NCERT Solutions Ganita Prakash Chapter 3 Figure it Out — Supercells

Book page 57 & 58 Updated on2026-09-05

Q1.

Colour or mark the supercells in the table below. 6828 | 670 | 9435 | 3780 | 3708 | 7308 | 8000 | 5583 | 52

Answer

A cell is a supercell if its number is bigger than the numbers in the cells touching it. Compare each number with its left and right neighbours:

Cell	6828	670	9435	3780	3708	7308	8000	5583	52
Neighbours	670	6828, 9435	670, 3780	9435, 3708	3780, 7308	3708, 8000	7308, 5583	8000, 52	5583
Supercell?	Yes	No	Yes	No	No	No	Yes	No	No

Answer: 6828, 9435 and 8000 are the supercells.

Why 52 is not a supercell: it has only one neighbour, 5583, but 52 is smaller than 5583. Compare it with 6828 at the other end — 6828 also has one neighbour (670) but 6828 is bigger, so 6828 is a supercell.

Q2.

Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells. (The row has 9 cells; 5346 is in the 1st cell, 1258 in the 4th coloured cell and 9635 in the 8th cell; the 2nd, 4th and 9th cells are coloured.)

Answer

The coloured cells are the 2nd, 4th and 9th. So exactly those three must beat their neighbours, and no other cell may.

Cell	1	2 ✓	3	4 ✓	5	6	7	8	9 ✓
Number	5346	5678	1000	1258	1100	1200	1300	9635	9800

Check the three coloured cells:

5678 > 5346 and 5678 > 1000 ✓
1258 > 1000 and 1258 > 1100 ✓
9800 > 9635 (its only neighbour) ✓

And no other cell wins: 1300 loses to 9635, 9635 loses to 9800, 1200 loses to 1300, and so on.

Tip: the last cell is coloured and touches only 9635, so the number you write there must be bigger than 9635 — any 4-digit number from 9636 to 9999 will do.

Q3.

Fill the table below such that we get as many supercells as possible. Use numbers between 100 and 1000 without repetitions.

Answer

The table has 9 cells, so put the big numbers in the 1st, 3rd, 5th, 7th and 9th cells and small numbers in between:

Cell	1 ✓	2	3 ✓	4	5 ✓	6	7 ✓	8	9 ✓
Number	500	100	600	200	700	300	800	400	900

500 > 100 ✓ 600 > 100, 200 ✓ 700 > 200, 300 ✓ 800 > 300, 400 ✓ 900 > 400 ✓

All nine numbers are different and each lies between 100 and 1000, so the conditions are satisfied. This gives 5 supercells, the most that 9 cells can have.

Q4.

Out of the 9 numbers, how many supercells are there in the table above? ___________

Answer

5 supercells — the numbers written in the 1st, 3rd, 5th, 7th and 9th cells.

Why 5 is the maximum: two supercells can never be side by side (of two neighbours, only the bigger one can win). So the supercells must be separated by at least one ordinary cell. In a row of 9 cells, the most you can mark with a gap in between is 1, 3, 5, 7, 9 — that is 5 cells.

Q5.

Find out how many supercells are possible for different numbers of cells. Do you notice any pattern? What is the method to fill a given table to get the maximum number of supercells? Explore and share your strategy.

Answer

Fill the biggest numbers in the 1st, 3rd, 5th, … cells (alternate cells starting from the first) and the small numbers in between. Then:

Number of cells	2	3	4	5	6	7	8	9	10
Most supercells	1	2	2	3	3	4	4	5	5

The pattern:

If the number of cells n is even → most supercells = n ÷ 2
If the number of cells n is odd → most supercells = (n + 1) ÷ 2

Why this is the best possible: no two neighbouring cells can both be supercells. So between any two supercells there must be at least one other cell. Marking cells 1, 3, 5, 7 … packs them as tightly as the rule allows.

Math Talk: Try it with 6 cells — 600, 100, 700, 200, 800, 300 gives supercells at cells 1, 3 and 5, that is 6 ÷ 2 = 3 supercells.

Q6.

Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?

Answer

No — it is impossible. There will always be at least one supercell.

Why it happens: look at the largest number you have written anywhere in the table. Since no number is repeated, every neighbour of that cell holds a smaller number. So that cell is greater than all its neighbours — it is a supercell, wherever you place it.

Try This: write any 5 different numbers in a row and hunt for the biggest one. It is a supercell every single time.

Q7.

Will the cell having the largest number in a table always be a supercell? Can the cell having the smallest number in a table be a supercell? Why or why not?

Answer

The largest number is always a supercell. The smallest number can never be a supercell (as long as the table has more than one cell).

Largest: every other number in the table is smaller, so in particular all its neighbours are smaller. Being greater than all its neighbours is exactly what a supercell means.
Smallest: every neighbour holds a bigger number, so the cell loses the comparison at once.

Tip: this is why question 6 has the answer “no” — a table always contains its largest number, and that cell is always a supercell.

Q8.

Fill a table such that the cell having the second largest number is not a supercell.

Answer

Simply write the second largest number right next to the largest number.

Cell	1	2	3	4	5	6	7	8	9
Number	110	120	130	140	150	160	170	190	180

Largest = 190 (cell 8) → supercell
Second largest = 180 (cell 9), and its only neighbour is 190
180 < 190, so 180 is not a supercell ✓

Q9.

Fill a table such that the cell having the second largest number is not a supercell but the second smallest number is a supercell. Is it possible?

Answer

Yes, it is possible. Put the second largest number beside the largest, and put the second smallest number beside the smallest one at the far end.

Cell	1	2	3	4	5	6	7	8	9
Number	120	110	130	140	150	160	170	190	180

Second smallest = 120 (cell 1). Its only neighbour is 110, and 120 > 110 → supercell ✓
Second largest = 180 (cell 9). Its only neighbour is 190, and 180 < 190 → not a supercell ✓

The trick: a cell at the end of the row has only one neighbour, so it is very easy to control. Keep the smallest number at one end and the largest number next to the other end.

Q10.

Make other variations of this puzzle and challenge your classmates.

Answer

Here are some puzzles you can set for your friends — all of them use a row of 9 cells:

Fill the table with 3-digit numbers so that there are exactly 2 supercells.
Fill the table so that the middle cell (5th) is the only supercell.
Fill the table so that both end cells are supercells but no cell in between is.
Can you get 6 supercells out of 9 cells? (Answer: no — and asking your friend to explain why is the real puzzle.)
Use only even numbers between 100 and 1000 and get the maximum number of supercells.

Try This: make the same puzzles for a table with 2 rows and 4 columns. Remember, there the neighbours are the cells to the left, right, above and below.

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