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

We will find a pattern for the sum of page numbers on any single sheet. Then we will check if 6000 fits this pattern.

Step 1 — Page numbers on one sheet

Let us think about how pages are printed on a sheet of paper in a book. A sheet is a single piece of paper. It has two sides. Each side has a page number. The page numbers on a single sheet are always consecutive. For example, the first sheet has page 1 on one side and page 2 on the other side. The second sheet has page 3 on one side and page 4 on the other side. The smaller page number on any sheet is always an odd number. So, if a sheet is the $k$ -th sheet in the book, its pages are $2k-1$ and $2k$ . Let us find the sum of page numbers on any one sheet. Let the sheet number be $k$ . The page numbers on this sheet are $2k-1$ and $2k$ . Sum of page numbers on one sheet $= (2k-1) + 2k$ $= 4k - 1$ This means the sum of page numbers on any sheet is always one less than a multiple of 4. For example, for the first sheet ( $k=1$ ), the sum is $4(1)-1 = \mathbf{3}$ . For the second sheet ( $k=2$ ), the sum is $4(2)-1 = \mathbf{7}$ .

Step 2 — Total sum for 50 sheets

Liswini counted 50 loose sheets. Let us call the sheet numbers of these loose sheets $k_1, k_2, \dots, k_{50}$ . Each $k_i$ is a whole number, like 1, 2, 3, and so on. The sum of page numbers for each sheet $k_i$ is $4k_i - 1$ . The total sum of page numbers for all 50 sheets is $S$ . $S = (4k_1 - 1) + (4k_2 - 1) + \dots + (4k_{50} - 1)$ We can group the terms with 4 and the terms with -1. $S = 4k_1 + 4k_2 + \dots + 4k_{50} - (1 + 1 + \dots + 1 \text{ (50 times)})$ $S = 4(k_1 + k_2 + \dots + k_{50}) - 50$ Let us call the sum of all these sheet numbers $K_{total}$ . $K_{total} = k_1 + k_2 + \dots + k_{50}$ Since each $k_i$ is a whole number, $K_{total}$ must also be a whole number. So, the total sum $S$ can be written as: $S = 4 \times K_{total} - 50$ This means that if we add 50 to the total sum $S$ , the result must be a multiple of 4.

Step 3 — Check if 6000 is possible

The question asks if the sum of page numbers can be 6000. Let us put $S = 6000$ into our formula. $6000 = 4 \times K_{total} - 50$ We need to find the value of $K_{total}$ . Let us add 50 to both sides of the equation. $6000 + 50 = 4 \times K_{total}$ $6050 = 4 \times K_{total}$ Now, let us divide both sides by 4 to find $K_{total}$ . $K_{total} = \frac{6050}{4}$ $K_{total} = \frac{3025}{2}$

$\boxed{K_{total} = 1512.5}$ We found that $K_{total}$ must be 1512.5. But we know that $K_{total}$ must be a whole number. This is because it is the sum of 50 sheet numbers, and sheet numbers are always whole numbers (like 1st sheet, 2nd sheet, etc.). Since 1512.5 is not a whole number, it is impossible for the sum of page numbers to be 6000.

Answer

No, the sum of the page numbers of the loose sheets cannot be 6000. This is because the sum of page numbers on any single sheet is of the form $4k-1$ . For 50 sheets, the total sum $S$ would be $4 \times K_{total} - 50$ , where $K_{total}$ is the sum of the sheet numbers. If $S=6000$ , then $K_{total}$ would be $1512.5$ , which is not a whole number. Since $K_{total}$ must be a whole number, a sum of 6000 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) even + even = even

(b) odd + odd = even

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?

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 × 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 × 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}