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Question 9

What other operations can be performed on a magic square to yield another magic square?

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Solution
Understand the Question
  • A magic square of order n×nn \times n is a grid of numbers where the sum of every row, column, and both main diagonals equals the same value, known as the magic constant (SS).
  • To produce a new magic square, operations must preserve the equal-sum property across all rows, columns, and diagonals.
  • These operations fall into two categories:
    • Arithmetic transformations: Uniformly modifying every entry (adding, subtracting, multiplying, or dividing by a constant).
    • Geometric transformations: Rearranging grid positions through rotations and reflections (symmetries of the square).

Step 1 · Adding or Subtracting a Constant

Diagram 1

Let the original magic sum of an n×nn \times n magic square be SS.

If a constant kk is added to every entry in the grid, each row, column, and diagonal contains nn elements that are each increased by kk.

New sum=S+n×k\text{New sum} = S + n \times k

Since this new sum is identical for all rows, columns, and diagonals, the resulting grid is also a magic square. Subtracting a constant kk corresponds to adding −k-k and similarly yields a magic square with sum S−n×kS - n \times k.

Step 2 · Multiplying or Dividing by a Non-zero Constant

Diagram 2

Let the original magic sum be SS.

If every entry is multiplied by a non-zero constant kk, the sum of each row, column, and diagonal is multiplied by kk:

New sum=S×k\text{New sum} = S \times k

Since the sum remains uniform across all rows, columns, and diagonals, the new grid is a magic square. Dividing by a non-zero constant kk is equivalent to multiplying by 1k\dfrac{1}{k}, resulting in a new sum of Sk\dfrac{S}{k}.

Step 3 · Rotating the Grid

Diagram 3

Rotating the entire square by 90∘90^\circ, 180∘180^\circ, or 270∘270^\circ:

  • Rows transform into columns (and vice versa).
  • The main diagonals map to each other.
  • The numbers in each line are preserved, keeping the magic sum SS unchanged.

Step 4 · Reflecting the Grid (Mirroring)

Diagram 4

Reflecting the magic square across its symmetry axes:

  • Horizontal reflection: Swaps left and right columns.
  • Vertical reflection: Swaps top and bottom rows.
  • Diagonal reflection: Transposes elements across the main or secondary diagonals.

In each reflection, the set of numbers comprising every row, column, and diagonal remains unchanged, preserving the sum SS.

Answer

The following operations yield a new magic square:

  1. Adding or subtracting a constant to every number.
  2. Multiplying or dividing every number by a non-zero constant.
  3. Rotating the entire grid by 90∘90^\circ, 180∘180^\circ, or 270∘270^\circ.
  4. Reflecting the grid horizontally, vertically, or diagonally.
Common Mistakes
  • Multiplying by Zero: Multiplying all entries by 00 results in a trivial square where all entries are equal to 00, losing the distinct positive integer property of a standard magic square.
  • Non-Uniform Operations: Applying an operation (such as adding or multiplying a number) to only a single row or column destroys the equal-sum balance across all directions; it must be applied to every cell in the grid.
  • Arbitrary Swapping: Swapping arbitrary individual rows or columns (other than symmetric reflections) disrupts the diagonal sums.

More questions in FIO

Q1

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

Q2

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?

Q3

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

Q4

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?

Q5

We know that:

(a) even+even=even\text{even} + \text{even} = \text{even}

(b) odd+odd=even\text{odd} + \text{odd} = \text{even}

(c) even+odd=odd\text{even} + \text{odd} = \text{odd}

Similarly, find out the parity for the scenarios below:

(d) even−even=_________\text{even} - \text{even} = \text{\_\_\_\_\_\_\_\_\_}

(e) odd−odd=_________\text{odd} - \text{odd} = \text{\_\_\_\_\_\_\_\_\_}

(f) even−odd=_________\text{even} - \text{odd} = \text{\_\_\_\_\_\_\_\_\_}

(g) odd−even=_________\text{odd} - \text{even} = \text{\_\_\_\_\_\_\_\_\_}

Q6

How many different magic squares can be made using the numbers 1−91 - 9?

Q7

Create a magic square using the numbers 2−102 - 10. What strategy would you use for this? Compare it with the magic squares made using 1−91 - 9.

Q8

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?

Q9

What other operations can be performed on a magic square to yield another magic square?

Q10

Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).

Q11

Using this generalised form, find a magic square if the centre number is 25.

Q12

What is the expression obtained by adding the 3 terms of any row, column or diagonal?

Q13

Write the result obtained by— (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form

Q14

Create a magic square whose magic sum is 60.

Q15

Is it possible to get a magic square by filling nine non-consecutive numbers?

Q16

A light bulb is ON. Dorjee toggles its switch 77 times. Will the bulb be on or off? Why?

Q17

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?

Q18

Here is a 2×32 \times 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.

Q19

Make a 3×33 \times 3 magic square with 00 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.

Q20

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 ______

Q21

What is the parity of the sum of the numbers from 11 to 100100?

Q22

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?

Q23

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?

Q24

What is the parity of the 20th term of the Virahāṅka sequence?

Q25

Identify the statements that are true.

(a) The expression 4m−14m - 1 always gives odd numbers. (b) All even numbers can be expressed as 6j−46j - 4. (c) Both expressions 2p+12p + 1 and 2q−12q - 1 describe all odd numbers. (d) The expression 2f+32f + 3 gives both even and odd numbers.

Q26

Solve this cryptarithm:

UT+TATAT\begin{array}{rcc} & \text{U} & \text{T} \\ + & \text{T} & \text{A} \\ \hline \text{T} & \text{A} & \text{T} \end{array}
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