NCERT Solutions Ganita Prakash (Part 1) Chapter 6 In-text Questions — Some Explorations in Grids
Book page 136 Updated on2026-09-05
Q1.
If yes, then there should exist three ways of adding 1 with two other numbers to give 15. We have 1 + 5 + 9 = 1 + 6 + 8 = 15. Is any other combination possible?
Answer
No, there is no third combination. Only two pairs add to 14.
1 + ? + ? = 15 → the other two must add to 14
14 = 5 + 9 ✓
14 = 6 + 8 ✓
14 = 7 + 7 ✗ (a number cannot repeat)
So only two lines through 1 are possible.
A corner cell lies on three lines — one row, one column and one diagonal — so it needs three such combinations. Since 1 has only two, 1 cannot occur in a corner.
Why it happens: a corner is a busy cell. It must join two other numbers three separate times, all making 15. A small number like 1 needs very large partners, and there are simply not enough large numbers to do it three times.
Q2.
Similarly, can 9 can be placed in a corner position?
Answer
No. The same counting rules it out.
9 + ? + ? = 15 → the other two must add to 6
6 = 1 + 5 ✓
6 = 2 + 4 ✓
6 = 3 + 3 ✗ (repeat not allowed)
Only two lines through 9 are possible, but a corner needs three.
Observation 3: The numbers 1 and 9 cannot occur in any corner, so they should occur in one of the middle positions.
Why it happens: 9 is so large that its two partners must be very small, and there are only two such small pairs available.
Q3.
Can you find the other possible positions for 1 and 9?
Answer
1 and 9 must sit in middle cells of the edges, and they must be opposite each other, with 5 between them.
1 + 5 + 9 = 15, and 5 is at the centre.
So 1 and 9 lie at the two ends of a line through the centre.
A corner is ruled out, so that line is a middle row or a middle column.
The four possibilities are:
1 in the middle of the left edge, 9 in the middle of the right edge
9 in the middle of the left edge, 1 in the middle of the right edge
1 in the middle of the top edge, 9 in the middle of the bottom edge
9 in the middle of the top edge, 1 in the middle of the bottom edge
Why it happens: once 5 is fixed at the centre and 1 and 9 are pushed out of the corners, the only cells left for them are the four edge-middles — and 1, 5, 9 must line up because 1 + 5 + 9 = 15.
Q4.
Now, we have one full row or column of the magic square! Try completing it! [Hint: First fill the row or columns containing 1 and 9]
Answer
Start with 1, 5, 9 filling the middle row, then finish the grid.
C1
C2
C3
Sum
R1
8
3
4
15
R2
1
5
9
15
R3
6
7
2
15
Sum
15
15
15
—
Diagonals: 8 + 5 + 2 = 15 ✓ and 4 + 5 + 6 = 15 ✓
Column 1: 1 must join two numbers adding to 14 → 8 and 6
Column 3: 9 must join two numbers adding to 6 → 4 and 2
Column 2: 3 + 5 + 7 = 15 ✓
Turning and flipping this grid gives the familiar form of the same magic square: