Number Play | FIO

Question 5

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.

Solution

Understand the Question

A supercell is a cell in a grid whose value is strictly greater than all of its adjacent neighbouring cells. If a cell has no neighbours, it is considered a supercell.
Two adjacent cells cannot both be supercells because one must be less than or equal to the other.
To find the maximum number of supercells, we place larger numbers such that they do not share borders, alternating them with smaller numbers (like a checkerboard pattern).

Step 1 · Explore Different Grid Sizes

Case 1: One cell Diagram 1

[ 5 ]

The cell has no neighbours, so it is a supercell.
Maximum supercells: $1$

Case 2: Two cells ( $1 \times 2$ ) Diagram 2

[ 5 | 3 ]

$5 > 3$ , so cell $5$ is a supercell; cell $3$ is not.
Maximum supercells: $1$

Case 3: Three cells ( $1 \times 3$ ) Diagram 3

[ 5 | 1 | 6 ]

$5 > 1$ and $6 > 1$ , so both $5$ and $6$ are supercells.
Maximum supercells: $2$

Case 4: Four cells ( $2 \times 2$ ) Diagram 4

[ 5 | 1 ] [ 2 | 6 ]

$5 > 1, 2$ and $6 > 1, 2$ , so cells $5$ and $6$ are supercells.
Maximum supercells: $2$

Case 5: Five cells ( $1 \times 5$ ) Diagram 5

[ 5 | 1 | 6 | 2 | 7 ]

Cells $5$ , $6$ , and $7$ are each greater than their adjacent neighbours.
Maximum supercells: $3$

Case 6: Six cells ( $2 \times 3$ ) Diagram 6

[ 7 | 1 | 8 ] [ 2 | 9 | 3 ]

Cells $7$ , $8$ , and $9$ are supercells.
Maximum supercells: $3$

Step 2 · Identify the Pattern

Summarising the maximum supercells for each total number of cells:

\begin{array}{|c|c|} \hline \text{Number of cells } (n) & \text{Maximum supercells} \\ \hline 1 & 1 \\ \hline 2 & 1 \\ \hline 3 & 2 \\ \hline 4 & 2 \\ \hline 5 & 3 \\ \hline 6 & 3 \\ \hline \end{array}

For an even number of cells $n$ , the maximum supercells is $\dfrac{n}{2}$ .
For an odd number of cells $n$ , the maximum supercells is $\dfrac{n + 1}{2}$ .

Step 3 · Strategy for Maximum Supercells

To maximise the number of supercells:

Checkerboard Arrangement: Alternate large numbers and small numbers so that no two large numbers are adjacent.
Surround with Smaller Values: Place the largest numbers in positions where they are completely surrounded by smaller numbers.

For example, in a $3 \times 3$ grid:

[ 1 | 2 | 3 ] [ 4 | 9 | 5 ] [ 6 | 7 | 8 ]

The central number $9$ is strictly greater than all $8$ of its neighbours, making it a supercell.

Answer

Number of Supercells: For $n$ cells, the maximum number of supercells is $\left\lceil \dfrac{n}{2} \right\rceil$ .

Pattern: The maximum number of supercells is half the total cells (for even $n$ ) or one more than half (for odd $n$ ).
Strategy: Use an alternating checkerboard pattern placing large numbers surrounded by smaller numbers so that adjacent cells do not compete with each other.

Common Mistakes

Adjacent Supercells: Assuming two adjacent cells can both be supercells. Since one number must be smaller than the other, at most one of two adjacent cells can be a supercell.
Ignoring Diagonal/All Neighbours: Forgetting that in a 2D grid, a cell must be compared with all adjacent neighbours surrounding it.

More questions in FIO

Colour or mark the supercells in the table below.

Fill the table below with only 4-digit numbers such that the supercells are exactly the coloured cells.

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

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

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.

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

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?

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

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?

Q10

Make other variations of this puzzle and challenge your classmates.

Q11

Identify the numbers marked on the number lines below, and label the remaining positions.

Put a circle around the smallest number and a box around the largest number in each of the sequences above.

Q12

Digit sum 14 a. Write other numbers whose digits add up to 14. b. What is the smallest number whose digit sum is 14? c. What is the largest 5-digit whose digit sum is 14? d. How big a number can you form having the digit sum of 14? Can you make an even bigger number?

Q13

Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.

Q14

Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?

Q15

Pratibha uses the digits '4', '7', '3' and '2', and makes the smallest and largest 4-digit numbers with them: $2347$ and $7432$ . The difference between these two numbers is $7432 - 2347 = 5085$ . The sum of these two numbers is $9779$ . Choose 4-digits to make:

a. the difference between the largest and smallest numbers greater than $5085$ .

b. the difference between the largest and smallest numbers less than $5085$ .

c. the sum of the largest and smallest numbers greater than $9779$ .

d. the sum of the largest and smallest numbers less than $9779$ .

Q16

What is the sum of the smallest and largest 5-digit palindrome? What is their difference?

Q17

The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?

Q18

How many rounds does the number 5683 take to reach the Kaprekar constant?

Q19

Write an example for each of the scenarios shown in the diagram whenever possible.

Q20

Always, Sometimes, Never?

Below are some statements. Think, explore and find out if each of the statement is 'Always true', 'Only sometimes true' or 'Never true'. Why do you think so? Write your reasoning and discuss this with the class.

a. 5-digit number + 5-digit number gives a 5-digit number

b. 4-digit number + 2-digit number gives a 4-digit number

c. 4-digit number + 2-digit number gives a 6-digit number

d. 5-digit number – 5-digit number gives a 5-digit number

e. 5-digit number – 2-digit number gives a 3-digit number

Q21

Steps you would take to walk: a. From the place you are sitting to the classroom door b. Across the school ground from start to end c. From your classroom door to the school gate d. From your school to your home

Q22

Number of times you blink your eyes or number of breaths you take: a. In a minute b. In an hour c. In a day

Q23

Name some objects around you that are: a. a few thousand in number b. more than ten thousand in number

Q24

There is only one supercell (number greater than all its neighbours) in this grid. If you exchange two digits of one of the numbers, there will be 4 supercells. Figure out which digits to swap.

Q25

How many rounds does your year of birth take to reach the Kaprekar constant?

Q26

We are the group of 5-digit numbers between 35,000 and 75,000 such that all of our digits are odd. Who is the largest number in our group? Who is the smallest number in our group? Who among us is the closest to 50,000?

Q27

Estimate the number of holidays you get in a year including weekends, festivals and vacation. Then, try to get an exact number and see how close your estimate is.

Q28

Estimate the number of liters a mug, a bucket and an overhead tank can hold.

Q29

Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.

Q30

Choose a number between 210 and 390. Create a number pattern similar to those shown in Section 3.9 that will sum up to this number.

Q31

Recall the sequence of Powers of 2 from Chapter 1, Table 1. Why is the Collatz conjecture correct for all the starting numbers in this sequence?

Q32

Check if the Collatz Conjecture holds for the starting number 100.

Q33

Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?