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

Book page 134 Updated on2026-09-05

Q1.

Make a couple of questions like this on your own and challenge your peers.

Answer

Start from a filled grid and hide the numbers — then only the circles remain.

Here is one you can set:

	C1	C2	C3	Row sum
R1	2	9	4	15
R2	7	5	3	15
R3	6	1	8	15
Column sum	15	15	15	45

Give your friend only the six circled numbers 15, 15, 15 (rows) and 15, 15, 15 (columns), plus one filled cell as a clue.

Tip for setting a good puzzle: always check that your three row sums add to 45 and your three column sums add to 45, and that no circle is below 6 or above 24. Otherwise your puzzle will have no answer at all.

Q2.

You might have realised that it is not possible to find a solution for this grid. Why is this the case?

Answer

Because two of the circled numbers lie outside the possible range.

Smallest sum of three different numbers from 1 – 9 = 1 + 2 + 3 = 6
Largest sum of three different numbers from 1 – 9 = 9 + 8 + 7 = 24
So every circle must satisfy 6 ≤ circle ≤ 24.

This grid shows the circles 5, 21, 19 (rows) and 9, 11, 26 (columns).

Circle	Allowed?	Reason
5	No	less than the smallest possible sum 6
26	No	more than the largest possible sum 24
21, 19, 9, 11	Yes	between 6 and 24

So the grid is impossible.

Why it happens: you did not have to try filling a single cell. Just testing the range of the circles settles the question at once.

Check it yourself: the row sums here are 5 + 21 + 19 = 45 and the column sums are 9 + 11 + 26 = 46. The column total is not even 45, which is a second reason the grid cannot work.

Q3.

Why should the row sums and column sums always add to 45?

Answer

Because adding all three row sums means adding every number in the grid exactly once.

Row sums added together
= (all numbers of row 1) + (all of row 2) + (all of row 3)
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
= 45

The same argument works for columns, because the three columns also cover the whole grid once.

Grid solved earlier	Row sums	Total	Column sums	Total
4 7 5 / 6 1 2 / 3 9 8	16, 9, 20	45	13, 17, 15	45
9 1 3 / 8 2 4 / 7 6 5	13, 14, 18	45	24, 9, 12	45
7 8 9 / 4 6 5 / 1 2 3	24, 15, 6	45	12, 16, 17	45

Why it happens: the rows do not overlap and together they use up all nine cells. So their total is fixed at 1 + 2 + … + 9 = 45, whatever arrangement you choose. That also explains Kishor's observation that all six circles add to 45 + 45 = 90.

Q4.

What can the magic sum be? Can it be any number?

Answer

No. For a magic square made from the numbers 1 – 9 the magic sum must be 15.

The three row sums always add to 45.
In a magic square all three row sums are equal, say S.
S + S + S = 45
3S = 45
S = 15

Observation 1: In a magic square made using the numbers 1 – 9, the magic sum must be 15.

Why it happens: the magic sum is not free to choose — it is forced by the numbers you are allowed to use. Use 2 – 10 instead and the total becomes 54, so the magic sum becomes 54 ÷ 3 = 18.

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