Class 7MathematicsChapter 6NCERT • Ganita Prakash

Number Play

Explore number patterns through parity, puzzles, magic squares, sequences and cryptarithms.

Chapter Snapshot

Number patterns can reveal information without full calculation.
Parity tells whether a number is odd or even.
Odd and even numbers follow predictable operation rules.
Magic squares have equal row, column and diagonal sums.
The Virahanka sequence grows by adding the previous two terms.
Cryptarithms replace digits with letters and are solved by logic.

What You Will Learn

Explore patterns and relationships among numbers
Understand odd and even numbers through parity
Predict whether sums and products are odd or even
Use simple algebra to explain parity patterns
Understand the structure of 3 × 3 magic squares
Recognise and extend the Virahanka number pattern
Solve simple cryptarithms using place value and logical reasoning
Explain whether a number statement is always, sometimes or never true

Exam PriorityBoardsVery HighFoundationHigh

Chapter Overview

Number Play is about looking beyond routine calculation. We observe patterns, make predictions and explain why they work.

1. Parity

An even number can be written as:

2n

An odd number can be written as:

2n+1

for a whole number $n$ .

2. Odd and Even Rules

Parity of Sums and Differences

Operation	Result
even + even	even
odd + odd	even
even + odd	odd
even − even	even
odd − odd	even
even − odd	odd
odd − even	odd

For multiplication:

even × any whole number → even;
odd × odd → odd.

Quick Check

Without multiplying, is 135 × 654 odd or even?

Show answer

Even, because one factor (654) is even.

3. Parity of Expressions

We can often predict the parity of an algebraic expression.

For example:

2n+4

is always even because both terms are even.

Similarly:

2n+1

is always odd.

Can the Parity Change?

Consider:

3n+4

If $n=2$ :

3(2)+4=10

which is even.

If $n=3$ :

3(3)+4=13

which is odd.

So this expression can be even or odd, depending on $n$ .

4. Magic Squares

A famous $3\times3$ magic square using 1 to 9 is:

| | | | |---|---|---| | 8 | 1 | 6 | | 3 | 5 | 7 | | 4 | 9 | 2 |

Every row, column and diagonal adds to:

15

Changing a Magic Square

If the same number is added to every entry, the new grid is still a magic square.

If every entry is multiplied by the same number, it also remains a magic square.

Quick Check

If 1 is added to every entry of a 3 × 3 magic square with magic sum 15, what is the new magic sum?

Show answer

18, because each row contains three entries, so its sum increases by 3.

5. Virahanka Numbers

A beautiful sequence begins:

1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ldots

From the third term onward, each term is the sum of the previous two.

Sequence Rule

\text{next term}=\text{previous term}+\text{term before it}

For example:

5+8=13

and:

8+13=21

6. Cryptarithms

To solve one:

begin from the ones column;
consider possible carries;
use place value carefully;
reject any choice that creates a contradiction.

Simple Letter Puzzle

Suppose:

A+A=10

Then:

2A=10

so:

A=5

Real cryptarithms often involve several letters and carrying between columns.

7. Always, Sometimes or Never?

Many number statements should be tested as:

Always True
Sometimes True
Never True

Reasoning with Parity

Statement: “The sum of two odd numbers is even.”

Write odd numbers as:

2a+1

and:

2b+1

Their sum is:

2a+1+2b+1=2(a+b+1)

This is divisible by 2.

Therefore the statement is always true.

8. Understanding Parity More Deeply

Parity tells whether a whole number is odd or even. Every even number can be written as $2n$ for some whole number $n$ . Every odd number can be written as $2n+1$ .

This simple representation explains many patterns.

For example, the sum of two odd numbers is:

(2a+1)+(2b+1)=2a+2b+2

=2(a+b+1)

The result is divisible by 2, so it is even.

Similarly, an odd number multiplied by an odd number has the form:

(2a+1)(2b+1)

which simplifies to an odd number.

Parity Rules

Operation	Result
even + even	even
odd + odd	even
even + odd	odd
even × any whole number	even
odd × odd	odd

9. Why a 3 × 3 Magic Square Is Special

In a $3 \times 3$ magic square, every row, column and main diagonal has the same sum, called the magic sum.

A useful feature is that the centre entry plays a special role. In a normal 3 × 3 magic square made from nine consecutive numbers, opposite cells balance around the centre.

Missing Entry in a Magic Row

Question

The magic sum is 24. A row contains 7, 11 and x. Find x.

Solution

7+11+x=24

10. Virahanka Numbers and Recursive Patterns

A recursive pattern uses earlier terms to create later terms. In the Virahanka-type sequence, each new term is obtained by adding the previous two terms.

For example:

1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ldots

because:

1+1=2,\quad 1+2=3,\quad 2+3=5

The important idea is not only to continue the sequence but to explain how the rule generates every next term.

11. Cryptarithms and Place-value Logic

In a cryptarithm, digits are replaced by symbols or letters. The same symbol must represent the same digit each time, and different symbols usually represent different digits.

The best strategy is to work column by column from the right, just as in ordinary addition or subtraction. Carries are especially useful clues.

12. Always, Sometimes or Never

Number play often asks whether a statement is always true, sometimes true or never true.

Example: The sum of two odd numbers is even. This is always true.

Example: The sum of two numbers is odd. This is sometimes true because it happens when one number is odd and the other is even.

Example: The product of an even number and any whole number is odd. This is never true.

13. Worked Understanding and Pattern Strategies

Building Parity Rules from Examples

Try a few additions:

4+8=12,\quad 6+10=16

Both are even + even, and both results are even. Now test odd + odd:

3+5=8,\quad 7+9=16

Again the result is even. But even + odd gives odd:

6+5=11

Examples suggest a rule; algebra explains why the rule always works.

Parity Without Calculating the Whole Number

Is $127+468+913$ odd or even?

127 is odd, 468 is even and 913 is odd.

Odd + even = odd, and odd + odd = even.

Therefore the total is even, even without finding the exact sum.

Parity of Products

A product is even if at least one factor is even. This is because an even factor already contains a factor of 2.

A product is odd only when every factor is odd.

For example:

17\times25\times9

is odd, but:

17\times25\times10

is even.

Magic-square Checking

When a square is claimed to be magic, do not check only the rows. Check:

all rows;
all columns;
both main diagonals.

If even one sum differs, the arrangement is not a magic square.

Magic Sum

If each row of a 3 × 3 magic square sums to 27, the total of all nine entries is:

3\times27=81

because the three rows together contain every entry exactly once.

Pattern Rules and Position

A pattern such as:

2,5,8,11,14,\ldots

adds 3 each time. The $n$ th term can be written as:

3n-1

Check:

n=1\Rightarrow3(1)-1=2

n=4\Rightarrow3(4)-1=11

A position rule is useful because it finds a distant term directly without listing all earlier terms.

Cryptarithm Checklist

When solving a letter-digit puzzle:

write possible carries above each column;
start with the most restricted column;
reject any digit already used by another letter;
remember that a leading letter cannot be zero;
verify the completed addition or subtraction.

Quick Check

Is the product of 37 odd numbers always odd or even?

Show answer

Always odd, because a product of only odd factors is odd.

Number play is less about fast arithmetic and more about noticing structure, testing conjectures and giving reasons.

Important Terms

Key Terms

ParityEven NumberOdd NumberMagic SquareMagic SumVirahanka SequenceCryptarithmPattern

Common Mistakes

Exam Focus

Important Exam Topics

parity rules;
parity of products and algebraic expressions;
always/sometimes/never reasoning;
properties of magic squares;
continuing the Virahanka sequence;
simple cryptarithms and number puzzles.

Quick Revision

Parity means odd or even.
Even numbers can be written as $2n$ .
Odd numbers can be written as $2n+1$ .
even + even = even.
odd + odd = even.
even + odd = odd.
A product is even if at least one factor is even.
A magic square has equal row, column and diagonal sums.
Virahanka numbers follow the rule “add the previous two terms”.
Cryptarithms use letters in place of digits.

Practice Questions

Q1MCQEasyClass 7

2n

View Solution

2n+1

Q2Short AnswerEasyClass 7

What is the parity of the sum of five odd numbers?

View Solution

Odd + odd = even. Four odd numbers make an even sum; adding the fifth odd number makes the result odd .

Q3NumericalModerateClass 7

1,\ 2,\ 3,\ 5,\ 8,\ 13,\ \_\_,\ \_\_

View Solution

8+13=21

Q4Short AnswerModerateFoundation

Why is the product of an even number and an odd number always even?

View Solution

An even number contains a factor 2. Multiplying it by any whole number keeps that factor 2, so the product remains even.

Additional Practice

Q5Short AnswerEasyClass 7

Is the sum of three odd numbers odd or even?

View Solution

Odd + odd = even, and even + odd = odd. Therefore the sum is odd .

Q6Short AnswerModerateClass 7

Explain why the square of an even number is even.

View Solution

2n

Q7NumericalModerateClass 7

1,1,2,3,5,8

View Solution

13,21,34,55

Q8Short AnswerHardFoundation

Is “the sum of two whole numbers is even” always, sometimes or never true?

View Solution

Sometimes true. It is even when both numbers have the same parity, but odd when one is even and the other is odd.

Chapter Summary

Number play develops reasoning through patterns and puzzles.
Parity gives quick information about odd and even results.
Algebra can describe all even and odd numbers.
Magic squares show strong numerical structure and symmetry.
The Virahanka sequence is built by adding previous terms.
Cryptarithms require place-value reasoning and logical elimination.

Class 7MathematicsChapter 6NCERT • Ganita Prakash

Number Play

Explore number patterns through parity, puzzles, magic squares, sequences and cryptarithms.

Chapter Snapshot

Number patterns can reveal information without full calculation.
Parity tells whether a number is odd or even.
Odd and even numbers follow predictable operation rules.
Magic squares have equal row, column and diagonal sums.
The Virahanka sequence grows by adding the previous two terms.
Cryptarithms replace digits with letters and are solved by logic.

What You Will Learn

Explore patterns and relationships among numbers
Understand odd and even numbers through parity
Predict whether sums and products are odd or even
Use simple algebra to explain parity patterns
Understand the structure of 3 × 3 magic squares
Recognise and extend the Virahanka number pattern
Solve simple cryptarithms using place value and logical reasoning
Explain whether a number statement is always, sometimes or never true

Exam PriorityBoardsVery HighFoundationHigh

Chapter Overview

Number Play is about looking beyond routine calculation. We observe patterns, make predictions and explain why they work.

1. Parity

An even number can be written as:

2n

An odd number can be written as:

2n+1

for a whole number $n$ .

2. Odd and Even Rules

Parity of Sums and Differences

Operation	Result
even + even	even
odd + odd	even
even + odd	odd
even − even	even
odd − odd	even
even − odd	odd
odd − even	odd

For multiplication:

even × any whole number → even;
odd × odd → odd.

Quick Check

Without multiplying, is 135 × 654 odd or even?

Show answer

Even, because one factor (654) is even.

3. Parity of Expressions

We can often predict the parity of an algebraic expression.

For example:

2n+4

is always even because both terms are even.

Similarly:

2n+1

is always odd.

Can the Parity Change?

Consider:

3n+4

If $n=2$ :

3(2)+4=10

which is even.

If $n=3$ :

3(3)+4=13

which is odd.

So this expression can be even or odd, depending on $n$ .

4. Magic Squares

A famous $3\times3$ magic square using 1 to 9 is:

| | | | |---|---|---| | 8 | 1 | 6 | | 3 | 5 | 7 | | 4 | 9 | 2 |

Every row, column and diagonal adds to:

15

Changing a Magic Square

If the same number is added to every entry, the new grid is still a magic square.

If every entry is multiplied by the same number, it also remains a magic square.

Quick Check

If 1 is added to every entry of a 3 × 3 magic square with magic sum 15, what is the new magic sum?

Show answer

18, because each row contains three entries, so its sum increases by 3.

5. Virahanka Numbers

A beautiful sequence begins:

1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ldots

From the third term onward, each term is the sum of the previous two.

Sequence Rule

\text{next term}=\text{previous term}+\text{term before it}

For example:

5+8=13

and:

8+13=21

6. Cryptarithms

To solve one:

begin from the ones column;
consider possible carries;
use place value carefully;
reject any choice that creates a contradiction.

Simple Letter Puzzle

Suppose:

A+A=10

Then:

2A=10

so:

A=5

Real cryptarithms often involve several letters and carrying between columns.

7. Always, Sometimes or Never?

Many number statements should be tested as:

Always True
Sometimes True
Never True

Reasoning with Parity

Statement: “The sum of two odd numbers is even.”

Write odd numbers as:

2a+1

and:

2b+1

Their sum is:

2a+1+2b+1=2(a+b+1)

This is divisible by 2.

Therefore the statement is always true.

8. Understanding Parity More Deeply

Parity tells whether a whole number is odd or even. Every even number can be written as $2n$ for some whole number $n$ . Every odd number can be written as $2n+1$ .

This simple representation explains many patterns.

For example, the sum of two odd numbers is:

(2a+1)+(2b+1)=2a+2b+2

=2(a+b+1)

The result is divisible by 2, so it is even.

Similarly, an odd number multiplied by an odd number has the form:

(2a+1)(2b+1)

which simplifies to an odd number.

Parity Rules

Operation	Result
even + even	even
odd + odd	even
even + odd	odd
even × any whole number	even
odd × odd	odd

9. Why a 3 × 3 Magic Square Is Special

In a $3 \times 3$ magic square, every row, column and main diagonal has the same sum, called the magic sum.

A useful feature is that the centre entry plays a special role. In a normal 3 × 3 magic square made from nine consecutive numbers, opposite cells balance around the centre.

Missing Entry in a Magic Row

Question

The magic sum is 24. A row contains 7, 11 and x. Find x.

Solution

7+11+x=24

10. Virahanka Numbers and Recursive Patterns

A recursive pattern uses earlier terms to create later terms. In the Virahanka-type sequence, each new term is obtained by adding the previous two terms.

For example:

1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ldots

because:

1+1=2,\quad 1+2=3,\quad 2+3=5

The important idea is not only to continue the sequence but to explain how the rule generates every next term.

11. Cryptarithms and Place-value Logic

In a cryptarithm, digits are replaced by symbols or letters. The same symbol must represent the same digit each time, and different symbols usually represent different digits.

The best strategy is to work column by column from the right, just as in ordinary addition or subtraction. Carries are especially useful clues.

12. Always, Sometimes or Never

Number play often asks whether a statement is always true, sometimes true or never true.

Example: The sum of two odd numbers is even. This is always true.

Example: The sum of two numbers is odd. This is sometimes true because it happens when one number is odd and the other is even.

Example: The product of an even number and any whole number is odd. This is never true.

13. Worked Understanding and Pattern Strategies

Building Parity Rules from Examples

Try a few additions:

4+8=12,\quad 6+10=16

Both are even + even, and both results are even. Now test odd + odd:

3+5=8,\quad 7+9=16

Again the result is even. But even + odd gives odd:

6+5=11

Examples suggest a rule; algebra explains why the rule always works.

Parity Without Calculating the Whole Number

Is $127+468+913$ odd or even?

127 is odd, 468 is even and 913 is odd.

Odd + even = odd, and odd + odd = even.

Therefore the total is even, even without finding the exact sum.

Parity of Products

A product is even if at least one factor is even. This is because an even factor already contains a factor of 2.

A product is odd only when every factor is odd.

For example:

17\times25\times9

is odd, but:

17\times25\times10

is even.

Magic-square Checking

When a square is claimed to be magic, do not check only the rows. Check:

all rows;
all columns;
both main diagonals.

If even one sum differs, the arrangement is not a magic square.

Magic Sum

If each row of a 3 × 3 magic square sums to 27, the total of all nine entries is:

3\times27=81

because the three rows together contain every entry exactly once.

Pattern Rules and Position

A pattern such as:

2,5,8,11,14,\ldots

adds 3 each time. The $n$ th term can be written as:

3n-1

Check:

n=1\Rightarrow3(1)-1=2

n=4\Rightarrow3(4)-1=11

A position rule is useful because it finds a distant term directly without listing all earlier terms.

Cryptarithm Checklist

When solving a letter-digit puzzle:

write possible carries above each column;
start with the most restricted column;
reject any digit already used by another letter;
remember that a leading letter cannot be zero;
verify the completed addition or subtraction.

Quick Check

Is the product of 37 odd numbers always odd or even?

Show answer

Always odd, because a product of only odd factors is odd.

Number play is less about fast arithmetic and more about noticing structure, testing conjectures and giving reasons.

Important Terms

Key Terms

ParityEven NumberOdd NumberMagic SquareMagic SumVirahanka SequenceCryptarithmPattern

Common Mistakes

Exam Focus

Important Exam Topics

parity rules;
parity of products and algebraic expressions;
always/sometimes/never reasoning;
properties of magic squares;
continuing the Virahanka sequence;
simple cryptarithms and number puzzles.

Quick Revision

Parity means odd or even.
Even numbers can be written as $2n$ .
Odd numbers can be written as $2n+1$ .
even + even = even.
odd + odd = even.
even + odd = odd.
A product is even if at least one factor is even.
A magic square has equal row, column and diagonal sums.
Virahanka numbers follow the rule “add the previous two terms”.
Cryptarithms use letters in place of digits.

Practice Questions

Q1MCQEasyClass 7

2n

View Solution

2n+1

Q2Short AnswerEasyClass 7

What is the parity of the sum of five odd numbers?

View Solution

Odd + odd = even. Four odd numbers make an even sum; adding the fifth odd number makes the result odd .

Q3NumericalModerateClass 7

1,\ 2,\ 3,\ 5,\ 8,\ 13,\ \_\_,\ \_\_

View Solution

8+13=21

Q4Short AnswerModerateFoundation

Why is the product of an even number and an odd number always even?

View Solution

An even number contains a factor 2. Multiplying it by any whole number keeps that factor 2, so the product remains even.

Additional Practice

Q5Short AnswerEasyClass 7

Is the sum of three odd numbers odd or even?

View Solution

Odd + odd = even, and even + odd = odd. Therefore the sum is odd .

Q6Short AnswerModerateClass 7

Explain why the square of an even number is even.

View Solution

2n

Q7NumericalModerateClass 7

1,1,2,3,5,8

View Solution

13,21,34,55

Q8Short AnswerHardFoundation

Is “the sum of two whole numbers is even” always, sometimes or never true?

View Solution

Sometimes true. It is even when both numbers have the same parity, but odd when one is even and the other is odd.

Chapter Summary

Number play develops reasoning through patterns and puzzles.
Parity gives quick information about odd and even results.
Algebra can describe all even and odd numbers.
Magic squares show strong numerical structure and symmetry.
The Virahanka sequence is built by adding previous terms.
Cryptarithms require place-value reasoning and logical elimination.