Question 9
What other operations can be performed on a magic square to yield another magic square?
A magic square is a grid of numbers. Numbers in every row add up to the same total. Numbers in every column also add up to this same total. Numbers along both main diagonals also add up to this total. This total is called the magic constant.
Step 1 — Changing all numbers by adding or subtracting
Let us imagine a magic square. Let its magic total be S. Let us add a constant number k to every number. Consider any row in the square. Each number in that row gets k added to it. If there are n numbers in a row. The row's total sum changes. The new row sum is S plus n times k. This new sum is S + n × k. This new sum will be the same for all rows. It will also be the same for all columns. It will be the same for both main diagonals too. So, the new square is also a magic square. Subtracting a constant is just like adding a negative constant.
Step 2 — Changing all numbers by multiplying or dividing
Let us again start with a magic square. Its magic total is S. Let us multiply every number by a constant k. This constant k must not be zero. Consider any row in the square. Each number in that row gets multiplied by k. The row's total sum will also change. The new row sum is S multiplied by k. This new sum is S × k. This new sum will be the same for all rows. It will also be the same for all columns. It will be the same for both main diagonals too. So, the new square is also a magic square. Dividing by a constant is like multiplying by its reciprocal.
Step 3 — Turning the entire grid
Let us take the whole magic square. We can turn it by 90 degrees. This means rows become columns. Columns become rows. The numbers themselves do not change. Only their positions change. The numbers forming each row, column, and diagonal are unchanged. So, their sums will still be the same. We can turn it by 180 degrees. This is like turning it twice by 90 degrees. We can turn it by 270 degrees. This is like turning it three times by 90 degrees. All these rotations will give a new magic square.
Step 4 — Mirroring the entire grid
Let us imagine holding a mirror to the magic square. We can reflect it horizontally. The left side swaps with the right side. The numbers in each row stay in that row. Their order reverses. So, the row sums remain the same. The numbers in each column also stay the same. Their column position changes. So, the column sums remain the same. The diagonal sums also remain the same. We can reflect it vertically. The top side swaps with the bottom side. We can also reflect it diagonally. Numbers swap across a diagonal line. In all these reflections, the numbers themselves do not change. Only their positions change. The numbers forming each row, column, and diagonal are unchanged. So, the sums will still be the same. This means the new square is still a magic square.
Answer
(i) Adding or subtracting a constant to every number. (ii) Multiplying or dividing every number by a non-zero constant. (iii) Rotating the entire grid by 90, 180, or 270 degrees. (iv) Reflecting the grid horizontally, vertically, or diagonally.
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 = _________
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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
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What other operations can be performed on a magic square to yield another magic square?
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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?
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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 ______
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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: