Chapter 6 Number Play (Class 7 - Latest Maths NCERT (Ganita Prakash I) NCERT Solutions)

Question. Write down the number each child should say based on this rule for the arrangement shown below.

Question 1. 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) $0, 0, 0, 3, 3, 3, 3$

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?

Question 1. 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

Question 2. Lakpa has an odd number of $\textsf{₹}1$ coins, an odd number of $\textsf{₹}5$ coins and an even number of $\textsf{₹}10$ coins in his piggy bank. He calculated the total and got $\textsf{₹}205$. Did he make a mistake? If he did, explain why. If he didn’t, how many coins of each type could he have?

Question 3. We know that:

(a) $\text{even} + \text{even} = \text{even}$

(b) $\text{odd} + \text{odd} = \text{even}$

(c) $\text{even} + \text{odd} = \text{odd}$

Similarly, find out the parity for the scenarios below:

(d) $\text{even} - \text{even} =$ ___________________

(e) $\text{odd} - \text{odd} =$ ___________________

(f) $\text{even} - \text{odd} =$ ___________________

(g) $\text{odd} - \text{even} =$ ___________________

Question. In a $3 \times 3$ grid, there are $9$ small squares, which is an odd number. Meanwhile, in a $3 \times 4$ grid, there are $12$ small squares, which is an even number.

Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product?

Question. Find the parity of the number of small squares in these grids:

(a) $27 \times 13$

(b) $42 \times 78$

(c) $135 \times 654$

Question. Consider the algebraic expression: $3n + 4$. For different values of $n$, the expression has different parity:

Come up with an expression that always has even parity.

Question. Come up with expressions that always have odd parity.

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

Question. Are there expressions using which we can list all the even numbers?

Question. Are there expressions using which we can list all odd numbers?

Question. What would be the $n^{th}$ term for multiples of $2$? Or, what is the $n^{th}$ even number?

Question. Observe this $3 \times 3$ grid. It is filled following a simple rule — use numbers from $1 – 9$ without repeating any of them. There are circled numbers outside the grid.

The numbers in the yellow circles are the sums of the corresponding rows and columns.

Fill the grids below based on the rule mentioned above:

Question 1. How many different magic squares can be made using the numbers $1 - 9$?

Question 2. Create a magic square using the numbers $2 - 10$. What strategy would you use for this? Compare it with the magic squares made using $1 - 9$.

Question 3. 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?

Question 4. What other operations can be performed on a magic square to yield another magic square?

Question 5. Discuss ways of creating a magic square using any set of $9$ consecutive numbers (like $2 - 10$, $3 - 11$, $9 - 17$, etc.).

Question. Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$, how much more or less than $m$.

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Question 1. Using this generalised form, find a magic square if the centre number is $25$.

Question 2. What is the expression obtained by adding the $3$ terms of any row, column or diagonal?

Question 3. Write the result obtained by—

(a) adding $1$ to every term in the generalised form.

(b) doubling every term in the generalised form

Question 4. Create a magic square whose magic sum is $60$.

Question 5. Is it possible to get a magic square by filling nine non-consecutive numbers?

Question. Write the next $3$ numbers in the sequence: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \_\_\_\_, \_\_\_\_, \_\_\_\_, \dots$

If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Question. What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?

Question 1. A light bulb is ON. Dorjee toggles its switch $77$ times. Will the bulb be on or off? Why?

Question 2. 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?

Question 3. Here is a $2 \times 3$ 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.

Question 4. Make a $3 \times 3$ magic square with $0$ as the magic sum. All numbers can not be zero. Use negative numbers, as needed.

Question 5. 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 ______

Question 6. What is the parity of the sum of the numbers from $1$ to $100$?

Question 7. 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?

Question 8. 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?

Question 9. What is the parity of the $20^{th}$ term of the Virahāṅka sequence?

Question 10. Identify the statements that are true.

(a) The expression $4m - 1$ always gives odd numbers.

(b) All even numbers can be expressed as $6j - 4$.

(c) Both expressions $2p + 1$ and $2q - 1$ describe all odd numbers.

(d) The expression $2f + 3$ gives both even and odd numbers.

Question 11. Solve this cryptarithm:

$$\begin{array}{cccc} & & U & T \\ + & & T & A \\ \hline & T & A & T \\ \hline \end{array}$$