Number Play | FIO

Question 24

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.

Solution

Understand the Question

A supercell is a cell whose value is greater than all its adjacent neighbours (top, bottom, left, and right).
In the given $3 \times 3$ grid, the center cell $62,871$ is initially the only supercell because it is greater than all surrounding numbers.
Notice that each of the four edge cells ( $39,344$ , $23,609$ , $45,306$ , and $50,319$ ) is already strictly greater than its neighboring corner cells.
The only neighbor preventing all four edge cells from being supercells is the large center cell $62,871$ . By swapping digits in $62,871$ to make it smaller than all four edge cells, all four will simultaneously become supercells.

Step 1 · Analyze the Original Grid

Consider the given $3 \times 3$ grid: Question diagram

\begin{array}{|c|c|c|} \hline 16,200 & 39,344 & 29,765 \\ \hline 23,609 & 62,871 & 45,306 \\ \hline 19,381 & 50,319 & 38,408 \\ \hline \end{array}

Comparing each edge cell with its adjacent corner neighbours:

Top-middle ( $39,344$ ): $39,344 > 16,200$ and $39,344 > 29,765$
Middle-left ( $23,609$ ): $23,609 > 16,200$ and $23,609 > 19,381$
Middle-right ( $45,306$ ): $45,306 > 29,765$ and $45,306 > 38,408$
Bottom-middle ( $50,319$ ): $50,319 > 19,381$ and $50,319 > 38,408$

Currently, none of these four are supercells only because each is adjacent to the center cell $62,871$ , which is larger than all of them.

Step 2 · Swap Digits in the Center Cell

To make all four edge cells supercells at the same time, the center cell must become smaller than the smallest edge cell ( $23,609$ ).

Swap the digits $6$ and $1$ in $62,871$ : $62,871 \xrightarrow{\text{swap } 6 \text{ and } 1} 12,876$

The new grid is:

\begin{array}{|c|c|c|} \hline 16,200 & \mathbf{39,344} & 29,765 \\ \hline \mathbf{23,609} & 12,876 & \mathbf{45,306} \\ \hline 19,381 & \mathbf{50,319} & 38,408 \\ \hline \end{array}

Now verify the four supercells:

$39,344 > 16,200,\, 29,765,\, 12,876$
$23,609 > 16,200,\, 19,381,\, 12,876$
$45,306 > 29,765,\, 38,408,\, 12,876$
$50,319 > 19,381,\, 38,408,\, 12,876$

There are now exactly $4$ supercells.

Answer

Swap the digits $6$ and $1$ in the center number $62,871$ (to make it $12,876$ ).

Common Mistakes

Trying to increase corner cells: Trying to swap digits inside corner cells to make them larger than $62,871$ is impossible (e.g., the largest from $16,200$ is only $61,200$ ).
Incorrect digit swap: Swapping $6$ and $2$ in $62,871$ gives $26,871$ , which is still larger than $23,609$ , so the left cell would not become a supercell. Swapping $6$ and $1$ gives $12,876 < 23,609$ , successfully unlocking all $4$ supercells.

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?