NCERT Solutions Ganita Prakash Chapter 10 An amazing grid of numbers! — Figure it Out

Book page 265 Updated on2026-09-05

Q1.

Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!

Answer

The sum is always – 1, no matter which numbers you circle. Here are three fresh games played on the grid of page 264 (3, 4, 0, 9 / – 2, – 1, – 5, 4 / 1, 2, – 2, 7 / – 7, – 6, – 10, – 1):

Game	Circled numbers (one from each row and each column)	Sum
The book's game	(– 1) + 9 + (– 7) + (– 2)	– 1
Game 2	3 + 4 + 2 + (– 10)	– 1
Game 3	0 + (– 2) + 7 + (– 6)	– 1
Game 4	9 + (– 5) + 1 + (– 6)	– 1

In Game 2, for example, 3 comes from row 1 column 1, 4 from row 2 column 4, 2 from row 3 column 2 and – 10 from row 4 column 3 — the striking-out rule makes sure you use every row once and every column once.

3 + 4 + 2 + (– 10) = – 1 0 + (– 2) + 7 + (– 6) = – 1 9 + (– 5) + 1 + (– 6) = – 1

Answer: the sum is – 1 every single time. It is never different, however you play.

Q2.

Play the same game with the grids below. What answer did you get?

Answer

First grid (7, 10, 13, 16 / – 2, 1, 4, 7 / – 11, – 8, – 5, – 2 / – 20, – 7, – 14, – 11): the answer is – 8 every time.

16 + 4 + (– 8) + (– 20) = – 8
10 + 7 + (– 11) + (– 14) = – 8
16 + 1 + (– 11) + (– 14) = – 8

Second grid (– 11, – 10, – 9, – 8 / – 7, – 6, – 5, – 4 / – 3, – 2, – 1, 0 / 1, 2, 3, 4):

(– 11) + (– 6) + (– 1) + 4 = – 14 (– 8) + (– 5) + (– 2) + 1 = – 14 (– 9) + (– 4) + (– 3) + 2 = – 14

Answer: the first grid always gives – 8 and the second grid always gives – 14.

Note on a printing slip: in the first grid the last row is printed as – 20, – 7, – 14, – 11. To fit the pattern of the grid (each row going up in steps of 3) that second entry should be – 17. If you happen to circle the printed – 7, your total will come out + 2 instead of – 8. With – 17 in that cell, every game gives – 8.

Q3.

What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

Answer

The magic is in the arrangement — the numbers are built from a row list and a column list.

Look at the second grid. Write 1, 2, 3, 4 above the columns and – 11, – 7, – 3, 1 beside the rows. Every entry is simply row number + column number — for instance the entry in row 2, column 3 is – 7 + 2 = – 5. ✔

+	0	1	2	3
– 11	– 11	– 10	– 9	– 8
– 7	– 7	– 6	– 5	– 4
– 3	– 3	– 2	– 1	0
1	1	2	3	4

Why the sum can never change: the striking-out rule forces you to take exactly one number from each row and one from each column. So the four circled numbers use every row number once and every column number once:

Sum = (– 11 – 7 – 3 + 1) + (0 + 1 + 2 + 3) = (– 20) + 6 = – 14

It does not matter which cells you pick — the same eight numbers always get added.

Make your own: choose any four row numbers and any four column numbers, add them in a grid, and the answer of your game will always be (sum of the rows) + (sum of the columns). For example rows 5, 0, – 2, 10 and columns 1, – 1, 4, 0 give a grid whose game always totals 13 + 4 = 17.

Try This: Make a 4 × 4 grid whose game always gives 0. Just pick row numbers and column numbers that together add up to 0 — say rows 3, – 1, 2, – 4 and columns 5, – 5, 1, – 1.

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