Number Play Class 7 Notes Maths Chapter 6 Free PDF

Table of Contents
1. Numbers Tell us Things
2. Picking Parity
3. Some Explorations in Grids
4. What is a Magic Square?
5. The First-ever 4 × 4 Magic Square
6. Magic Squares in History and Culture
7. What is a Kubera Yantra?
8. Nature's Favourite Sequence: The Virahāṅka-Fibonacci Numbers
9. Digits in Disguise (Cryptarithms / Alphametics)
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Numbers Tell us Things

Numbers are more than tools for counting. They represent relationships and often follow rules that let us infer information about a situation. For example, when people in a line call out numbers that do not match the order they are standing in, those numbers may still be encoding something meaningful about their positions.

Observe the two pictures below and notice that the numbers are not random; they relate to each child's position and some other fixed property (for example, height).

Children are standing in a straight line and saying numbers: (0,1,0,2,2,2,1)

The pattern may look unclear at first. Look now at the second picture - the numbers change according to the children's positions.

Now the same children are in a new order, and they say:  (0,0, 1, 2, 2, 5, 6)

If you observe carefully, the numbers reflect information based on the children's positions in the line.

What Do These Numbers Actually Mean?

One useful interpretation is that each child is saying how many children in front of them are taller than they are. This assumes the height of each child is fixed and known relative to others.

So, if the tallest child is red, then in an arrangement each child may call out the number of children taller than them who stand ahead in the line. In other situations the number might mean how many children came before them in line, or how many taller or shorter children are around them. The exact meaning depends on the rule the problem gives, but the key idea is that numbers can encode positional relationships.

Picking Parity

Parity is the property of a number being either even or odd.

This section explores rules and patterns connected with parity.

Kishor's Puzzle

Kishor has number cards (1, 3, 5, 7, 9, 11, 13) and 5 empty boxes. He needs to place exactly one card in each box such that the numbers sum to 30. Can he do it?

We will use parity (even/odd reasoning) to answer this.

Properties of Even and Odd Numbers

• Even numbers: Can be paired with no leftover; examples: 2, 4, 6, 8, 10.
• Odd numbers: Always leave one unpaired object when grouped; examples: 1, 3, 5, 7, 9.
• The sum of two odd numbers is even.
• The sum of three odd numbers is odd.

No pair is left

One unpaired left

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The Puzzle (Detailed)

The Puzzle:

Kishor has some number cards. He needs to place 5 cards in boxes so that their total equals 30.

The number cards available are:

13, 9, 7, 11, 5, 3

All these are odd numbers.

Question: Can the sum of 5 odd numbers be 30 (an even number)?

Reasoning and Solution

Consider what happens when you add odd numbers together.

When you add two odd numbers, the result is even.

When you add four odd numbers, the result is even (because (odd + odd) + (odd + odd) = even + even = even).

When you add an odd number of odd numbers (3, 5, 7, ...), the result is always odd.

Therefore, the sum of 5 odd numbers must be odd.

30 is even, so it is impossible to make 30 by adding 5 odd numbers.

So, the answer to Kishor's Puzzle is no.

Quick Parity Facts (for reference)

• 2 odd numbers → even
• 4 odd numbers → even
• 5 odd numbers → odd

Real-Life Example: Consecutive Ages

Martin and Maria are siblings born one year apart. If their total age is 112, is that possible?

Try values for their ages to check parity.

Martin = 55, Maria = 56 → 55 + 56 = 111 (odd).

Martin = 56, Maria = 57 → 56 + 57 = 113 (odd).

Any two consecutive numbers consist of one even and one odd; even + odd = odd.

So, the sum of two consecutive numbers is always odd. Therefore 112 (even) is not possible.

Small Squares in Grids

• A 3 × 3 grid has 9 squares (an odd number).

A 3 × 4 grid has 12 squares (even).

Can you tell the parity of total squares from the grid dimensions?

Rule: The product (total squares) is odd only if both dimensions are odd.

• Odd × Odd = Odd
• Odd × Even = Even
• Even × Odd = Even
• Even × Even = Even

Parity of Expressions

Consider the expression 3n + 4. Its parity depends on n.

• If n is odd, 3n is odd, and 3n + 4 (odd + even) is odd.
• If n is even, 3n is even, and 3n + 4 (even + even) is even.
• Expressions always even: Any expression having a factor 2, for example 2n, 100p, 48w - 2, 6k + 2, n + n.
• Expressions always odd: An even expression plus or minus 1, for example 2n + 1, 2n - 1.

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Finding the nth Even and Odd Number

• The nth even number is given by the formula 2n. Example: 100th even number = 2 × 100 = 200.
• The nth odd number is given by the formula 2n - 1. Example: 100th odd number = 2 × 100 - 1 = 199.

Some Explorations in Grids

We now look at patterns and puzzles using small grids filled with numbers.

The 3 × 3 Sum Grid Puzzle

Observe the 3 × 3 grid filled with numbers 1 to 9, with no repeats. The circled numbers outside represent the sums of numbers in the corresponding rows or columns. Verify the row and column sums.

Challenge: Fill the grids using numbers 1-9 without repetition so that row and column sums match the circled numbers.

Grid 1:

Ans:

Grid 2:

Ans:

Impossible Grid?

Is it possible to find a solution? Why or why not?

• Reasoning: The smallest possible sum for a row or column using three distinct numbers from 1-9 is 1 + 2 + 3 = 6. The largest is 7 + 8 + 9 = 24. Any required circled sum must therefore be between 6 and 24, inclusive. If a circled sum lies outside this range (for example 5), the grid has no solution.

Total Sum Check

What is the sum of all numbers from 1 to 9? (1 + 2 + ... + 9 = 45).

What is the sum of all the row sums (or column sums) in a completed 3 × 3 grid puzzle?

Each number in the grid is counted exactly once in the row sums, so the total of row sums equals the sum of numbers 1-9, which is 45. The same is true for column sums.

What is a Magic Square?

A magic square is a special grid of numbers where the sum of numbers in every row, every column, and both diagonals is the same. This common total is called the magic sum.

In a 3 × 3 magic square using numbers 1 to 9 without repetition, the magic sum is fixed.

Example that is not a magic square

The following filled 3 × 3 grid is not a magic square because row sums and column sums are not equal, and the diagonals do not give the same sum.

Row sums: 16, 9, 20 (not equal).

Column sums: 4 + 6 + 3 = 13, 7 + 1 + 9 = 17, 5 + 2 + 8 = 15 (not equal).

A correct 3 × 3 magic square

Check the magic:

• Rows: 2 + 7 + 6 = 15; 9 + 5 + 1 = 15; 4 + 3 + 8 = 15.
• Columns: 2 + 9 + 4 = 15; 7 + 5 + 3 = 15; 6 + 1 + 8 = 15.
• Diagonals: 2 + 5 + 8 = 15; 6 + 5 + 4 = 15.

This is a true magic square with magic sum = 15.

Why is the magic sum 15 in a 3 × 3 magic square using 1-9?

The total of numbers 1 to 9 is 1 + 2 + ... + 9 = 45.

This total is split evenly across 3 rows (or 3 columns), so each row/column must sum to 45 ÷ 3 = 15. Hence the magic sum is 15.

Observations about the 3 × 3 magic square

• Observation 1: All rows, columns, and diagonals must add up to 15.

Can 9 be at the centre?

Try putting 9 in the centre. If a row already has 8 and 9, then 8 + 9 + ? = 15, which gives 17 + ? = 15 - impossible because 17 > 15. So 9 cannot be at the centre.

Conclusion: 9 cannot be at the centre.

Can 1 be at the centre?

Try putting 1 in the centre and 2 in the same row: 1 + 2 + ? = 15 ⇒ 3 + ? = 15 ⇒ ? = 12, which is not allowed because we must use numbers 1-9. Therefore 1 cannot be at the centre.

Conclusion: 1 cannot be at the centre.

Which number can be at the centre?

If you test all numbers from 1 to 9, you find that only 5 can be at the centre. With 5 in the centre, there are four different pairs that add with 5 to make 15:

• 5 + 1 + 9 = 15
• 5 + 2 + 8 = 15
• 5 + 3 + 7 = 15
• 5 + 4 + 6 = 15

Observation 2: The number 5 must be at the centre of the 3 × 3 magic square using numbers 1-9.

Where do 1 and 9 go?

The smallest number (1) and the largest number (9) cannot be at the centre and, by testing positions, they are best placed on the middle positions on the boundary (edge centres), not corners. If 1 or 9 is placed in a corner it becomes difficult to find enough valid combinations that sum to 15.

For instance, if 1 occupies a corner, possible combinations that make 15 are 1 + 5 + 9 and 1 + 6 + 8 - two valid combinations. Similarly for 9: 9 + 1 + 5 and 9 + 2 + 4 are valid combinations. With careful placement of these numbers in edge centres and 5 at the centre, we can complete the square.

How to start filling a 3 × 3 magic square

One approach is to place three numbers that already add to 15 in one row or column (for example 1, 5, 9 in a row). Then fill the remaining positions using the remaining numbers (2, 3, 4, 6, 7, 8), ensuring each number is used once and that every row, column, and diagonal adds to 15.

A classic 3 × 3 magic square

• Rows: 2 + 7 + 6 = 15; 9 + 5 + 1 = 15; 4 + 3 + 8 = 15.
• Columns: 2 + 9 + 4 = 15; 7 + 5 + 3 = 15; 6 + 1 + 8 = 15.
• Diagonals: 2 + 5 + 8 = 15; 6 + 5 + 4 = 15.

Every direction adds to the magic sum of 15.

Generalising a 3 × 3 magic square (using algebra)

When we generalise in mathematics we look for a pattern or rule that works for all similar cases. For a 3 × 3 magic square made with consecutive numbers (like 1-9), the centre number plays a key role. Call the centre number m. For the 1-9 case, m = 5 and the magic sum = 3m = 15.

We can express other entries relative to m (for example m ± k), and there is a standard arrangement that always gives a 3 × 3 magic square when using consecutive numbers centred on m.

Consider the magic square

We can express it using the letter-number m as:

The First-ever 4 × 4 Magic Square

This famous 4 × 4 magic square from ancient India is called the Chautīśā Yantra, found in the Parshvanath Jain temple in Khajuraho. "Chautīśā" means 34.

The first ever recorded 4 × 4 magic square, the Chautīsā Yantra, at Khajuraho, India

It is a 4 × 4 grid in which each row, each column and both diagonals add up to the same number, here 34.

• Row 1: 7 + 12 + 1 + 14 = 34
• Column 1: 7 + 2 + 16 + 9 = 34
• Diagonal (top-left to bottom-right): 7 + 13 + 10 + 4 = 34
• Diagonal (top-right to bottom-left): 14 + 8 + 3 + 9 = 34

Every row, column, and diagonal gives the same magic sum = 34.

Other patterns that add up to 34

The Chautīśā Yantra is special because many other combinations of four numbers in it also add to 34. Examples to check:

• Corners: 7 + 14 + 4 + 9 = 34
• Centre 2 × 2: 13 + 8 + 3 + 10 = 34
• Same position in different rows: 12 + 13 + 3 + 6 = 34

There are many hidden combinations in this magic square.

Magic Squares in History and Culture

Lo Shu Square - the oldest known magic square

The first recorded magic square is called the Lo Shu Square, from ancient China, more than 2000 years ago. According to legend a turtle from the Lo River had a 3 × 3 grid on its back containing numbers 1-9 in a special pattern:

• Any row: 2 + 7 + 6 = 15
• Any column: 2 + 9 + 4 = 15
• Any diagonal: 2 + 5 + 8 = 15

So the magic sum = 15 in this case.

Where else are magic squares found?

• Magic squares have been studied in India, Japan, China, Central Asia, and Europe.
• Ancient India produced 3 × 3, 4 × 4, 5 × 5 and larger magic squares using clever methods.
• They appear in art, architecture and religious diagrams (yantras).

Ancient magic squares in Indian temples

• A 3 × 3 magic square appears on a pillar in a temple in Palani, Tamil Nadu, dating back to the 8th century CE.
• Magic squares are used in Navagraha Yantras and other household or religious symbols, connecting them with planets and rituals.

What is a Kubera Yantra?

The Kubera Yantra is a sacred diagram associated with Lord Kubera, the deity of wealth. It often contains a magic square inside it.

Checking the Kubera Yantra as a magic square

We check the sums of rows, columns and diagonals to confirm it is a magic square.

Rows:

• Row 1: 27 + 20 + 25 = 72
• Row 2: 22 + 24 + 26 = 72
• Row 3: 23 + 28 + 21 = 72

Columns:

• Column 1: 27 + 22 + 23 = 72
• Column 2: 20 + 24 + 28 = 72
• Column 3: 25 + 26 + 21 = 72

Diagonals:

• 27 + 24 + 21 = 72
• 25 + 24 + 23 = 72

All rows, columns, and diagonals add to 72, so 72 is the magic sum here.

This demonstrates that magic squares can be built with different sets of numbers (not only 1-9) leading to different magic sums.

Nature's Favourite Sequence: The Virahāṅka-Fibonacci Numbers

A well-known sequence begins

1, 2, 3, 5, 8, 13, 21, 34, ...

Each number after the first two is the sum of the two preceding numbers. This list is called the Fibonacci sequence in the West and the Virahāṅka numbers in India. It appears widely in plants, art and music.

Discovery and origin

Surprisingly, this sequence first appeared in the study of poetry (Sanskrit, Tamil, Telugu and other Indian traditions). Poets composed rhythms using short syllables (1 beat) and long syllables (2 beats). They asked questions such as: "In how many ways can we fill a line of poetry with n beats using short and long syllables?"

Example: n = 4 beats

List the ways to write 4 using 1s and 2s (short and long syllables). The count equals the 4th Fibonacci number.

There are 5 ways for n = 4, which corresponds to the Fibonacci number at position 4 (if we start counting from n = 1).

How to build the next Fibonacci number

To get the number of ways for n beats, consider the first syllable:

If the rhythm starts with 1, the remaining (n - 1) beats can be arranged in all the ways for n - 1.

If the rhythm starts with 2, the remaining (n - 2) beats can be arranged in all the ways for n - 2.

Therefore: Ways(n) = Ways(n - 1) + Ways(n - 2).

Example: n = 5 beats

All ways to write 5 using 1s and 2s are:

• 1 + 1 + 1 + 1 + 1
• 1 + 1 + 1 + 2
• 1 + 1 + 2 + 1
• 1 + 2 + 1 + 1
• 2 + 1 + 1 + 1
• 1 + 2 + 2
• 2 + 1 + 2
• 2 + 2 + 1

There are 8 rhythms - the 5th Virahāṅka-Fibonacci number is 8.

History fun facts

• Virahāṅka introduced the numbers in India around 700 CE.
• Fibonacci described them in Europe in 1202 CE.
• The sequence appears in nature: sunflower spirals, pinecones, pineapples and flowers often show counts equal to Fibonacci numbers (for example 13, 21 or 34 petals).

Digits in Disguise (Cryptarithms / Alphametics)

In cryptarithms each letter stands for a digit (0-9) and you must find which digit each letter represents to make the arithmetic correct.

Consider the simple cryptarithm where T + T + T = UT (a two-digit number with tens digit U and units digit T). This means 3 × T = UT, and the last digit of the product must be the same as T.

Solution (showing the reasoning)

Try values for T to satisfy 3 × T ending with digit T.

Try T = 5.

3 × 5 = 15.

The product is 15, which ends with 5, so U = 1 and T = 5 works.

Next cryptarithm:

Here "K2" means a two-digit number whose tens digit is K and units digit is 2. We have K2 + K2 = HMM (a three-digit number whose last two digits are the same, M).

Solution

Try K = 6: 62 + 62 = 124 → gives digits 1,2,4 (not HMM).

Try K = 7: 72 + 72 = 144 → this gives H = 1 and M = 4, so HMM = 144 works.