Question 1

Q1: 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,...

Accepted Answer

Rule: Each person says the number of people in front of them who are taller than them. Let's represent the heights of the 7 people using numbers 1 to 7, where 1 is the shortest and 7 is the tallest. The arrangement is a line from left to right (person 1 to person 7). (a) 0, 1, 1, 2, 4, 1, 5 One poss...

Question 2

Q2: 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 pe...

Accepted Answer

(a) If a person says '0', then they are the tallest in the group. Answer: Only Sometimes True. Reasoning: A person saying '0' means there is no one taller than them in front of them. This could be because they are the tallest person in the group, or simply the tallest among those standing in front o...

Question 3

Q1: 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...

Accepted Answer

Parity Rules for Addition: - even + even = even - odd + odd = even - even + odd = odd (a) Sum of 2 even numbers and 2 odd numbers - (even + even) + (odd + odd) - (even) + (even) = even (b) Sum of 2 odd numbers and 3 even numbers - (odd + odd) + (even + even + even) - (even) + (even) = even (c) Sum o...

Question 4

Q2: 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...

Accepted Answer

Given: - Number of ₹1 coins = odd - Number of ₹5 coins = odd - Number of ₹10 coins = even - Total amount = ₹205 To Find: Whether Lakpa made a mistake. Solution: Let's determine the parity of the total amount based on the coins. 1.  Value from ₹1 coins:     (Number of coins) $\times$ (Value of coin)...

Question 5

Q3: 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...

Accepted Answer

Parity Rules for Subtraction: Subtraction follows the same parity rules as addition. (d) even - even = _____ Answer: even Example: $8 - 4 = 4$ (even) (e) odd - odd = _____ Answer: even Example: $9 - 5 = 4$ (even) (f) even - odd = _____ Answer: odd Example: $10 - 3 = 7$ (odd) (g) odd - even = _____ A...

Question 6

Q1: Come up with an expression that always has even parity.

Accepted Answer

Expression: $2n$ Reasoning: Any integer multiplied by 2 is, by definition, an even number. For any integer value of $n$, the expression $2n$ will always result in an even number. Other Examples: - $4n$ (since $4n = 2(2n)$) - $2n + 4$ (sum of two even numbers) - $6k - 2$ (difference of two even numbe...

Question 7

Q2: Come up with expressions that always have odd parity.

Accepted Answer

Expression: $2n + 1$ Reasoning: An even number ($2n$) plus an odd number (1) always results in an odd number. For any integer value of $n$, the expression $2n+1$ will always result in an odd number. This expression generates all positive odd numbers for $n=0, 1, 2, ...$ Other Examples: - $2n - 1$ -...

Question 8

Q3: Come up with other expressions, like $3n+4$, which could have either odd or even parity.

Accepted Answer

Expression: $n + 1$ Reasoning: The parity of the expression depends on the parity of $n$. - If $n$ is even, then $n+1$ is even + odd = odd. - If $n$ is odd, then $n+1$ is odd + odd = even. Since the expression can be either odd or even, it fits the condition. Other Examples: - $5n$ (If $n$ is even,...

Question 9

Q1: Find the parity of the number of small squares in these grids: (a) $27 \times 13$ (b) $42 \times 78$ (c) $135 \times 654$

Accepted Answer

Parity Rules for Multiplication: - even $\times$ even = even - even $\times$ odd = even - odd $\times$ odd = odd (a) $27 \times 13$ - 27 is odd. - 13 is odd. - Parity = odd $\times$ odd = odd (b) $42 \times 78$ - 42 is even. - 78 is even. - Parity = even $\times$ even = even (c) $135 \times...

Question 10

Q1: How many different magic squares can be made using the numbers 1-9?

Accepted Answer

Answer: 8 Explanation: Using the numbers 1 through 9, there is essentially only one unique $3 \times 3$ magic square. However, this unique square can be rotated and reflected to produce a total of 8 different-looking magic squares. All 8 of these are considered variations of the same fundamental so...

Question 11

Q2: 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.

Accepted Answer

Strategy: 1.  Start with a known magic square using the numbers 1-9. The magic sum for this square is 15.     One such square is:     $$\begin{vmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{vmatrix}$$ 2.  To create a magic square using the numbers 2-10, we can simply add 1 to each number in...

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