Number Play | IT

Question 8

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

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

Solution

Understand the Question

Each child in a line looks only at their immediate left and right neighbors and counts how many are strictly taller.
The children at the two ends have only $1$ neighbor, so their count can only be $0$ or $1$ .
The three children in the middle each have $2$ neighbors, so their count can be $0, 1,$ or $2$ .
To determine if the sequence $0, 1, 2, 1, 0$ is possible, we establish the relative height conditions between adjacent children and check if a valid height assignment exists.

Step 1 · Analyze Neighbor Relations

Let the five children in line from left to right be $C_1, C_2, C_3, C_4, C_5$ . Diagram 1

Given the sequence of taller neighbors $0, 1, 2, 1, 0$ :

Child $C_3$ says $2$ : Both adjacent neighbors ( $C_2$ and $C_4$ ) are taller than $C_3$ . $\text{Height}(C_2) > \text{Height}(C_3) \quad \text{and} \quad \text{Height}(C_4) > \text{Height}(C_3)$
Child $C_2$ says $1$ : Only one neighbor is taller. Since $C_3$ is shorter than $C_2$ , neighbor $C_1$ must be taller: $\text{Height}(C_1) > \text{Height}(C_2)$
Child $C_4$ says $1$ : Only one neighbor is taller. Since $C_3$ is shorter than $C_4$ , neighbor $C_5$ must be taller: $\text{Height}(C_5) > \text{Height}(C_4)$
Children $C_1$ and $C_5$ say $0$ : Neither neighbor is taller, which matches $\text{Height}(C_1) > \text{Height}(C_2)$ and $\text{Height}(C_5) > \text{Height}(C_4)$ .

Step 2 · Verify with an Example Height Assignment

Combining the inequality relations: $\text{Height}(C_1) > \text{Height}(C_2) > \text{Height}(C_3) < \text{Height}(C_4) < \text{Height}(C_5)$

Assign positive integer heights: $C_1 = 3, \quad C_2 = 2, \quad C_3 = 1, \quad C_4 = 2, \quad C_5 = 3$

Verification:

$C_1$ (height $3$ ): Neighbor $C_2$ (height $2$ ) is shorter $\implies 0$ taller neighbors.
$C_2$ (height $2$ ): Neighbors are $3$ (taller) and $1$ (shorter) $\implies 1$ taller neighbor.
$C_3$ (height $1$ ): Neighbors are $2$ and $2$ (both taller) $\implies 2$ taller neighbors.
$C_4$ (height $2$ ): Neighbors are $1$ (shorter) and $3$ (taller) $\implies 1$ taller neighbor.
$C_5$ (height $3$ ): Neighbor $C_4$ (height $2$ ) is shorter $\implies 0$ taller neighbors.

Answer

Yes, the sequence $0, 1, 2, 1, 0$ is possible. For example, heights $(3, 2, 1, 2, 3)$ satisfy all conditions.

Common Mistakes

Comparing with All Children: Thinking each child counts everyone taller in the entire line rather than just their immediate neighbors.
End-Position Constraint: Forgetting that the first and last children only have one neighbor, so they can never say '2'.

More questions in IT

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.

Some children in a park are standing in a line. Each one says a number.

Q. What do you think these numbers mean?

group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

Q. Can the children rearrange themselves so that the children standing at the ends say '2'?

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

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

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

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

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

Q. 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?

A group of children stand in a line. Each child says a number (' $0$ ', ' $1$ ', or ' $2$ ') representing the number of taller neighbors standing next to them.

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

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

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

A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.

Q. How would you rearrange the five children so that the maximum number of children say '2'?

Q10

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

Q11

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?

Q12

Consider the digits $1$ , $2$ , $3$ .

Q. Write all possible 3-digit palindromes using these digits.

Q13

Try the same procedure for some other numbers, and perform the same steps. Stop if you get a palindrome. There are numbers for which you have to repeat this a large number of times. Are there numbers for which you do not reach a palindrome at all?

Q14

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? _____________

Q15

What happens if we continue doing this?

Q16

Try and find out all possible times on a 12-hour clock of each of these types.

Q17

Manish has his birthday on 20/12/2012 where the digits '2', '0', '1', '2' repeat in that order.

Q. Find some other dates of this form from the past.

Q18

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.

Q. Find all possible dates of this form from the past.

Q19

But, will any year's calendar repeat again after some years? Will all dates and days in a year match exactly with that of another year?

Q20

Observe the figure below. What can you say about the numbers and the lines drawn?

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!

$38,800 = 25,000 + 400 \times 2 + 13,000$

$3400 = 1500 + 1500 + 400$

Q21

Can we make 1,000 using the numbers in the middle? Why not? What about 14,000, 15,000 and 16,000? Yes, it is possible. Explore how. What thousands cannot be made?

Q22

Using the numbers in the boxes, we are allowed to use both addition and subtraction to get the required number. Complete the equations shown in the diagram.

Q23

3.9 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?

Share and discuss in class the different methods each one of you used to solve these questions.

Q24

3.10 An Unsolved Mystery—the Collatz Conjecture!

Look at the sequences below—the same rule is applied in all the sequences:

a. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1

b. 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

c. 21, 64, 32, 16, 8, 4, 2, 1

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?

Q25

Estimate the answer

Try to guess within 30 seconds. Check your guess with your friends.

Number of words in your maths textbook: a. More than 5000 b. Less than 5000

Q26

Estimate the answer

Try to guess within 30 seconds. Check your guess with your friends.

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

Q27

Estimate the answer

Try to guess within 30 seconds. Check your guess with your friends.

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?

Q28

Estimate the answer

Try to guess within 30 seconds. Check your guess with your friends.

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

Q29

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?

Q30

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.

Q31

Make some estimation questions and challenge your classmates!