Number Play | FIO

Question 17

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?

Solution

Understand the Question

Each physical sheet has two sides with consecutive page numbers: an odd page on the front ( $2k - 1$ ) and an even page on the back ( $2k$ ), where $k$ represents the sheet number.
The sum of page numbers on any $k$ -th sheet is $(2k - 1) + 2k = 4k - 1$ .
For $50$ loose sheets, the total sum must equal $4K_{\text{total}} - 50$ , where $K_{\text{total}}$ is the sum of the sheet numbers (which must be a whole number).
We can set the total sum to $6000$ and check if $K_{\text{total}}$ results in an integer.

Step 1 · Find Sum of Page Numbers on One Sheet

For any $k$ -th sheet in a book, the front page is odd ( $2k - 1$ ) and the back page is even ( $2k$ ).

\begin{aligned} \text{Sum of page numbers on one sheet} &= (2k - 1) + 2k \\[0.6em] &= 4k - 1 \end{aligned}

For example:

Sheet $1$ ( $k = 1$ ): $4(1) - 1 = 3$ (pages $1$ and $2$ )
Sheet $2$ ( $k = 2$ ): $4(2) - 1 = 7$ (pages $3$ and $4$ )

Step 2 · Find Total Sum Expression for 50 Sheets

Let the sheet numbers of the $50$ loose sheets be $k_1, k_2, \dots, k_{50}$ , where each $k_i$ is a positive integer.

The total sum $S$ of all page numbers is:

\begin{aligned} S &= (4k_1 - 1) + (4k_2 - 1) + \dots + (4k_{50} - 1) \\[0.6em] &= 4(k_1 + k_2 + \dots + k_{50}) - (1 + 1 + \dots + 1 \text{ [50 times]}) \\[0.6em] &= 4(k_1 + k_2 + \dots + k_{50}) - 50 \end{aligned}

Let $K_{\text{total}} = k_1 + k_2 + \dots + k_{50}$ . Since each $k_i$ is an integer, $K_{\text{total}}$ must also be an integer: $S = 4K_{\text{total}} - 50$

Step 3 · Check if Total Sum Can Be 6000

Substitute $S = 6000$ :

\begin{aligned} 6000 &= 4K_{\text{total}} - 50 \\[0.6em] 6000 + 50 &= 4K_{\text{total}} \\[0.6em] 6050 &= 4K_{\text{total}} \\[0.6em] K_{\text{total}} &= \dfrac{6050}{4} \\[0.6em] &= \dfrac{3025}{2} = 1512.5 \end{aligned}

Since $K_{\text{total}} = 1512.5$ is not a whole number, it is impossible for the sum to be $6000$ .

Answer

No, the sum of the page numbers cannot be $6000$ because $S + 50 = 6050$ is not divisible by $4$ .

Common Mistakes

Treating Pages Independently: Assuming loose pages can have any arbitrary combination of even and odd numbers, ignoring the fact that both sides of a single physical sheet are linked as consecutive integers ( $2k-1$ and $2k$ ).
Odd/Even Parity Shortcut: Noting that the sum of each sheet is odd ( $4k - 1$ ), the sum of $50$ odd numbers must be even. While $6000$ is even, checking divisibility modulo $4$ ( $S \equiv -50 \equiv 2 \pmod 4$ , whereas $6000 \equiv 0 \pmod 4$ ) proves it is impossible.

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

(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'.

(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

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

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

Similarly, find out the parity for the scenarios below:

(d) $\text{even} - \text{even} = \text{\_\_\_\_\_\_\_\_\_}$

(e) $\text{odd} - \text{odd} = \text{\_\_\_\_\_\_\_\_\_}$

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

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

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?

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

Q18

Here is a $2 \times 3$ 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 $3 \times 3$ magic square with $0$ 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 ______

(d) Sum of an odd number of odd numbers is ______

Q21

What is the parity of the sum of the numbers from $1$ to $100$ ?

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

Q26

Solve this cryptarithm:

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