Number Play


 Figure it out ( Page: 57)

1. Colour or mark the supercells in the table below.

Answer:

The supercells in the table are marked below:

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

Answer: The table below is filled with only 4-digit numbers such that the supercells are exactly the coloured cells:

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

Answer:

The table below is filled with as many supercells as possible using numbers between 100 and 1000 without repetitions:


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

Answer:  9

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:

Yes, the cell having the largest number in a table will always be a supercell because that is the definition of a supercell.

The slimmest number in a table can never be a supercell because the adjacent numbers will be greater.

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

Answer:

The table filled below has the second largest number 815 that is not a supercell:


Page 59:

We are quite familiar with number lines now. Let’s 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.

Answer:

Figure it Out

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.

Answer:

Page 60:

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

1-digit numbers From 1–92-digit numbers3-digit numbers4-digit numbers5-digit numbers
9


Answer:

1-digit numbers From 1–92-digit numbers3-digit numbers4-digit numbers5-digit numbers
990900900090000

Figure it Out (Page 60):

1. 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 5-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?

Answer:

a. Write other numbers whose digits add up to 14.

Some other numbers whose digits add up to 14 are as follows:

1 + 9 + 4 = 14

5 + 5 + 4 = 14

11 + 2 + 1 = 14

10 + 4 = 14

12 + 2 = 14

13 + 1 = 14

7 + 7 = 14

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

We observe,

1 + 13 = 13

2 + 12 = 14

3 + 11 = 14

4 + 10 = 14

5 + 9 = 14

6 + 8 = 14

7 + 7 = 15

8 + 6 = 14

9 + 5 = 15

10 + 4 = 14

11 + 3 = 14

12 + 2 = 14

13 + 1 = 14

Out of all these sums, the sum 5 + 9 = 14 is of interest. The smallest number whose digit’s sum is 14 is 59.

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

To get the largest 5-digit number, we start with the digit 9, followed by the digit 5 and fill the rest of the places with zeroes.

The largest five-digit number whosedigit sum is 14 is 95,000.

d. How big a number can you form having the digit sum 14? Can you make an even bigger number?

We can keep adding zeroes to the end of 14 consecutive 1’s to get bigger and bigger numbers as shown below:

11111111111111, 1111111111111100, 11111111111111000 and so on

Therefore, you can always make a bigger number having the digit sum 14.

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:

123 = 1 + 2 + 3 = 6

234 = 2 + 3 + 4 = 9

345 = 3 + 4 + 5 = 12

456 = 4 + 5 + 6 = 15

567 = 5 + 6 + 7 = 18

678 = 6 + 7 + 8 = 21

789 = 7 + 8 + 9 = 24

The pattern is consecutive multiples of 3 starting from 6.

This pattern cannot continue after 789 because the next digit after 9 is 10, which cannot give us a three-digit number.

Figure it Out (Page 64):

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.

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.

Answer:

a. the difference between the largest and smallest numbers greater than 5085.

Let us take the digits 7, 8, 9, 1.

Largest 4-digit number with the digits 7, 8, 9, 1 = 9871.

Smallest 4-digit number with the digits 7, 8, 9, 1 = 1789.

The difference between these two numbers = 9871 – 1789 = 8082 which is greater than 5085.

b. the difference between the largest and smallest numbers less than 5085.

Let us take the digits 6, 2, 1, 0.

Largest 4-digit number with the digits 6, 2, 3, 0 = 6230.

Smallest 4-digit number with the digits 6, 2, 3, 0 = 2360.

The difference between these two numbers = 6230 – 2360 = 3870.

c. the sum of the largest and smallest numbers greater than 9779.

Let us take the digits 7, 8, 3, 2.

Largest 4-digit number with the digits 8, 7, 3, 2 = 8732.

Smallest 4-digit number with the digits 8, 7, 3, 2 = 2378.

The sum of the largest and smallest numbers = 8732 + 2378 = 1110 which is greater than 9779.

d. the sum of the largest and smallest numbers less than 9779.

Let us take the digits 2, 3, 1, 4.

Largest 4-digit number with the digits 2, 3, 1, 4 = 4321.

Smallest 4-digit number with the digits 1, 2, 3, 4 = 1234.

The sum of the largest and smallest numbers = 4321 + 1234 = 5555 which is less than 9779.

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

Answer:

The smallest 5-digit palindrome = 10001.

The largest 5-digit palindrome = 99999.

Sum = 10001 + 99999 = 110,000.

Difference = 99999 – 10001 = 89998.

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:

The time now is 10:01.

The next palindromic time is 11:11.

Therefore, it is 70 minutes until the clock shows the next palindromic time

The next palindromic time after 11:11 is 12:21.

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

Answer:

Follow these steps:

Step 1: Take a 4-digit number.

Step 2: Make the largest number from these digits. Call it A.

Step 3: Make the smallest number from these digits. Call it B.

Step 4: Subtract B from A. Call it C. C = A – B

Step 5: Continue doing this until you reach 6174.

Number = 5683

First Try:

A = 8653

B = 3568

C = A – B = 8653 – 3568 = 5085

Second Try:

A = 8550

B = 558

C = A – B = 8550 – 558 = 7992

Third Try:

A = 9972

B = 2799

C = A – B = 9972 – 2799 = 7173

Fourth Try:

A = 7731

B = 1377

C = A – B = 7731 – 1377 = 6354

Fifth Try:

A = 6543

B = 3456

C = A – B = 6543 – 3456 = 3087

Sixth Try:

A = 8730

B = 3780

C = A – B = 8730 – 378 = 8352

Seventh Try:

A = 8532

B = 2358

C = A – B = 8532 – 2358 = 6174 (Kaprekar constant)

Therefore, it takes 7 tries to reach the Kaprekar constant.

Figure it Out (Page 66):

1. Write an example for each of the below scenarios whenever possible.

Could you find examples for all the cases? If not, think and discuss what could be the reason. Make other such questions and challenge your classmates.

Answer:

5-digit + 5-digit to give a 5-digit sum more than 90,250

First number = 70,000

Second number = 25,000

Sum = 70,000 + 25,000 = 95,000 which is greater than 90,250

5-digit + 3-digit to give a 6-digit sum

First number = 99,500

Second number = 600

Sum = 99,500 + 600 = 100,100 (6 digits)

4-digit + 4-digit to give a 6-digit sum

Not possible

Largest 4-digit number = 9999

9999 + 9999 = 19,998 (5 digits, not 6 digits)

5-digit + 5-digit to give a 6-digit sum

First number = 50,000

Second number = 50,000

Sum = 50,000 + 50,000 = 100,000 (6 digits)

5-digit + 5-digit to give 18,500

Not possible

The smallest possible 5-digit number = 10,000

Sum = 10,000 + 10,000 = 20,000 which is less than 18,500

5-digit – 5-digit to give a difference less than 56,503

First number = 90,000

Second number = 34,500

Difference = 90,000 – 34,500 = 55,500 which is less than 56,503

5-digit – 3-digit to give a 4-digit difference

First number = 10,000

Second number = 500

Difference = 10,000 – 500 = 9500 (4 digits)

5-digit − 4-digit to give a 4-digit difference

First number = 10,999

Second number = 1,999

Difference = 10,999 – 1,999 = 9000 (4 digits)

5-digit − 5-digit to give a 3-digit difference

First number = 99,900

Second number = 99,000

Difference = 99,900 – 99,000 = 900 (3 digits)

5-digit − 5-digit to give 91,500

Not possible.

Explanations: We take the largest 5-digit possible number which is 99,999. We have to subtract 8,499 from 99,999 to give 91,500. 8,499 is a 4-digit number.

Figure it Out (Page 72):

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.

Answer: The digits ‘6’ and ‘1’ in 62,871 can be swapped to form the number 12,876.

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:

We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd.

The largest number in our group is 73,999.

The smallest number in our group is 35,111.

The number closest to 50,000 is 51,111.

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

Answer:

5-digit number: 18,000

3-digit number = 300

3-digit number = 370

Sum = 18,000 + 300 + 370 = 18,670

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:

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. The sequence should always reach 1.

The Collatz conjecture is correct for all the starting numbers in this sequence: 1, 2, 4, 8, 16, 32, 64, … because each time you divide by 2 you get an even number, until you reach 1.

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:

The winning strategy is as follows:

We have to find a position from which the opponent cannot win, which can be called a losing position.

losing position is a total from which no matter what number (1, 2, or 3) your opponent adds, they cannot stop you from reaching 22 first.

To identify losing positions, we start backward from 22.

18 is a losing position, because from 18, any move (whether 1, 2 or 3 is added) your opponent makes (at 19, 20, or 21) leaves you in control of the game because you can just add 3, 2 or 1 respectively and reach 22.

Continuing this process, we find the other losing positions and force our opponent to land on these positions.

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