits sum .

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.

Example: digits sum .

2. What is the sum of the smallest and largest 5-digit palindrome? What is their difference?

Smallest 5-digit palindrome is , largest is . Sum is . Difference is .

3. The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?

Next palindromic time after is , which is minutes later. The one after that is , which is minutes after (or minutes after ).

4. How many rounds does the number 5683 take to reach the Kaprekar constant?

. It reaches the Kaprekar constant in rounds.

Figure it Out

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

works for first. works. -digit + -digit can never be -digit (max ). works for 5-digit minus 4-digit = 4-digit. works for 5-digit+3-digit=6-digit. works for 5-digit-3-digit=4-digit. works for 5-digit+5-digit=6-digit. works for 3-digit difference. -digit+5-digit=18,500 impossible (minimum sum ). -digit-5-digit=91,500 impossible (max difference ).

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

Only sometimes true. Example is -digit, while is still -digit.

2. Always, Sometimes, Never? b. 4-digit number + 2-digit number gives a 4-digit number

Only sometimes true. becomes -digit, but stays -digit.

2. Always, Sometimes, Never? c. 4-digit number + 2-digit number gives a 6-digit number

Never true. Maximum is which is only -digit, never -digit.

2. Always, Sometimes, Never? d. 5-digit number – 5-digit number gives a 5-digit number

Only sometimes true. is -digit, but is -digit.

2. Always, Sometimes, Never? e. 5-digit number – 2-digit number gives a 3-digit number

Never true. Smallest -digit is , minus largest -digit is still -digit.

Figure it Out

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

Estimates vary. a) about to steps. b) about to steps. c) about to steps. d) depends on distance, maybe to steps. Method: measure average step length ~ m and divide distance.

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

Blinks ~ – per minute, so – per hour, per waking day. Breaths ~ per minute, per hour, per day.

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

a) Few thousand: pages in library books, grains in a small bag, words in a chapter. b) More than ten thousand: grains of rice in a sack, hairs on head, population of a town. For textbook words: more than is correct, a Class 6 maths book has well over words.

2. Number of students in your school who travel to school by bus: a. More than 200 b. Less than 200

Depends on school. In a medium school with students, maybe less than travel by bus if many walk or cycle. Estimate by counting buses × capacity.

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?

No, likely too low. Milk ₹ per litre plus three fruits even in small quantity costs more than today. With cheap seasonal fruits and small servings it might be just enough, but generally – is more realistic.

4. Estimate the distance between Gandhinagar (in Gujarat) to Kohima (in Nagaland). Hint: Look at the map of India to locate these cities.

About km by road/aerial distance across India from west to far east.

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?

No. At ~ hours per day × days per year = hours per year. Over years (Nursery to Grade 6) that is ~ hours, so would imply ~ years, too high.

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.

Walking speed ~ km/h. a) Nearby place km → ~ minutes. b) Neighbouring capital km → ~ hours walking, about – days. c) Kanyakumari to Srinagar ~ km → ~ hours, about days non-stop or – months at h per day.

Figure it Out

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

Swap digits and in to make . Then supercells become , , and (or similar four) depending on placement; the grid then has four numbers larger than their orthogonal neighbours.

2. How many rounds does your year of birth take to reach the Kaprekar constant?

Example for birth year : . So takes rounds to reach the Kaprekar constant . (Try your own birth year the same way.)

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?

With repetitions allowed: largest , smallest , closest to is . Without repetition: largest , smallest , closest to is .

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.

Estimate: weekends days + festivals ~ + summer/winter vacation ~ = ~ holidays. Exact count varies by school calendar, check your school almanac.

5. Estimate the number of liters a mug, a bucket and an overhead tank can hold.

Mug ~ to litre, bucket ~ to litres, overhead tank ~ to litres.

6. Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.

Example: . Many solutions such as .

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.

Choose . Pattern: two on top, three in middle, two at bottom = . Any arrangement summing to chosen number using repeated blocks is acceptable.

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?

Powers of are All are even, so Collatz rule just halves them repeatedly: . Hence they always reach without any odd step.

9. Check if the Collatz Conjecture holds for the starting number 100.

. Yes, it reaches after steps.

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?

This is a take-away game. Aim for multiples of plus ? Since target is and moves are –, the winning positions are . First player can win by starting with and then always making the sum go to next target (add minus opponent's move). So first player has winning strategy.