NCERT Solutions Ganita Prakash (Part 1) Chapter 6 Generalising a 3 × 3 Magic Square — Figure it Out

Book page 137 Updated on2026-09-05

Q1.

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

Answer

Put m = 25 in the generalised form.

Generalised	m + 3	m – 4	m + 1
Value	28	21	26
Generalised	m – 2	m	m + 2
Value	23	25	27
Generalised	m – 1	m + 4	m – 3
Value	24	29	22

The magic square is:

28	21	26
23	25	27
24	29	22

Magic sum = 3m = 3 × 25 = 75
Rows: 28 + 21 + 26 = 75, 23 + 25 + 27 = 75, 24 + 29 + 22 = 75 ✓
Columns: 28 + 23 + 24 = 75, 21 + 25 + 29 = 75, 26 + 27 + 22 = 75 ✓
Diagonals: 28 + 25 + 22 = 75, 26 + 25 + 24 = 75 ✓

Check it yourself: the nine entries are 21 to 29 — nine consecutive numbers with 25 in the middle, exactly as expected.

Q2.

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

Answer

The expression is 3m.

Row 1: (m + 3) + (m – 4) + (m + 1) = 3m + 3 – 4 + 1 = 3m
Row 2: (m – 2) + m + (m + 2) = 3m – 2 + 2 = 3m
Column 2: (m – 4) + m + (m + 4) = 3m – 4 + 4 = 3m
Diagonal: (m + 1) + m + (m – 1) = 3m

Why it happens: each line has three m's, and the added and subtracted numbers always cancel out. That is exactly why the square is magic — and it also shows the magic sum is always three times the centre number.

Q3.

Write the result obtained by— (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form

Answer

(a) Adding 1 to every term

m + 4	m – 3	m + 2
m – 1	m + 1	m + 3
m	m + 5	m – 2

Line sum = (m + 4) + (m – 3) + (m + 2) = 3m + 3
So the magic sum increases by 3, and the new centre is m + 1.

(b) Doubling every term

2m + 6	2m – 8	2m + 2
2m – 4	2m	2m + 4
2m – 2	2m + 8	2m – 6

Line sum = (2m + 6) + (2m – 8) + (2m + 2) = 6m
So the magic sum is doubled, and the new centre is 2m.

Why it happens: both grids are still magic because every cell was changed in the same way. Notice that after doubling the entries are no longer consecutive — they jump in steps of 2.

Q4.

Create a magic square whose magic sum is 60.

Answer

Magic sum = 3m, so choose m = 20.

3m = 60
m = 60 ÷ 3 = 20

23	16	21
18	20	22
19	24	17

Rows: 23 + 16 + 21 = 60, 18 + 20 + 22 = 60, 19 + 24 + 17 = 60 ✓
Columns: 23 + 18 + 19 = 60, 16 + 20 + 24 = 60, 21 + 22 + 17 = 60 ✓
Diagonals: 23 + 20 + 17 = 60, 21 + 20 + 19 = 60 ✓

Tip: to make a magic square with any magic sum S that is a multiple of 3, simply take m = S ÷ 3 and fill in the generalised form.

Q5.

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

Answer

Yes, it is possible. Multiply every entry of a known magic square by the same number.

Multiplying 8 1 6 / 3 5 7 / 4 9 2 by 3 gives nine multiples of 3 — which are certainly not consecutive:

24	3	18
9	15	21
12	27	6

Rows: 24 + 3 + 18 = 45, 9 + 15 + 21 = 45, 12 + 27 + 6 = 45 ✓
Columns: 24 + 9 + 12 = 45, 3 + 15 + 27 = 45, 18 + 21 + 6 = 45 ✓
Diagonals: 24 + 15 + 6 = 45, 18 + 15 + 12 = 45 ✓
Magic sum = 15 × 3 = 45

Why it happens: what a magic square really needs is the pattern of differences from the centre, not consecutive numbers. Here every difference has simply been tripled: 3 × 5 = 15 is the centre, and the entries are 15 ± 3, 15 ± 6, 15 ± 9, 15 ± 12.

Try This: any nine numbers of the form m – 4d, m – 3d, …, m + 4d work for any step d. Take m = 20 and d = 5 to get 35 0 25 / 10 20 30 / 15 40 5, with magic sum 60.

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