NCERT Solutions Ganita Prakash (Part 1) Chapter 6 Figure it Out — Some Explorations in Grids

Book page 136 Updated on2026-09-05

Q1.

How many different magic squares can be made using the numbers 1 – 9?

Answer

There is essentially only one magic square using 1 – 9. Counting turns and mirror images as different, there are 8.

8	1	6
3	5	7
4	9	2

Rotations: 4 (turn by 0°, 90°, 180°, 270°)
Each rotation also has a mirror image: 4 × 2 = 8 grids
But all 8 are the same square seen differently.

Why it happens: we proved that 5 must be at the centre, and that 1 and 9 must be opposite edge-middles. Those two facts leave no freedom at all except for turning the paper or looking at it in a mirror.

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.

Answer

Strategy: the numbers 2 – 10 are just the numbers 1 – 9 with 1 added to each. So add 1 to every entry of the 1 – 9 magic square.

Magic square with 1 – 9	Add 1 to each	Magic square with 2 – 10
8 1 6	→	9 2 7
3 5 7	→	4 6 8
4 9 2	→	5 10 3

Rows: 9 + 2 + 7 = 18, 4 + 6 + 8 = 18, 5 + 10 + 3 = 18
Columns: 9 + 4 + 5 = 18, 2 + 6 + 10 = 18, 7 + 8 + 3 = 18
Diagonals: 9 + 6 + 3 = 18, 7 + 6 + 5 = 18
Magic sum = 15 + 3 × 1 = 18

Comparison: the pattern is exactly the same. The centre is now 6 (the middle number of 2 – 10), and the magic sum grew by 3, because each of the three numbers in a line gained 1.

Q3.

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?

Answer

Yes — in both cases the result is again a magic square. Start from 8 1 6 / 3 5 7 / 4 9 2 with magic sum 15.

(a) Increase each number by 1

9	2	7
4	6	8
5	10	3

Every line gains 1 + 1 + 1 = 3.
New magic sum = 15 + 3 = 18

(b) Double each number

16	2	12
6	10	14
8	18	4

Every line is doubled.
New magic sum = 15 × 2 = 30
Check: 16 + 2 + 12 = 30, 6 + 10 + 14 = 30, 8 + 18 + 4 = 30 ✓

Why it happens: adding the same number k to all nine cells raises every line by 3k; multiplying all nine cells by k multiplies every line by k. Since all lines change in the same way, they stay equal — so the grid stays magic.

Q4.

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

Answer

Operation	Still magic?	New magic sum (if old sum is S)
Add a number k to every cell	Yes	S + 3k
Subtract a number k from every cell	Yes	S – 3k
Multiply every cell by k	Yes	k × S
Divide every cell by k (k ≠ 0)	Yes	S ÷ k
Rotate the grid by 90°, 180° or 270°	Yes	S (unchanged)
Reflect the grid in a mirror	Yes	S (unchanged)
Swap only two numbers	Usually no	the lines stop being equal

Why it happens: an operation keeps a square magic if it treats every cell in the same way. Adding, subtracting, multiplying, dividing and turning all do that; changing just one or two cells does not.

Try This: multiply 8 1 6 / 3 5 7 / 4 9 2 by 3 to get 24 3 18 / 9 15 21 / 12 27 6 with magic sum 45.

Q5.

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

Answer

Take the 1 – 9 magic square and add the same number to every cell. The number to add is (smallest number of the new set) – 1.

Set of 9 consecutive numbers	Add to each cell of 1 – 9 square	Centre	Magic sum
1 – 9	0	5	15
2 – 10	1	6	18
3 – 11	2	7	21
9 – 17	8	13	39

For 9 – 17, adding 8 to each entry gives:

16	9	14
11	13	15
12	17	10

Rows: 16 + 9 + 14 = 39, 11 + 13 + 15 = 39, 12 + 17 + 10 = 39 ✓
Columns: 16 + 11 + 12 = 39, 9 + 13 + 17 = 39, 14 + 15 + 10 = 39 ✓
Diagonals: 16 + 13 + 10 = 39, 14 + 13 + 12 = 39 ✓

Why it happens: nine consecutive numbers have exactly the same spacing pattern as 1 – 9 — each one is a fixed amount bigger. The middle number always goes at the centre and the magic sum is 3 × centre.

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