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NCERT Solutions Ganita Prakash Chapter 10 & 264Section 10.4 Explorations with Integers — Figure it Out

Book page 263 Updated on2026-09-05

Q1.
Do the calculations for the second grid above and find the border sum.
Answer

The second grid is

5– 3– 5
0– 5
– 8– 27
Top row: 5 + (– 3) + (– 5) = – 3
Bottom row: (– 8) + (– 2) + 7 = – 3
Left column: 5 + 0 + (– 8) = – 3
Right column: (– 5) + (– 5) + 7 = – 3

Answer: the border sum of the second grid is – 3.

Why it is called a ‘hollow’ grid: only the eight numbers on the border are used — the middle cell plays no part in any of the four sums.
Q2.
Complete the grids to make the required border sum: (i) – 10, ___, ___ / ___, ___, – 5 / 9, ___, ___ with border sum + 4 (ii) 6, 8, ___ / ___, ___, – 5 / ___, – 2, ___ with border sum – 2 (iii) 7, ___, ___ / ___, ___, – 5 / ___, ___, ___ with border sum – 4
Answer

Fill in whichever line has only one number missing, then move to the next.

(i) Border sum + 4 — start with the left column: – 10 + ? + 9 = 4, so the middle-left cell is 5.

– 10104
5– 5
9– 105
Top: – 10 + 10 + 4 = 4 ✔   Right: 4 + (– 5) + 5 = 4 ✔   Bottom: 9 + (– 10) + 5 = 4 ✔   Left: – 10 + 5 + 9 = 4 ✔

(ii) Border sum – 2 — the top row has only one gap: 6 + 8 + ? = – 2, so it is – 16. Everything else then follows, and this grid has only one answer.

68– 16
11– 5
– 19– 219
Right: – 16 + (– 5) + 19 = – 2 ✔   Bottom: – 19 + (– 2) + 19 = – 2 ✔   Left: 6 + 11 + (– 19) = – 2 ✔

(iii) Border sum – 4 — here you may choose freely. One filling is:

7– 2– 9
– 3– 5
– 8– 610
Top: 7 + (– 2) + (– 9) = – 4 ✔   Right: – 9 + (– 5) + 10 = – 4 ✔   Bottom: – 8 + (– 6) + 10 = – 4 ✔   Left: 7 + (– 3) + (– 8) = – 4 ✔
Q3.
For the last grid above, find more than one way of filling the numbers to get border sum – 4.
Answer

In the last grid only two numbers are fixed — 7 in the top-left corner and – 5 in the middle of the right column. That leaves plenty of freedom. Here are two more fillings:

Way 2

7– 110
– 11– 5
0– 51
Top: 7 – 11 + 0 = – 4 ✔   Right: 0 – 5 + 1 = – 4 ✔   Bottom: 0 – 5 + 1 = – 4 ✔   Left: 7 – 11 + 0 = – 4 ✔

Way 3

7– 6– 5
– 9– 5
– 2– 86
Top: 7 – 6 – 5 = – 4 ✔   Right: – 5 – 5 + 6 = – 4 ✔   Bottom: – 2 – 8 + 6 = – 4 ✔   Left: 7 – 9 – 2 = – 4 ✔
How to make your own: pick any two numbers for the top-right and bottom-left corners. The rest of the grid is then forced — each remaining cell sits in a line with only one gap left.
Q4.
Which other grids can be filled in multiple ways? What could be the reason?
Answer

The first grid (border sum + 4) and the third grid (border sum – 4) can be filled in many ways. The second grid (border sum – 2) has only one answer.

GridNumbers already givenFree choices leftHow many fillings
First (+ 4)3 numbers1many
Second (– 2)4 numbers0exactly one
Third (– 4)2 numbers2many
The reason: there are four conditions (top row, bottom row, left column, right column). Each condition can fill in one unknown, so four conditions can pin down four unknowns. In the second grid exactly four numbers are missing on the border, so everything is forced. In the other two grids more numbers are missing than there are conditions, and the extra ones can be chosen as you like — after that the rest is decided.
Another way to say it: in the first grid you may choose the top-right corner freely; in the third grid you may choose two of the corners freely.
Q5.
Make a border integer square puzzle and challenge your classmates.
Answer

Here is a simple recipe for making a puzzle that is sure to work.

  1. Choose a border sum, say – 6.
  2. Choose the four corner numbers first, for example 2, – 4 (top) and – 3, 1 (bottom).
  3. Now fill the middle cell of each line so that the line adds up to – 6:
    Top: 2 + ? + (– 4) = – 6 → ? = – 4
    Bottom: – 3 + ? + 1 = – 6 → ? = – 4
    Left: 2 + ? + (– 3) = – 6 → ? = – 5
    Right: – 4 + ? + 1 = – 6 → ? = – 3
  4. The complete grid is:
2– 4– 4
– 5– 3
– 3– 41

Now rub out three or four of the numbers, write ‘Border sum is – 6’ underneath, and hand it to a friend.

Try This: Rub out exactly four numbers and your puzzle will have one answer. Rub out five or more and your friend will find several answers — both kinds are fun.
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