Number Play

CBSE Class 7 Mathematics: Chapter 6 - Number Play (Ganita Prakash)

6.1 Numbers Tell us Things

**Key Concept:** Numbers can represent relationships and properties in different arrangements.

The Height Rule

**Definition:** Each person calls out the number of children in front of them who are taller than them.

**How it Works:**

Arrange children in a line

Each child looks at all children in front of them

Count how many of those children are taller

That count is the number they call out

**Important Observations:**

The first child in line always says **0** (no one is in front)

The last child says the total count of children taller than them in the entire line

Different arrangements produce different number sequences

**Example:**

If we have children arranged by height: Short, Medium, Tall, Very Tall

Short child (1st position): sees no one in front → calls **0**

Medium child (2nd position): sees Short in front (not taller) → calls **0**

Tall child (3rd position): sees Short and Medium in front (not taller) → calls **0**

Very Tall child (4th position): sees all 3 in front (all shorter) → calls **3**

**Analysis of Statements:**

(a) "If a person says '0', then they are the tallest" → **Only Sometimes True**

Could be tallest, or just happen to have no one taller in front

(b) "If a person is the tallest, their number is '0'" → **Only Sometimes True**

Tallest person says 0 only if standing in front; if at back, number depends on arrangement

(c) "The first person's number is '0'" → **Always True**

No one is in front of the first person

(d) "If in between (not first or last), cannot say '0'" → **Only Sometimes True**

Can say 0 if all people in front are shorter

(e) "Person with largest number is the shortest" → **Only Sometimes True**

Depends on arrangement; largest number means many taller people are in front

(f) "Largest number in 8 people" → **7**

Maximum when shortest person is last (all 7 in front are taller)

---

6.2 Picking Parity

**Definition of Parity:** The property of a number being **even** or **odd**.

Even and Odd Numbers Explained

**Even Numbers:** Can be arranged perfectly in pairs with no leftovers

Examples: 2, 4, 6, 8, 10, 12, 14...

Formula: **2n** (where n = 1, 2, 3...)

All multiples of 2

**Odd Numbers:** Leave one item unpaired (always one extra)

Examples: 1, 3, 5, 7, 9, 11, 13...

Formula: **2n - 1** (where n = 1, 2, 3...)

Can be written as 2n + 1 also

Key Parity Rules (Proven with Pictures)

**Rule 1: Even + Even = Even**

Adding two collections that are perfectly paired gives a perfectly paired result

Example: 4 + 6 = 10 (all pairs)

**Rule 2: Odd + Odd = Even**

Two odd numbers each have one unpaired item

Those two unpaired items pair together

Result is even

Example: 3 + 5 = 8 (the 1 from 3 pairs with 1 from 5, making a pair)

**Rule 3: Even + Odd = Odd**

Even (perfect pairs) + Odd (perfect pairs + 1) = perfect pairs + 1

Result is odd

Example: 4 + 5 = 9

**Rule 4: Sum of Any Number of Even Numbers = Even**

No matter how many even numbers you add, the result is always even

Example: 2 + 4 + 6 + 8 = 20 (even)

**Rule 5: Sum of ODD Count of Odd Numbers = Odd**

Add 1 odd number: result is odd

Add 3 odd numbers: result is odd

Add 5 odd numbers: result is odd

Pattern: 1, 3, 5, 7... odd count of odd numbers gives ODD sum

**Rule 6: Sum of EVEN Count of Odd Numbers = Even**

Add 2 odd numbers: result is even

Add 4 odd numbers: result is even

Add 6 odd numbers: result is even

Pattern: 2, 4, 6, 8... even count of odd numbers gives EVEN sum

Subtraction Rules with Parity

**(d) Even - Even = Even**

Example: 10 - 4 = 6 (even)

Proof: Pairs - Pairs = Pairs

**(e) Odd - Odd = Even**

Example: 9 - 5 = 4 (even)

Proof: (Pairs + 1) - (Pairs + 1) = Pairs

**(f) Even - Odd = Odd**

Example: 8 - 3 = 5 (odd)

Proof: Pairs - (Pairs + 1) = unpaired item left

**(g) Odd - Even = Odd**

Example: 7 - 2 = 5 (odd)

Proof: (Pairs + 1) - Pairs = Pairs + 1

Application: Kishor's Puzzle

**Problem:** 5 odd-numbered cards. Do they sum to 30?

5 is an **odd count**

Sum of odd count of odd numbers = **odd**

30 is **even**

Conclusion: **Impossible!**

Application: Martin and Maria's Ages

**Problem:** Two siblings born 1 year apart. Ages sum to 112. Possible?

Consecutive numbers: one is even, one is odd

Even + Odd = **Odd**

112 is **even**

Conclusion: **Impossible!** Their ages must sum to an odd number.

Parity in Grid Problems

**Small Squares in Grids:**

For an a × b grid:

If both a and b are **even**: number of squares = **even**

If both a and b are **odd**: number of squares = **odd**

If one is even, one is odd: number of squares = **even**

**Rule:** Parity of (a × b) depends on whether both factors are odd:

**Odd × Odd = Odd**

**Even × Anything = Even**

**Examples:**

27 × 13: Both odd → **odd** number of squares

42 × 78: Both even → **even** number of squares

135 × 654: 135 is odd, 654 is even → **even** number of squares

Parity of Algebraic Expressions

**Expressions with Constant Even Parity:**

Expression: **2n, 4k, 100p, 48w - 2**

These always give even results regardless of the value

Proof: Any multiple of 2 is even

**Expressions with Constant Odd Parity:**

Expression: **2n + 1, 4k + 3, 2m + 5**

These always give odd results

Proof: 2(something) always even, plus odd number = odd

**Expressions with Variable Parity:**

Expression: **3n + 4** (from the chapter)

If n = 3: 3(3) + 4 = 13 (odd)

If n = 8: 3(8) + 4 = 28 (even)

Parity depends on the value of n

Formula: The nth Even Number

**Formula:** **2n** (where n = 1, 2, 3, ...)

1st even number: 2(1) = 2

2nd even number: 2(2) = 4

100th even number: 2(100) = 200

Formula: The nth Odd Number

**Formula:** **2n - 1** (where n = 1, 2, 3, ...)

Method:

1. Find the nth even number: 2n

2. Subtract 1: 2n - 1

**Examples:**

1st odd number: 2(1) - 1 = 1

2nd odd number: 2(2) - 1 = 3

100th odd number: 2(100) - 1 = 199

50th odd number: 2(50) - 1 = 99

---

6.3 Some Explorations in Grids

Basic Grid Puzzles (3×3)

**Rules:**

Use numbers 1-9 without repeating any

Numbers in yellow circles = sum of that row or column

Find the missing numbers

**Key Pattern:** Sum of all numbers 1-9 = **45**

This means: sum of all row sums = 45

And: sum of all column sums = 45

**Example Analysis:**

If we have a grid where some numbers are given, we:

1. Check which numbers 1-9 are already placed

2. Find sums for incomplete rows/columns

3. Determine which numbers can fit

Impossible Grids

**Why Some Grids Are Impossible:**

Minimum sum of any 3 numbers from 1-9: **1 + 2 + 3 = 6**

Maximum sum of any 3 numbers from 1-9: **9 + 8 + 7 = 24**

Any circle value must be between 6 and 24

If a circle shows 5 or 26, **impossible!**

---

Magic Squares: Definition and Properties

**Definition:** A square grid of numbers where:

Each row sum = same number (magic sum)

Each column sum = same number (magic sum)

Each diagonal sum = same number (magic sum)

**Example: 3×3 Magic Square**

```

4 9 2

3 5 7

8 1 6

```

All rows, columns, diagonals sum to 15 (the magic sum)

Key Observations for 3×3 Magic Squares (using 1-9)

**Observation 1: The Magic Sum Must Be 15**

Total of all numbers 1-9 = 45

In a magic square, row sums are all equal

If magic sum = S, then 3S = 45

Therefore: **S = 15**

**Observation 2: The Centre Number Must Be 5**

Why?

If 9 is in center with 8 nearby: 8 + 9 + ? = 15 → need ?= -2 (impossible)

If 1 is in center with 2 nearby: 2 + 1 + ? = 15 → need ? = 12 (out of range)

Testing all numbers, only **5 works in the center**

Why 5 is special:

5 is the middle number of 1-9

It appears in multiple combinations that sum to 15

Examples: 1+5+9=15, 2+5+8=15, 3+5+7=15

**Observation 3: The Numbers 1 and 9 Cannot Be in Corners**

Why?

Corner positions have higher connections (part of row, column, AND diagonal)

1 + 5 + 9 = 15 (only one way with 1)

1 + 6 + 8 = 15 (another way with 1)

1 + 4 + 10 = 15 (impossible, 10 not available)

Numbers 1 and 9 have limited pairing options

They must go in middle edge positions

**Observation 4: The Numbers 1 and 9 Are on Opposite Middle Edges**

**Example:** One valid position set:

```

6 1 8

7 5 3

2 9 4

```

1 is at top middle

9 is at bottom middle

5 is at center

Creating Magic Squares Systematically

**Steps:**

1. Place 5 in the center (middle position)

2. Place 1 and 9 in opposite middle edge positions

3. Using the constraint that each row/column = 15:

If row has 5, need two numbers summing to 10

If row has 1, need two numbers summing to 14 (like 6+8)

If row has 9, need two numbers summing to 6 (like 2+4)

4. Fill remaining positions using complementary pairs

Generalised 3×3 Magic Square Form

**Let m = center number**

**General Structure (in terms of distance from center):**

```

m-4 m+3 m-2

m-1 m m+1

m+2 m-3 m+4

```

OR equivalently with consecutive numbers starting at (m-4):

```

(m-4) (m+3) (m-2)

(m-1) m (m+1)

(m+2) (m-3) (m+4)

```

**Verification:**

Any row sums to: (m-4) + (m+3) + (m-2) = 3m

Any column sums to: 3m

Diagonals also sum to: 3m

Magic sum = **3m**

**Example with m = 5:**

```

1 8 6

4 5 6

9 2 4

```

Wait, recalculating: m=5

m-4 = 1, m+3 = 8, m-2 = 3

m-1 = 4, m = 5, m+1 = 6

m+2 = 7, m-3 = 2, m+4 = 9

Gives:

```

1 8 3

4 5 6

7 2 9

```

Row sums: 1+8+3=12? No, this should be 15. Let me verify the pattern...

Actually, the general form should work as: 3m = magic sum.

If numbers are 1-9 and m=5, magic sum = 15. ✓

**Creating Magic Square with Any Center m:**

1. Magic sum = 3m

2. Use the formula pattern above

3. Place numbers according to the relationships

**Example: Center = 25**

Magic sum = 3 × 25 = 75

Numbers needed: 21, 28, 23, 24, 25, 26, 27, 22, 29

(or: 25-4 through 25+4)

Transformations of Magic Squares

**Operation 1: Add a constant to every number**

Starting square (magic sum S) + Add k to each number

Result: Still a magic square

New magic sum = **S + 3k**

Reason: Each row sum increases by 3k

**Operation 2: Multiply every number by a constant**

Starting square (magic sum S) × Multiply by k

Result: Still a magic square

New magic sum = **k × S**

Reason: Each row sum multiplies by k

**Operation 3: Rotate the entire square**

90°, 180°, 270° rotation

Result: Still a magic square

Magic sum stays the same

**Operation 4: Reflect the square**

Mirror horizontally or vertically

Result: Still a magic square

Magic sum stays the same

Creating Magic Squares from Different Sets

**Using Numbers 2-10 (instead of 1-9):**

These are not 1-9, but they are 9 consecutive numbers

Magic sum = (2+3+4+5+6+7+8+9+10)/3 = 54/3 = 18

Center should be 6 (middle of 2-10)

Pattern same as 1-9 magic square, but add 1 to each number

**General Rule for 9 Consecutive Numbers (a to a+8):**

Sum of all 9 numbers = 9a + (0+1+2+3+4+5+6+7+8) = 9a + 36

Magic sum = (9a + 36)/3 = **3a + 12**

Center number = **a + 4** (middle of the sequence)

Structure: similar to 1-9 pattern

---

The Chautīsā Yantra: The First 4×4 Magic Square

**Historical Significance:**

Found in 10th century CE inscription at Pārśhvanath Jain temple, Khajuraho, India

One of the earliest recorded magic squares

Name: "Chautīsā" means 34 in Hindi

**The Square:**

```

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

```

**Properties:**

Magic sum = **34**

Uses numbers 1-16 (no repeats)

Every row, column, and diagonal sums to 34

Multiple other combinations of 4 numbers also sum to 34 (hidden patterns)

**Why Magic Sum = 34?**

Sum of 1-16 = (16 × 17)/2 = 136

Magic sum = 136/4 = **34**

---

Historical Context: Magic Squares Worldwide

**China: Lo Shu Square (~2000+ years ago)**

Legend: A turtle emerged from flood waters with 3×3 magic square on shell

Numbers 1-9 in magical arrangement

Saved the people from catastrophic Lo River flood

**India:**

Extensively studied and developed by Indian mathematicians

Developed systematic methods for construction

Worked on 3×3, 4×4, 5×5, and larger squares

**Palani Temple, Tamil Nadu (8th century CE):** 3×3 magic square carved on pillar

Found in homes, temples, shops (like Navagraha Yantra, Kubera Yantra)

Each yantra associated with different graha (celestial body) with own magic sum

**Japan, Central Asia, Europe:** Also studied and applied in various contexts

---

6.4 Nature's Favourite Sequence: The Virahāṅka–Fibonacci Numbers

**The Sequence:** 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

**Alternative notation:** F(1)=1, F(2)=2, F(3)=3, F(4)=5, F(5)=8, ...

**Name Origins:**

**Virahāṅka:** Ancient Sanskrit poet and mathematician (from India)

**Fibonacci:** Leonardo of Pisa (Italian mathematician, 12th-13th century)

Both discovered the same sequence independently

Indian discovery came **much earlier** in poetic context

**Amazing Fact:** First discovered in poetry and music theory, not in nature!

Shows deep connection between Art, Science, and Mathematics

The Poetic Origin: Metrical Patterns

**Background:**

In Sanskrit, Prakrit, and other Indian languages

Each syllable is either **short (1 beat)** or **long (2 beats)**

In poetry/singing: short = 1 unit time, long = 2 units time

Poets studied rhythmic combinations

**The Core Question:**

"In how many ways can we fill N beats using short (1 beat) and long (2 beat) syllables?"

**Mathematical Reformulation:**

"In how many ways can we write N as a sum of 1's and 2's?"

Building the Sequence Step-by-Step

**For N = 1:**

Only way: **1**

Number of ways: **1**

**For N = 2:**

Ways: 1+1, 2

Number of ways: **2**

**For N = 3:**

Ways: 1+1+1, 1+2, 2+1

Number of ways: **3**

**For N = 4:**

Ways: 1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2

Number of ways: **5**

**For N = 5:**

Start with 1: Add 1 in front of all 4-beat rhythms (5 ways)

Start with 2: Add 2 in front of all 3-beat rhythms (3 ways)

Total: 5 + 3 = **8 ways**

**For N = 6:**

Start with 1: All 5-beat rhythms (8 ways)

Start with 2: All 4-beat rhythms (5 ways)

Total: 8 + 5 = **13 ways**

**Sequence:** 1, 2, 3, 5, 8, 13, 21, ...

The Recursive Pattern (Recurrence Relation)

**Key Insight:** Every N-beat rhythm MUST start with either:

**1 (short syllable)** → remaining N-1 beats

**2 (long syllable)** → remaining N-2 beats

**Formula:** f(N) = f(N-1) + f(N-2)

**Proof of Formula:**

All N-beat rhythms starting with "1+" = all (N-1)-beat rhythms

All N-beat rhythms starting with "2+" = all (N-2)-beat rhythms

Every rhythm starts with one of these, no overlap

Total = f(N-1) + f(N-2)

**Example:**

f(6) = f(5) + f(4) = 8 + 5 = 13 ✓

f(7) = f(6) + f(5) = 13 + 8 = 21 ✓

f(8) = f(7) + f(6) = 21 + 13 = 34 ✓

Computing f(8) for 8-Beat Rhythms

**Method 1: Using Recurrence**

f(1) = 1

f(2) = 2

f(3) = f(2) + f(1) = 2 + 1 = 3

f(4) = f(3) + f(2) = 3 + 2 = 5

f(5) = f(4) + f(3) = 5 + 3 = 8

f(6) = f(5) + f(4) = 8 + 5 = 13

f(7) = f(6) + f(5) = 13 + 8 = 21

f(8) = f(7) + f(6) = 21 + 13 = **34 ways**

**Method 2: Systematic Listing**

All 8-beat rhythms with (1 + 7-beat rhythms) = 13 rhythms

All 8-beat rhythms with (2 + 6-beat rhythms) = 21 rhythms

Total: 13 + 21 = 34 rhythms

Some Examples of 8-Beat Rhythms

1. 2+2+2+2 (four long syllables)

2. 1+1+1+1+1+1+1+1 (eight short syllables)

3. 1+2+2+1+2 (short, long, long, short, long)

4. 2+2+1+1+2 (long, long, short, short, long)

5. 1+1+2+2+2 (short, short, long, long, long)

... and 29 more!

Why This Sequence Is Special

**Nature's Occurrences:**

Flower petals: sunflower spiral (Fibonacci numbers)

Pine cones and pineapples: spiral patterns

Nautilus shell: spiral proportions

Tree branches: branching patterns

Leaf arrangement on stems

**Mathematical Properties:**

Adjacent ratios approach Golden Ratio (φ ≈ 1.618)

As N increases: f(N+1)/f(N) → φ

Example: 34/21 ≈ 1.619, 55/34 ≈ 1.618

**Connections:**

Appears in biology, physics, art, architecture

Used in financial analysis (Fibonacci retracements)

Computer algorithms and data structures

---

Summary of Important Formulas and Rules

**Parity Rules:**

Even + Even = Even

Odd + Odd = Even

Even + Odd = Odd

Even × Any = Even

Odd × Odd = Odd

Odd count of odd numbers sum = Odd

Even count of odd numbers sum = Even

**Number Formulas:**

nth even number = **2n**

nth odd number = **2n - 1**

**Magic Square (3×3 with 1-9):**

Magic sum = **15**

Center number = **5**

Formula with center m: magic sum = **3m**

**Virahāṅka–Fibonacci:**

f(N) = f(N-1) + f(N-2)

Sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

**Indian Mathematical Heritage:**

Magic squares found in Khajuraho (10th century) and Palani Temple (8th century)

Virahāṅka–Fibonacci numbers discovered in Sanskrit poetry

Indian mathematicians extended work to 5×5 and larger magic squares