Number Play

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 = _________

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?

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

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?

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

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?

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.

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 ______

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

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?

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?

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

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.

Solve this cryptarithm:

What do the numbers in the figure below tell us?

Remember the children from the Grade 6 textbook of mathematics? Now, they call out numbers using a different rule.

What do you think these numbers mean?

The children rearrange themselves and each one says a number based on the new arrangement.

Context: The children rearrange themselves and each one says a number based on the new arrangement.

Q. Could you figure out what these numbers convey? Observe and try to find out.

Write down the number each child should say based on this rule for the arrangement shown below.

Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it?

Can you figure out which 5 cards add to 30? Is it possible? There are many ways of choosing 5 cards from this collection. Is there a way to find a solution without checking all possibilities? Let us find out.

Context: Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it? Can you figure out which 5 cards add to 30? Is it possible? There are many ways of choosing 5 cards from this collection. Is there a way to find a solution without checking all possibilities? Let us find out.

Q. Add a few even numbers together. What kind of number do you get? Does it matter how many numbers are added?

Context: As we see in the figure, adding any number of even numbers will result in a number which can still be arranged in pairs without any leftovers. In other words, the sum will always be an even number.

Q. Now, add a few odd numbers together. What kind of number do you get? Does it matter how many odd numbers are added?

Context: Can we also think of an odd number as one less than a collection of pairs? This figure shows that the sum of two odd numbers must always be even! This along with the other figures here are more examples of a proof!

Q. What about adding 3 odd numbers? Can the resulting sum be arranged in pairs?

Explore what happens to the sum of: (a) 4 odd numbers (b) 5 odd numbers (c) 6 odd numbers

Two siblings, Martin and Maria, were born exactly one year apart. Today they are celebrating their birthday. Maria exclaims that the sum of their ages is 112. Is this possible? Why or why not?

Context: Small Squares in Grids In a $3 \times 3$ grid, there are 9 small squares, which is an odd number. Meanwhile, in a $3 \times 4$ grid, there are 12 small squares, which is an even number.

Q. Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product?

Find the parity of the number of small squares in these grids:

(a) $27 \times 13$

(b) $42 \times 78$

(c) $135 \times 654$

Context: Consider the algebraic expression: $3n + 4$. For different values of $n$, the expression has different parity:

(a) Come up with an expression that always has even parity. Some examples are: $100p$ and $48w - 2$. Try to find more.

(b) Come up with expressions that always have odd parity.

(c) Come up with other expressions, like $3n + 4$, which could have either odd or even parity.

(d) The expression $6k + 2$ evaluates to $8, 14, 20, \dots$ (for $k = 1, 2, 3, \dots$) — many even numbers are missing.

(e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor 2.

(f) Are there expressions using which we can list all odd numbers?

Context: We saw earlier how to express the $n^{\text{th}}$ term of the sequence of multiples of $4$, where $n$ is the letter-number that denotes a position in the sequence (e.g., first, twenty third, hundred and seventeenth, etc.).

(1) What would be the $n^{\text{th}}$ term for multiples of $2$? Or, what is the $n^{\text{th}}$ even number?

Let us consider odd numbers.

(2) What is the 100th odd number?

To answer this question, consider the following question:

(3) What is the 100th even number?

(4) Write a formula to find the $n^{\text{th}}$ odd number.

Context: Observe this $3 \times 3$ grid. It is filled following a simple rule — use numbers from $1 - 9$ without repeating any of them. There are circled numbers outside the grid. The numbers in the yellow circles are the sums of the corresponding rows and columns.

Q. Fill the grids below based on the rule mentioned above:

Make a couple of questions like this on your own and challenge your peers.

Can 1 occur in a corner position? For example, can it be placed as follows?

Q. If yes, then there should exist three ways of adding 1 with two other numbers to give 15. We have 1 + 5 + 9 = 1 + 6 + 8 = 15. Is any other combination possible?

Q. Similarly, can 9 can be placed in a corner position?

Can you find the other possible positions for 1 and 9?

Now, we have one full row or column of the magic square! Try completing it!

[Hint: First fill the row or columns containing 1 and 9]

Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$, how much more or less than $m$.

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Context: Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$, how much more or less than $m$.

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Q. Once the generalised form is obtained, share your observations with the class.

Chautīsā means 34. Why do you think they called it the Chautīsā Yantra? Every row, column and diagonal in this magic square adds up to 34. Can you find other patterns of four numbers in the square that add up to 34?

How many rhythms are there with 8 beats consisting of short syllables (1 beat) and long syllables (2 beats)? That is, in how many ways can one fill 8 beats with short and long syllables, where a short syllable takes one beat of time and a long syllable takes two beats of time?

Context: A short syllable takes one beat of time and a long syllable takes two beats of time. Some possibilities to fill 8 beats are:

• long long long long
• short short short short short short short short
• short long long short long
• long long short short long

Q. Can you find others?

Context: We can write the number 8 as a sum of 1's and 2's in several ways, for example:

• 8 = 2 + 2 + 2 + 2
• 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
• 8 = 1 + 2 + 2 + 1 + 2
• 8 = 2 + 2 + 1 + 1 + 2

Q. Do you see other ways?

Try writing the number 5 as a sum of 1s and 2s in all possible ways in your notebook! How many ways did you find? (You should find 8 different ways!) Can you figure out the answer without listing down all the possibilities? Can you try it for n = 8?

Context: Thus, there are 8 rhythms having 5 beats! The reason this method works is that every 5-beat rhythm must begin with either a '1+' or a '2+'. If it begins with a '1+', then the remaining numbers must give a 4-beat rhythm, and we can write all those down. If it begins with a 2+, then the remaining number must give a 3-beat rhythm, and we can write all those down. Therefore, the number of 5-beat rhythms is the number of 4-beat rhythms, plus the number of 3-beat rhythms. How many 6-beat rhythms are there? By the same reasoning, it will be the number of 5-beat rhythms plus the number of 4-beat rhythms, i.e., $8 + 5 = 13$. Thus, there are 13 rhythms having 6 beats.

Q. Use the systematic method to write down all 6-beat rhythms, i.e., write 6 as the sum of 1's and 2's in all possible ways. Did you get 13 ways?

Write the next 3 numbers in the sequence: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \underline{\quad}, \underline{\quad}, \underline{\quad}, \dots$

If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Context: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$

What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?

Context: Let us look at one more example. Here $\text{K2}$ means that the number is a 2-digit number having the digit '2' in the units place and 'K' in the tens place. $\text{K2}$ is added to itself to give a 3-digit sum $\text{HMM}$:

$\begin{array}{r} \text{K2} \\ +\ \text{K2} \\ \hline \text{HMM} \end{array}$

Q. What digit should the letter $\text{M}$ correspond to? Both the tens place and the units place of the sum have the same digit. What about $\text{H}$? Can it be 2? Can it be 3?

Context: These types of questions can be interesting and fun to solve! Here are some more questions like this for you to try out. Find out what each letter stands for. Share how you thought about each question with your classmates; you may find some new approaches.

Q. Find out what each letter stands for:

(i) $\begin{array}{r} \text{YY} \\ +\quad \text{Z} \\ \hline \text{ZOO} \end{array}$

(ii) $\begin{array}{r} \text{B5} \\ +\quad \text{3D} \\ \hline \text{ED5} \end{array}$

(iii) $\begin{array}{r} \text{KP} \\ +\quad \text{KP} \\ \hline \text{PRR} \end{array}$

(iv) $\begin{array}{r} \text{C1} \\ +\quad \text{C} \\ \hline \text{1FF} \end{array}$