NCERT Solutions Ganita Prakash (Part 1) Chapter 6 In-text Questions — Some Explorations in Grids

Book page 135 Updated on2026-09-05

Q1.

What are the possible numbers that could occur at the centre of a magic square?

Answer

Only 5 can sit at the centre.

Test 9 at the centre. Then 8 must go somewhere else, and 8 shares a line with the centre or does not; wherever it goes, some line contains 8 and 9:

8 + 9 + (another number) = 15
another number = 15 – 17 = –2 — impossible with 1 – 9.

Test 1 at the centre. Then 2 must go somewhere, and some line contains 1 and 2:

2 + 1 + (another number) = 15
another number = 15 – 3 = 12 — impossible, we only have 1 – 9.

So neither 9 nor 1 can be at the centre.

Why it happens: the centre cell belongs to four lines — one row, one column and both diagonals. So the centre number must pair up with the other eight numbers in four different pairs, each pair adding to 15 – centre.

Q2.

Using such reasoning, find out which other numbers 1 – 9 cannot occur at the centre.

Answer

Every number except 5 fails. Count, for each choice of centre c, how many pairs from the remaining numbers add up to 15 – c. Four pairs are needed.

Centre c	Pairs must add to 15 – c	Pairs available	How many	Allowed?
1	14	5+9, 6+8	2	No
2	13	4+9, 5+8, 6+7	3	No
3	12	4+8, 5+7	2	No
4	11	2+9, 3+8, 5+6	3	No
5	10	1+9, 2+8, 3+7, 4+6	4	Yes
6	9	1+8, 2+7, 4+5	3	No
7	8	2+6, 3+5	2	No
8	7	1+6, 2+5, 3+4	3	No
9	6	1+5, 2+4	2	No

Observation 2: The number occurring at the centre of a magic square, filled using 1 – 9, must be 5.

Why it happens: the centre lies on four lines, so the other eight numbers must split into four pairs, each adding to 15 – c. Only c = 5 gives exactly four such pairs — and 5 is also the middle number of 1 – 9.

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