Question 19
Make a 3 × 3 magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.
A magic square is a grid where all rows, columns, and main diagonals add up to the same number.
Step 1 — Finding the Center Number
Let us draw our 3x3 magic square. We will use letters for each cell.
Let the magic sum be . We are told that . The sum of numbers in each row is . The sum of numbers in each column is . The sum of numbers in both main diagonals is .
Let us write down some of these sums. The sum of the numbers in the middle row is .
The sum of the numbers in the middle column is .
The sum of the numbers in the main diagonal is .
The sum of the numbers in the other diagonal is .
Now, let us add these four sums together. We are adding the middle row, middle column, and both diagonals.
Let us call the sum of all nine numbers in the square . So, . Our equation becomes:
We also know that the sum of all numbers can be found by adding the sums of the three rows. The sum of the first row is . The sum of the second row is . The sum of the third row is . So, the total sum of all numbers is:
Now, we can substitute into our earlier equation.
To find , we subtract from both sides.
To find , we divide by 3.
This tells us that the number in the center cell () of any 3x3 magic square must always be equal to the magic sum divided by 3.
Step 2 — Checking the Conditions
We are given that the magic sum must be 0. We just found that the center number must be . So, let us calculate the value of .
This means the number in the center of our 3x3 magic square must be 0.
However, the problem states a very important condition: "All numbers can not be zero." This means that no cell in the magic square can contain the number 0.
Our calculation shows that the center cell must be 0 for a 3x3 magic square with a magic sum of 0. This creates a contradiction. We cannot have a 0 in the center, but the rules of a 3x3 magic square say it must be 0.
Therefore, it is impossible to create such a magic square.
Answer
It is impossible to create a 3x3 magic square with a magic sum of 0 where all numbers are non-zero. This is because the center element of any 3x3 magic square must be equal to the magic sum divided by 3. For a magic sum of 0, the center element must be 0, which contradicts the condition that no number can be zero.
More questions in FIO
Arrange the stick figure cutouts given at the end of the book or draw a height arrangement such that the sequence reads:
(a) 0, 1, 1, 2, 4, 1, 5
(b) 0, 0, 0, 0, 0, 0, 0
(c) 0, 1, 2, 3, 4, 5, 6
(d) 0, 1, 0, 1, 0, 1, 0
(e) 0, 1, 1, 1, 1, 1, 1
(f) 0, 0, 0, 3, 3, 3, 3
For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.
(a) If a person says '0', then they are the tallest in the group.
(b) If a person is the tallest, then their number is '0'.
(c) The first person's number is '0'.
(d) If a person is not first or last in line (i.e., if they are standing somewhere in between), then they cannot say '0'.
(e) The person who calls out the largest number is the shortest.
(f) What is the largest number possible in a group of 8 people?
Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:
(a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd)
(b) Sum of 2 odd numbers and 3 even numbers
(c) Sum of 5 even numbers
(d) Sum of 8 odd numbers
Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. He calculated the total and got ₹205. Did he make a mistake? If he did, explain why. If he didn't, how many coins of each type could he have?
We know that:
(a) even + even = even
(b) odd + odd = even
(c) even + odd = odd
Similarly, find out the parity for the scenarios below:
(d) even - even = _________
(e) odd - odd = _________
(f) even - odd = _________
(g) odd - even = _________
How many different magic squares can be made using the numbers 1 – 9?
Create a magic square using the numbers 2 – 10. What strategy would you use for this? Compare it with the magic squares made using 1 – 9.
Take a magic square, and
(a) increase each number by 1
(b) double each number
In each case, is the resulting grid also a magic square? How do the magic sums change in each case?
What other operations can be performed on a magic square to yield another magic square?
Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).
Using this generalised form, find a magic square if the centre number is 25.
What is the expression obtained by adding the 3 terms of any row, column or diagonal?
Write the result obtained by— (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form
Create a magic square whose magic sum is 60.
Is it possible to get a magic square by filling nine non-consecutive numbers?
A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?
Liswini has a large old encyclopaedia. When she opened it, several loose pages fell out of it. She counted 50 sheets in total, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?
Here is a 2 × 3 grid. For each row and column, the parity of the sum is written in the circle; 'e' for even and 'o' for odd. Fill the 6 boxes with 3 odd numbers ('o') and 3 even numbers ('e') to satisfy the parity of the row and column sums.
Make a 3 × 3 magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.
Fill in the following blanks with 'odd' or 'even':
(a) Sum of an odd number of even numbers is ______
(b) Sum of an even number of odd numbers is ______
(c) Sum of an even number of even numbers is ______
(d) Sum of an odd number of odd numbers is ______
What is the parity of the sum of the numbers from 1 to 100?
Two consecutive numbers in the Virahāṅka sequence are 987 and 1597. What are the next 2 numbers in the sequence? What are the previous 2 numbers in the sequence?
Angaan wants to climb an 8-step staircase. His playful rule is that he can take either 1 step or 2 steps at a time. For example, one of his paths is 1, 2, 2, 1, 2. In how many different ways can he reach the top?
What is the parity of the 20th term of the Virahāṅka sequence?
Identify the statements that are true.
(a) The expression always gives odd numbers. (b) All even numbers can be expressed as . (c) Both expressions and describe all odd numbers. (d) The expression gives both even and odd numbers.
Solve this cryptarithm: