Question 6
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?
We need to find if such an arrangement of children is possible.
Step 1 — Understand the numbers
Each child says how many taller neighbors they have. A child at an end has one neighbor. A child in the middle has two neighbors. '0' means no taller neighbors. '1' means one taller neighbor. '2' means two taller neighbors.
Step 2 — The tallest child
Let us think about the tallest child. No one is taller than the tallest child. So, any neighbor must be shorter. If at an end, the tallest child has one shorter neighbor. If in the middle, the tallest child has two shorter neighbors. So, the tallest child always says '0'. This matches the problem's condition.
Step 3 — The shortest child
Let us think about the shortest child. Everyone else is taller than the shortest child. If the shortest child is in the middle, they have two neighbors. Both neighbors are taller than the shortest child. So, the shortest child would say '2'. But the problem says four children say '1'. The shortest child must be one of these four. So, the shortest child cannot say '2'. This means the shortest child must be at an end. If at an end, the shortest child has one neighbor. That neighbor is taller. So, the shortest child says '1'. This works for the shortest child.
Step 4 — Propose an arrangement
We know the tallest child says '0'. We know the shortest child says '1' and is at an end. Let us name children by their height rank. Child 1 is the shortest. Child 5 is the tallest. Let us try arranging them from shortest to tallest. The arrangement is: Child 1, Child 2, Child 3, Child 4, Child 5.
Step 5 — Verify the arrangement
Let us check what each child says. Child 1 (shortest): Its only neighbor is Child 2. Child 2 is taller than Child 1. So, Child 1 has one taller neighbor. Child 1 says '1'. (This is good)
Child 2: Its neighbors are Child 1 and Child 3. Child 1 is shorter than Child 2. Child 3 is taller than Child 2. So, Child 2 has one taller neighbor. Child 2 says '1'. (This is good)
Child 3: Its neighbors are Child 2 and Child 4. Child 2 is shorter than Child 3. Child 4 is taller than Child 3. So, Child 3 has one taller neighbor. Child 3 says '1'. (This is good)
Child 4: Its neighbors are Child 3 and Child 5. Child 3 is shorter than Child 4. Child 5 is taller than Child 4. So, Child 4 has one taller neighbor. Child 4 says '1'. (This is good)
Child 5 (tallest): Its only neighbor is Child 4. Child 4 is shorter than Child 5. So, Child 5 has zero taller neighbors. Child 5 says '0'. (This is good)
All conditions are met. Four children say '1'. One child says '0'.
Answer
Yes, they can.
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'?
Find out how many numbers have two digits, three digits, four digits, and five digits.
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?
Consider the digits '1', '2', '3'.
Q. Write all possible 3-digit palindromes using these digits.
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?
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? _____________
What happens if we continue doing this?
Try and find out all possible times on a 12-hour clock of each of these types.
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.
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.
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?
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 × 2 + 13,000 3400 = 1500 + 1500 + 400
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?
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.
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.
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?
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
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
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?
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.
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?
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.
Make some estimation questions and challenge your classmates!