Question 6
Can you fill a supercell table without repeating numbers such that there are no supercells? Why or why not?
We need to understand what a supercell is.
Step 1 — Understanding a Supercell
A supercell is a special box in our table. The number inside this box is bigger than all the numbers in the boxes right next to it. Imagine a box with the number 10 inside. If all its neighbors have numbers like 1, 2, 3, 4, then 10 is a supercell.
Step 2 — Finding the Biggest Number
We are filling the table using numbers that are all different. This means we cannot use the same number twice. For example, we might use numbers like 1, 2, 3, 4, 5, 6, 7, 8, 9. When all numbers are different, there will always be one number that is the biggest of all. In our example, 9 is the biggest number.
Step 3 — The Biggest Number is Always a Supercell
Let us imagine we have filled all the boxes in the table. There is one box that holds the biggest number. Let us call this box "The Biggest Box". All other numbers in the table are smaller than the number in "The Biggest Box". So, any box that is next to "The Biggest Box" must have a smaller number. This means the number in "The Biggest Box" is bigger than all its neighbors. So, "The Biggest Box" itself becomes a supercell.
Step 4 — Why No Supercells is Impossible
Since we always have a biggest number when we use different numbers, that biggest number will always create a supercell. It does not matter how we arrange the numbers. Whether we put the biggest number first, last, or somewhere in the middle, it will still be the biggest. Its box will always be a supercell. So, it is not possible to fill the table without repeating numbers and have no supercells. There will always be at least one supercell.
Answer
No, it is not possible to fill a supercell table without repeating numbers such that there are no supercells.
This is because when we use different numbers, there will always be one number that is the biggest. The box containing this biggest number will always be a supercell. Its value will be greater than all its neighbors, no matter how the numbers are arranged.
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?
Make other variations of this puzzle and challenge your classmates.
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.
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?
Find out the digit sums of all the numbers from 40 to 70. Share your observations with the class.
Calculate the digit sums of 3-digit numbers whose digits are consecutive (for example, 345). Do you see a pattern? Will this pattern continue?
Pratibha uses the digits '4', '7', '3' and '2', and makes the smallest and largest 4-digit numbers with them: and . The difference between these two numbers is . The sum of these two numbers is . Choose 4-digits to make:
a. the difference between the largest and smallest numbers greater than .
b. the difference between the largest and smallest numbers less than .
c. the sum of the largest and smallest numbers greater than .
d. the sum of the largest and smallest numbers less than .
What is the sum of the smallest and largest 5-digit palindrome? What is their difference?
The time now is 10:01. How many minutes until the clock shows the next palindromic time? What about the one after that?
How many rounds does the number 5683 take to reach the Kaprekar constant?
Write an example for each of the scenarios shown in the diagram whenever possible.
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
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
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
Name some objects around you that are: a. a few thousand in number b. more than ten thousand in number
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.
How many rounds does your year of birth take to reach the Kaprekar constant?
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?
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.
Estimate the number of liters a mug, a bucket and an overhead tank can hold.
Write one 5-digit number and two 3-digit numbers such that their sum is 18,670.
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.
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?
Check if the Collatz Conjecture holds for the starting number 100.
Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?