Question 1

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

Accepted Answer

The expression obtained = 3 × m, where m is the letter-number representing the number in the centre. — In a magic square, the sum of the three numbers in any row, column, or diagonal is always three times the middle number m.

Question 2

Write the result obtained by
(a) Adding 1 to every term in the generalised form.
(b) Doubling every term in the generalised form.

Accepted Answer

(a) Adding 1 to every term in the generalised form results in:

| m+4 | m-3 | m+2 |
|-----|-----|-----|
| m-1 | m+1 | m+3 |
| m   | m+5 | m-2 |

(b) Doubling every term in the generalised form results in:

| 2m+6 | 2m-8 | 2m+2 |
|-------|-------|-------|
| 2m-4 | 2m   | 2m+4 |
| 2m-2 | 2m+8 | 2m-6 | — For (a), add 1 to each term of the original magic square terms.
For (b), multiply each term of the original magic square terms by 2.

Question 3

Create a magic square whose magic sum is 60.

Accepted Answer

One such magic square is:

| 23 | 16 | 21 |
|----|----|----|
| 18 | 20 | 22 |
| 19 | 24 | 17 | — The sum of each row, column, and diagonal in this square is 60.

Question 4

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

Accepted Answer

Yes, it is possible.

One such magic square is:

| 24 | 3  | 18 |
|----|----|----|
| 9  | 15 | 21 |
| 12 | 27 | 6  | — The numbers are non-consecutive but arranged so that the sums of rows, columns, and diagonals are equal.

Question 5

Write the next 3 numbers in the sequence:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, __, __, __, ...
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)?

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

Accepted Answer

Next three numbers are:
144, 233, 377

The next number after these will be even.

Parity pattern:
Sequence: 1(O), 2(E), 3(O), 5(O), 8(E), 13(O), 21(O), 34(E), ...
Pattern: Odd, Even, Odd, Odd, Even, Odd, Odd, Even...
The group odd, even, odd repeats.
Examples:
3rd term = 1st + 2nd = O + E = O
4th term = 2nd + 3rd = E + O = O
5th term = 3rd + 4th = O + O = E — The sequence is the Fibonacci (Virahanka) sequence. The parity pattern repeats in groups of three: odd, even, odd.

Question 6

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

Accepted Answer

The bulb will be OFF.

Explanation: Each toggle changes the state of the bulb. Starting ON, after an odd number of toggles (77), the bulb will be OFF. — Since toggling 77 times (an odd number) changes the bulb's state from ON to OFF.

Question 7

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?

Accepted Answer

Yes, it is possible.

Explanation: Each sheet has two pages, one even-numbered and one odd-numbered. There are 50 sheets, so 50 even and 50 odd pages. Sum of even numbers is even, sum of even number of odd numbers is also even. Thus, total sum can be even like 6000. — Sum of even numbers is even. Sum of even count of odd numbers is even. Hence total sum can be 6000 (even).

Question 8

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.

Accepted Answer

One possible solution is shown in the textbook figure (img-41.jpeg). Try more combinations. — By placing 3 odd and 3 even numbers appropriately, the parity conditions of rows and columns can be satisfied.

Question 9

Make a 3 × 3 magic square with 0 as the magic sum. All numbers cannot be zero. Use negative numbers, as needed.

Accepted Answer

One such magic square is:

| 3  | -4 | 1  |
|----|----|----|
| -2 | 0  | 2  |
| -1 | 4  | -3 | — The sum of each row, column, and diagonal is zero, using negative and positive integers.

Question 10

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 _____

Accepted Answer

(a) Even.
(b) Even.
(c) Even.
(d) Odd. — Sum of even numbers is always even.
Sum of even count of odd numbers is even.
Sum of odd count of odd numbers is odd.

Question 11

What is the parity of the sum of numbers from 1 to 100?

Accepted Answer

The parity is even.

Explanation: Sum of numbers from 1 to 100 = (100 × 101)/2 = 5050, which is even. — Sum of first n natural numbers is n(n+1)/2. For n=100, sum is 5050 (even).

Question 12

Two consecutive numbers in the Virahanka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers in the sequence?

Accepted Answer

Next two numbers:
987 + 1597 = 2584
1597 + 2584 = 4181

Previous two numbers:
1597 - 987 = 610
987 - 610 = 377

Sequence: ..., 377, 610, 987, 1597, 2584, 4181, ... — Virahanka sequence (Fibonacci) numbers are formed by sum of previous two numbers. Previous numbers found by subtraction.

Question 13

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?

Accepted Answer

Total ways Angaan can reach the top = 34 ways.

Explanation:
Number of ways to climb 8 steps taking 1 or 2 steps at a time corresponds to the 9th Fibonacci number (starting with 1,2,...). The sum of ways with different combinations is 34. — The number of ways to climb n steps with 1 or 2 steps at a time is Fibonacci(n+1). For 8 steps, ways = 34.

Question 14

What is the parity of the 20th term of the Virahanka sequence?

Accepted Answer

The 20th term of the Virahanka sequence is even.

Explanation:
The parity pattern of Virahanka numbers is O, E, O, O, E, O,... repeating. The 20th term falls on an even parity position. — By observing the parity pattern of the sequence, the 20th term is even.

Question 15

Identify the true statements.
(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.

Accepted Answer

(a) True
(b) False
(c) False
(d) False — (a) 4m - 1 is always odd because 4m is even, subtracting 1 gives odd.
(b) Not all even numbers can be expressed as 6j - 4.
(c) 2p + 1 and 2q - 1 do not cover all odd numbers.
(d) 2f + 3 is always odd (since 2f is even, plus 3 is odd), so it cannot give even numbers.

Number Play

Number Play — Study Notes

Introduction

Data Collection

Representation of Data

Practice Questions — Number Play

All 8 Chapters in Ganita Prakash