Number Play | FIO

Question 7

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.

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Solution
Understand the Question
  • A magic square is a square grid where numbers in each row, column, and main diagonal add up to the same constant total, called the magic sum.
  • A standard 3×33 \times 3 magic square uses numbers 11 to 99 with a magic sum of 1515 and center number 55.
  • Since the numbers 2−102 - 10 are obtained by adding 11 to each number from 1−91 - 9, the simplest strategy is to add 11 to every cell of the classic 1−91 - 9 magic square.
  • Because each row, column, and diagonal contains 33 numbers, increasing each number by 11 increases the magic sum by 3×1=33 \times 1 = 3, giving a new magic sum of 15+3=1815 + 3 = 18.

Step 1 · Examine the Standard 1−91 - 9 Magic Square

Consider the standard 3×33 \times 3 magic square using numbers from 11 to 99, with middle number 55 at the center.Diagram 1

816357492\begin{array}{|c|c|c|} \hline 8 & 1 & 6 \\ \hline 3 & 5 & 7 \\ \hline 4 & 9 & 2 \\ \hline \end{array}

Calculating the magic sum (first row):

8+1+6=9+6=15\begin{aligned} 8 + 1 + 6 &= 9 + 6 \\ &= 15 \end{aligned}

Step 2 · Formulate Strategy for 2−102 - 10 Magic Square

The required numbers are 2,3,4,5,6,7,8,9,102, 3, 4, 5, 6, 7, 8, 9, 10.

Each number is exactly 11 more than the corresponding number in the set 11 to 99: nnew=nold+1n_{\text{new}} = n_{\text{old}} + 1

Strategy: Add 11 to every number in the classic 1−91 - 9 magic square to obtain the 2−102 - 10 magic square.

Step 3 · Construct the 2−102 - 10 Magic Square and Find Magic Sum

Original 1−91 - 9 square:

816357492\begin{array}{|c|c|c|} \hline 8 & 1 & 6 \\ \hline 3 & 5 & 7 \\ \hline 4 & 9 & 2 \\ \hline \end{array}

Adding 11 to each number:

  • Row 1: 8+1=98 + 1 = 9, 1+1=21 + 1 = 2, 6+1=76 + 1 = 7
  • Row 2: 3+1=43 + 1 = 4, 5+1=65 + 1 = 6, 7+1=87 + 1 = 8
  • Row 3: 4+1=54 + 1 = 5, 9+1=109 + 1 = 10, 2+1=32 + 1 = 3Diagram 2

The new magic square for numbers 22 to 1010 is:

9274685103\begin{array}{|c|c|c|} \hline 9 & 2 & 7 \\ \hline 4 & 6 & 8 \\ \hline 5 & 10 & 3 \\ \hline \end{array}

Calculating the new magic sum (first row):

9+2+7=11+7=18\begin{aligned} 9 + 2 + 7 &= 11 + 7 \\ &= 18 \end{aligned}

Step 4 · Compare the Two Magic Squares

  • Magic Sum: Increases from 1515 to 1818 (15+3=1815 + 3 = 18) because each of the 33 numbers in every row, column, and diagonal is increased by 11.
  • Center Cell: The center number shifts from 55 (median of 1−91-9) to 6=5+16 = 5 + 1 (median of 2−102-10).
  • Structure: The relative positions and layout of numbers remain identical, with every cell value in the 2−102-10 square being exactly 11 greater than the corresponding cell in the 1−91-9 square.
Answer

(i) Strategy: Add 11 to each number of the classic 1−91 - 9 magic square.

(ii) Magic Square (2−102 - 10):

9274685103\begin{array}{|c|c|c|} \hline 9 & 2 & 7 \\ \hline 4 & 6 & 8 \\ \hline 5 & 10 & 3 \\ \hline \end{array}

(iii) Comparison: The magic sum increases by 33 (from 1515 to 1818) because 11 is added to each of the 33 cells in every row, column, and diagonal. The center number changes from 55 to 66, while the relative positions remain identical.

Common Mistakes
  • Magic Sum Increase: Assuming the magic sum increases by only 11 instead of 33. Since each line contains 33 numbers and each number increases by 11, the total sum increases by 3×1=33 \times 1 = 3.
  • Center Number Error: Forgetting that in a 3×33 \times 3 magic square with consecutive numbers, the center cell must always be the median of the set (66 for 2−102-10).

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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