Question 2
For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
(a) If a person says '0', then they are the tallest in the group.
(b) If a person is the tallest, then their number is '0'.
(c) The first person's number is '0'.
(d) If a person is not first or last in line (i.e., if they are standing somewhere in between), then they cannot say '0'.
(e) The person who calls out the largest number is the shortest.
(f) What is the largest number possible in a group of 8 people?
Each person says a number based on taller people ahead in the line.
Step 1 — Understanding the game rules
Let us understand the game first. People stand in a line. Each person says a number. This number tells how many taller people are in front. We will use this rule for all parts.
Step 2 — Analyzing statement (a)
Statement (a) asks: If a person says '0', are they the tallest? Saying '0' means no one taller is ahead. The tallest person has no one taller at all. So, the tallest person always says '0'. But a shorter person can also say '0'. Imagine the shortest person is first in line. No one is ahead of them. So, they say '0'. This person is not the tallest. So, saying '0' does not mean they are tallest.
(a) Only Sometimes True
Step 3 — Analyzing statement (b)
Statement (b) asks: If a person is the tallest, is their number '0'? The tallest person has no one taller than them. So, no one taller can be ahead of them. The count of taller people ahead is zero. This is always true for the tallest person.
(b) Always True
Step 4 — Analyzing statement (c)
Statement (c) asks: Is the first person's number '0'? The first person is at the front of the line. No one is standing ahead of the first person. So, there are zero taller people ahead. Their number must always be '0'.
(c) Always True
Step 5 — Analyzing statement (d)
Statement (d) asks: Can a middle person not say '0'? A person in the middle has people both ahead and behind. Saying '0' means no one taller is ahead. Consider a line: Tallest, Shortest, Medium. The Tallest person is first. They say '0'. The Shortest person is in the middle. Ahead of the Shortest person is the Tallest person. So, the Shortest person says '1'. The Medium person is last. Ahead of the Medium person are Tallest and Shortest. The Tallest is taller than Medium. So, the Medium person says '1'. In this example, the middle person (Shortest) said '1', not '0'. Now consider: Medium, Tallest, Shortest. The Medium person is first. They say '1' (Tallest is ahead). The Tallest person is in the middle. No one taller than Tallest is ahead. So, the Tallest person says '0'. This shows a middle person can say '0'. So, the statement is not always true.
(d) Only Sometimes True
Step 6 — Analyzing statement (e)
Statement (e) asks: Is the person with the largest number the shortest? The largest number means many taller people are ahead. Consider 3 people: Shortest (S), Medium (M), Tallest (T). If the line is S, M, T: S says '0' (no one ahead). M says '1' (T is ahead). T says '0' (no one taller ahead). Here, M says the largest number (1). But S is the shortest person. So, the person with the largest number is not always the shortest. If the line is T, M, S: T says '0'. M says '0'. S says '0'. This example does not help. If the line is M, T, S: M says '1' (T is ahead). T says '0'. S says '0'. Again, M says the largest number (1), but S is the shortest. If the line is T, S, M: T says '0'. S says '1' (M is ahead). M says '0'. Here, S says the largest number (1), and S is the shortest. So, it can happen sometimes.
(e) Only Sometimes True
Step 7 — Analyzing statement (f)
Statement (f) asks: What is the largest number in a group of 8 people? The number is how many taller people are ahead. To get the largest number, a person needs many taller people ahead. Imagine a person at the very end of the line. There are 7 people ahead of them. If all 7 people ahead are taller, this person will say '7'. This happens if the shortest person is last in line. They see 7 taller people in front. No one can see more than 7 people ahead in an 8-person line. So, the maximum number of taller people ahead is 7.
(f) The largest number possible is 7.
Answer
(a) Only Sometimes True (b) Always True (c) Always True (d) Only Sometimes True (e) Only Sometimes True (f) 7
More questions in FIO
Arrange the stick figure cutouts given at the end of the book or draw a height arrangement such that the sequence reads:
(a) 0, 1, 1, 2, 4, 1, 5
(b) 0, 0, 0, 0, 0, 0, 0
(c) 0, 1, 2, 3, 4, 5, 6
(d) 0, 1, 0, 1, 0, 1, 0
(e) 0, 1, 1, 1, 1, 1, 1
(f) 0, 0, 0, 3, 3, 3, 3
For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
(a) If a person says '0', then they are the tallest in the group.
(b) If a person is the tallest, then their number is '0'.
(c) The first person's number is '0'.
(d) If a person is not first or last in line (i.e., if they are standing somewhere in between), then they cannot say '0'.
(e) The person who calls out the largest number is the shortest.
(f) What is the largest number possible in a group of 8 people?
Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:
(a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd)
(b) Sum of 2 odd numbers and 3 even numbers
(c) Sum of 5 even numbers
(d) Sum of 8 odd numbers
Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. He calculated the total and got ₹205. Did he make a mistake? If he did, explain why. If he didn't, how many coins of each type could he have?
We know that:
(a) even + even = even
(b) odd + odd = even
(c) even + odd = odd
Similarly, find out the parity for the scenarios below:
(d) even - even = _________
(e) odd - odd = _________
(f) even - odd = _________
(g) odd - even = _________
How many different magic squares can be made using the numbers 1 – 9?
Create a magic square using the numbers 2 – 10. What strategy would you use for this? Compare it with the magic squares made using 1 – 9.
Take a magic square, and
(a) increase each number by 1
(b) double each number
In each case, is the resulting grid also a magic square? How do the magic sums change in each case?
What other operations can be performed on a magic square to yield another magic square?
Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).
Using this generalised form, find a magic square if the centre number is 25.
What is the expression obtained by adding the 3 terms of any row, column or diagonal?
Write the result obtained by— (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form
Create a magic square whose magic sum is 60.
Is it possible to get a magic square by filling nine non-consecutive numbers?
A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?
Liswini has a large old encyclopaedia. When she opened it, several loose pages fell out of it. She counted 50 sheets in total, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?
Here is a 2 × 3 grid. For each row and column, the parity of the sum is written in the circle; 'e' for even and 'o' for odd. Fill the 6 boxes with 3 odd numbers ('o') and 3 even numbers ('e') to satisfy the parity of the row and column sums.
Make a 3 × 3 magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.
Fill in the following blanks with 'odd' or 'even':
(a) Sum of an odd number of even numbers is ______
(b) Sum of an even number of odd numbers is ______
(c) Sum of an even number of even numbers is ______
(d) Sum of an odd number of odd numbers is ______
What is the parity of the sum of the numbers from 1 to 100?
Two consecutive numbers in the Virahāṅka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers in the sequence?
Angaan wants to climb an 8-step staircase. His playful rule is that he can take either 1 step or 2 steps at a time. For example, one of his paths is 1, 2, 2, 1, 2. In how many different ways can he reach the top?
What is the parity of the 20th term of the Virahāṅka sequence?
Identify the statements that are true.
(a) The expression always gives odd numbers. (b) All even numbers can be expressed as . (c) Both expressions and describe all odd numbers. (d) The expression gives both even and odd numbers.
Solve this cryptarithm: