A Square and a Cube

CHAPTER 1: A SQUARE AND A CUBE

COMPREHENSIVE NOTES FOR CLASS 8 MATHEMATICS (GANITA PRAKASH, NCF 2023)

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INTRODUCTION: THE LOCKER PUZZLE

The chapter opens with the famous **Queen Ratnamanjuri's Locker Puzzle**, which introduces the concept of factors and their relationship to whether numbers are perfect squares.

Understanding the Puzzle

**The Setup:**

100 people (Person 1 to Person 100) and 100 lockers

Each person n toggles (opens or closes) every nth locker

Person 1 opens all lockers (1, 2, 3, ..., 100)

Person 2 toggles every 2nd locker (2, 4, 6, 8, ..., 100)

Person 3 toggles every 3rd locker (3, 6, 9, 12, ..., 99)

This pattern continues through Person 100

**Key Insight:** A locker remains open at the end if and only if it is toggled an **odd number of times**.

**Why This Matters:** The number of times a locker is toggled equals the **number of factors** of that locker's number.

Partner Factors and Factor Pairing

Every factor of a number has a "partner factor" such that their product equals the original number.

**Example for Locker #6:**

Factors: 1, 2, 3, 6

Factor pairs: (1 × 6) and (2 × 3)

Number toggled: 4 times (even) → locker closes

Person 1 opens it, Person 2 closes it, Person 3 opens it, Person 6 closes it

**Example for Locker #4:**

Factors: 1, 2, 4

Factor pairs: (1 × 4) and (2 × 2)

Notice: In the pair (2 × 2), both factors are the SAME

Number toggled: 3 times (odd) → locker opens

Person 1 opens it, Person 2 closes it, Person 4 opens it

**Critical Observation:** Most numbers have an EVEN number of factors because factors pair up. ONLY numbers that are **perfect squares** have an ODD number of factors because one factor (the square root) pairs with itself.

Perfect Squares: The Answer

**The Lockers That Open:**

Locker 1: 1 × 1 (1 factor: 1) → opens

Locker 4: 2 × 2 (3 factors: 1, 2, 4) → opens

Locker 9: 3 × 3 (3 factors: 1, 3, 9) → opens

Locker 16: 4 × 4 (5 factors: 1, 2, 4, 8, 16) → opens

Locker 25, 36, 49, 64, 81, 100 → all open

**Why Khoisnam Knew:** He realized that only the perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) would remain open.

The Passcode Discovery

The puzzle also asks: "Which 5 lockers were touched exactly twice?"

**Answer:** The **prime numbers** 2, 3, 5, 7, 11

Prime numbers have exactly 2 factors: 1 and the number itself

So they are toggled exactly twice

**Passcode: 2-3-5-7-11**

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1.1 SQUARE NUMBERS

Definition of Square Numbers

**A square number** (or perfect square) is a number that can be expressed as the product of a number with itself.

**Mathematical Definition:** A number n is a perfect square if n = a × a = a² for some number a.

**Why Called "Squares"?** Geometrically, if a square has sides of length a units, its area is a × a = a² square units. The number of unit squares that fit inside equals a².

Table of Squares from Geometric Interpretation

| Sidelength (units) | Area (sq. units) | Notation |

|---|---|---|

| 1 | 1 × 1 = 1 | 1² = 1 |

| 2 | 2 × 2 = 4 | 2² = 4 |

| 3 | 3 × 3 = 9 | 3² = 9 |

| 4 | 4 × 4 = 16 | 4² = 16 |

| 5 | 5 × 5 = 25 | 5² = 25 |

| 10 | 10 × 10 = 100 | 10² = 100 |

**General Notation:** For any number n, we write n × n = n² (read as "n squared")

Squares of Non-Integer Numbers

**Important:** Perfect squares are NOT limited to whole numbers.

**Example with Fractions:**

(3/5)² = (3/5) × (3/5) = 9/25

**Example with Decimals:**

(2.5)² = 2.5 × 2.5 = 6.25

**Example with Negative Numbers:**

(-6)² = (-6) × (-6) = 36

(Note: Any negative number squared becomes positive)

Perfect Squares: First 30 Natural Numbers

**Table of Squares:**

| n | n² |

|---|---|

| 1 | 1 |

| 2 | 4 |

| 3 | 9 |

| 4 | 16 |

| 5 | 25 |

| 6 | 36 |

| 7 | 49 |

| 8 | 64 |

| 9 | 81 |

| 10 | 100 |

| 11 | 121 |

| 12 | 144 |

| 13 | 169 |

| 14 | 196 |

| 15 | 225 |

| 16 | 256 |

| 17 | 289 |

| 18 | 324 |

| 19 | 361 |

| 20 | 400 |

| 21 | 441 |

| 22 | 484 |

| 23 | 529 |

| 24 | 576 |

| 25 | 625 |

| 26 | 676 |

| 27 | 729 |

| 28 | 784 |

| 29 | 841 |

| 30 | 900 |

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PATTERNS AND PROPERTIES OF PERFECT SQUARES

Pattern 1: Units Digits (Last Digit) of Perfect Squares

**Key Observation:** When you examine the squares of all numbers, the units digit (last digit) can ONLY be: **0, 1, 4, 5, 6, or 9**

Numbers ending in 2, 3, 7, or 8 can NEVER be perfect squares.

**Why This Works:**

If a number ends in 0, its square ends in 0

If a number ends in 1, its square ends in 1

If a number ends in 2, its square ends in 4

If a number ends in 3, its square ends in 9

If a number ends in 4, its square ends in 6

If a number ends in 5, its square ends in 5

If a number ends in 6, its square ends in 6

If a number ends in 7, its square ends in 9

If a number ends in 8, its square ends in 4

If a number ends in 9, its square ends in 1

**Important Limitation:** Just because a number ends in 0, 1, 4, 5, 6, or 9 does NOT mean it's a perfect square. For example:

26 ends in 6, but it's NOT a perfect square

16 = 4² ends in 6, and IS a perfect square

36 = 6² ends in 6, and IS a perfect square

**Use as a Screening Tool:** If a number ends in 2, 3, 7, or 8, you can IMMEDIATELY say it's NOT a perfect square without further calculation.

Pattern 2: Squares with Last Digit 1

**Numbers with units digit 1 or 9, when squared, always give units digit 1:**

1² = 1 (ends in 1) ✓

9² = 81 (ends in 1) ✓

11² = 121 (ends in 1) ✓

19² = 361 (ends in 1) ✓

21² = 441 (ends in 1) ✓

29² = 841 (ends in 1) ✓

31² = 961 (ends in 1) ✓

**Pattern:** Any number whose units digit is 1 or 9 will have a square ending in 1.

Pattern 3: Squares with Last Digit 6

**Numbers with units digit 4 or 6, when squared, always give units digit 6:**

4² = 16 (ends in 6) ✓

6² = 36 (ends in 6) ✓

14² = 196 (ends in 6) ✓

16² = 256 (ends in 6) ✓

24² = 576 (ends in 6) ✓

26² = 676 (ends in 6) ✓

**Pattern:** Any number whose units digit is 4 or 6 will have a square ending in 6.

**Worked Example:**

Which of the following have units digit 6 in their squares?

(i) 38² → 8 ends in 8, so 38² ends in 4 (NOT 6)

(ii) 34² → 4 ends in 4, so 34² ends in 6 ✓

(iii) 46² → 6 ends in 6, so 46² ends in 6 ✓

(iv) 56² → 6 ends in 6, so 56² ends in 6 ✓

(v) 74² → 4 ends in 4, so 74² ends in 6 ✓

(vi) 82² → 2 ends in 2, so 82² ends in 4 (NOT 6)

Pattern 4: Number of Zeros in Perfect Squares

**Key Rule:** If a number has an EVEN number of zeros at the end, its square will have an EVEN number of zeros at the end. Specifically, if a number ends with n zeros, its square ends with 2n zeros.

**Examples:**

| Number | Zeros | Square | Zeros in Square |

|---|---|---|---|

| 10 | 1 zero | 100 | 2 zeros |

| 100 | 2 zeros | 10,000 | 4 zeros |

| 20 | 1 zero | 400 | 2 zeros |

| 200 | 1 zero | 40,000 | 4 zeros |

| 40 | 1 zero | 1,600 | 2 zeros |

| 700 | 1 zero | 490,000 | 4 zeros |

| 900 | 1 zero | 810,000 | 4 zeros |

**Important Consequence:** Perfect squares can ONLY have an **even number of zeros** at the end. A number ending with an odd number of zeros (like 10, 100, 1,000) CAN be a perfect square, but ANY number ending in exactly 1 zero, 3 zeros, 5 zeros, etc., would need verification.

**Worked Example:**

Can 123,000 be a perfect square?

It ends in 3 zeros (odd number)

Perfect squares can only have an even number of zeros

Therefore, 123,000 is NOT a perfect square

Pattern 5: Parity and Perfect Squares

**Definition of Parity:** Whether a number is even or odd.

**Relationship Between a Number and Its Square:**

(even)² = even

(odd)² = odd

**Examples:**

2² = 4 (even squared = even)

4² = 16 (even squared = even)

3² = 9 (odd squared = odd)

5² = 25 (odd squared = odd)

7² = 49 (odd squared = odd)

**Pattern:** The parity of a perfect square depends only on the parity of its square root. If n is even, n² is even. If n is odd, n² is odd.

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PERFECT SQUARES AND ODD NUMBERS

The Relationship: Sum of Consecutive Odd Numbers

**Major Discovery:** Every perfect square can be expressed as the sum of consecutive odd numbers starting from 1.

**Proof Through Examples:**

1 = 1²

1 + 3 = 4 = 2²

1 + 3 + 5 = 9 = 3²

1 + 3 + 5 + 7 = 16 = 4²

1 + 3 + 5 + 7 + 9 = 25 = 5²

1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

**General Formula:** The sum of the first n odd numbers equals n².

**Why This Works (Visual Proof):** Imagine an L-shaped border around a square:

Start with 1 unit square

Add an L-shaped border of 3 more squares → makes 2×2

Add an L-shaped border of 5 more squares → makes 3×3

Add an L-shaped border of 7 more squares → makes 4×4

Each new L-shaped layer contains the next odd number (3, 5, 7, 9, ...) because you add one unit to the top and one to the right, then one in the corner.

Differences Between Consecutive Squares

**Key Pattern:** The difference between consecutive perfect squares equals the next odd number.

**Proof:**

4 - 1 = 3 (which is the 2nd odd number)

9 - 4 = 5 (which is the 3rd odd number)

16 - 9 = 7 (which is the 4th odd number)

25 - 16 = 9 (which is the 5th odd number)

36 - 25 = 11 (which is the 6th odd number)

**Algebraic Explanation:**

(n+1)² - n² = (n² + 2n + 1) - n² = 2n + 1

And 2n + 1 is exactly the formula for the nth odd number!

Finding the nth Odd Number

**Formula:** The nth odd number = 2n - 1

**Examples:**

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

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

3rd odd number: 2(3) - 1 = 5

4th odd number: 2(4) - 1 = 7

36th odd number: 2(36) - 1 = 71

Worked Example 1: Finding a Square Using Known Square

**Given:** 35² = 1225, find 36²

**Solution:**

We know 35² = 1225 (sum of first 35 odd numbers)

To find 36², we add the 36th odd number to 1225

The 36th odd number = 2(36) - 1 = 71

Therefore, 36² = 1225 + 71 = 1296

**Verification:** 36 × 36 = 1296 ✓

Using Consecutive Odd Numbers to Test for Perfect Squares

**Method:** Successively subtract consecutive odd numbers starting from 1. If you reach exactly 0, the number is a perfect square.

**Worked Example 2: Testing 25**

25 - 1 = 24

24 - 3 = 21

21 - 5 = 16

16 - 7 = 9

9 - 9 = 0 ✓

**Result:** 25 is a perfect square. We subtracted exactly 5 consecutive odd numbers, so 25 = 5²

**Worked Example 3: Testing 38 (Not a Perfect Square)**

38 - 1 = 37

37 - 3 = 34

34 - 5 = 29

29 - 7 = 22

22 - 9 = 13

13 - 11 = 2

2 - 13 = -11 (we crossed 0!)

**Result:** 38 is NOT a perfect square because we cannot express it as a sum of consecutive odd numbers starting from 1.

Counting Numbers Between Consecutive Squares

**Question:** How many numbers lie between n² and (n+1)²?

**Answer:** Between n² and (n+1)², there are 2n numbers.

**Proof:** (n+1)² - n² - 1 = n² + 2n + 1 - n² - 1 = 2n

**Worked Examples:**

Between 4² and 5² (between 16 and 25): 25 - 16 - 1 = 8 numbers

Check: 2n = 2(4) = 8 ✓

The numbers are: 17, 18, 19, 20, 21, 22, 23, 24

Between 6² and 7² (between 36 and 49): 49 - 36 - 1 = 12 numbers

Check: 2n = 2(6) = 12 ✓

Counting Perfect Squares in Intervals of 100

**Table of Perfect Squares in Ranges:**

| Range | Count | Squares |

|---|---|---|

| 1-100 | 10 | 1², 2², 3², ..., 10² |

| 101-200 | 9 | 11², 12², ..., 14² |

| 201-300 | 8 | 15², 16², ..., 17² |

| 301-400 | 7 | 18², 19², 20² |

| 401-500 | 7 | 21², 22² |

| 501-600 | 6 | 23², 24² |

| 601-700 | 5 | 25², 26² |

| 701-800 | 5 | 27², 28² |

| 801-900 | 5 | 29² |

| 901-1000 | 4 | 30², 31² |

**The Largest Perfect Square Less Than 1000:** 31² = 961

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PERFECT SQUARES AND TRIANGULAR NUMBERS

What Are Triangular Numbers?

**Triangular Numbers:** Numbers that can be arranged in the shape of an equilateral triangle.

Sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

**Formula:** The nth triangular number = n(n+1)/2

**Examples:**

1st triangular: 1(2)/2 = 1

2nd triangular: 2(3)/2 = 3

3rd triangular: 3(4)/2 = 6

4th triangular: 4(5)/2 = 10

5th triangular: 5(6)/2 = 15

Relationship Between Squares and Triangular Numbers

**Key Discovery:** The sum of two consecutive triangular numbers equals a perfect square.

**Pattern:**

T₁ + T₂ = 1 + 3 = 4 = 2²

T₂ + T₃ = 3 + 6 = 9 = 3²

T₃ + T₄ = 6 + 10 = 16 = 4²

T₄ + T₅ = 10 + 15 = 25 = 5²

**Algebraic Proof:**

Tₙ + Tₙ₊₁ = n(n+1)/2 + (n+1)(n+2)/2

= [(n+1)(n + n + 2)]/2

= [(n+1)(2n + 2)]/2

= [(n+1) × 2(n+1)]/2

= (n+1)²

**Interpretation:** Each perfect square is the sum of two consecutive triangular numbers.

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SQUARE ROOTS

Definition of Square Root

**Square Root Definition:** If y = x², then x is the square root of y.

In other words, the square root is the **inverse operation** of squaring.

**Example:** Since 7 × 7 = 49 or 7² = 49, we say that 7 is the square root of 49.

Positive and Negative Square Roots

**Key Fact:** Every positive perfect square has TWO square roots: one positive and one negative.

**Examples:**

8² = 64, so 8 is a square root of 64

(-8)² = 64, so -8 is also a square root of 64

Therefore, both 8 and -8 are square roots of 64

10² = 100, so 10 is a square root of 100

(-10)² = 100, so -10 is also a square root of 100

Therefore, both 10 and -10 are square roots of 100

**Notation:** √64 = ±8 and √100 = ±10

**General Rule:** √(n²) = ±n

**In This Course:** We focus primarily on **positive square roots** unless otherwise specified.

How to Determine if a Number is a Perfect Square

#### Method 1: Sequential Testing

**Procedure:** If you suspect a number might be a perfect square between n² and (n+1)², calculate the squares sequentially until you find the exact match or exceed your target.

**Worked Example: Is 576 a Perfect Square?**

Since 20² = 400 < 576 < 625 = 25², the square root must be between 20 and 25.

Let's calculate:

20² = 400

21² = 441

22² = 484

23² = 529

24² = 576 ✓

**Answer:** Yes, 576 is a perfect square, and √576 = 24

**Disadvantage:** This method becomes inefficient for very large numbers.

#### Method 2: Successive Subtraction of Odd Numbers

**Procedure:** Subtract consecutive odd numbers starting from 1 until you reach exactly 0.

**Worked Example: Is 81 a Perfect Square?**

81 - 1 = 80

80 - 3 = 77

77 - 5 = 72

72 - 7 = 65

65 - 9 = 56

56 - 11 = 45

45 - 13 = 32

32 - 15 = 17

17 - 17 = 0 ✓

**Result:** 81 is a perfect square. We subtracted 9 consecutive odd numbers, so 81 = 9²

**Disadvantage:** Very time-consuming for large numbers.

#### Method 3: Prime Factorisation (Most Reliable)

**Principle:** A number is a perfect square if and only if ALL prime factors appear an EVEN number of times in its prime factorisation.

**Procedure:**

1. Find the prime factorisation of the number

2. Count the frequency of each prime factor

3. If all frequencies are even, it's a perfect square

4. The square root is found by taking half the power of each prime

**Worked Example 1: Is 324 a Perfect Square?**

Prime factorisation of 324:

324 = 2 × 162

= 2 × 2 × 81

= 2 × 2 × 3 × 27

= 2 × 2 × 3 × 3 × 9

= 2 × 2 × 3 × 3 × 3 × 3

= 2² × 3⁴

**Analysis:**

Prime factor 2 appears 2 times (EVEN)

Prime factor 3 appears 4 times (EVEN)

All exponents are even → 324 is a perfect square

**Finding the Square Root:**

√324 = 2^(2/2) × 3^(4/2) = 2¹ × 3² = 2 × 9 = 18

**Verification:** 18 × 18 = 324 ✓

**Worked Example 2: Is 156 a Perfect Square?**

Prime factorisation of 156:

156 = 2 × 78

= 2 × 2 × 39

= 2 × 2 × 3 × 13

= 2² × 3¹ × 13¹

**Analysis:**

Prime factor 2 appears 2 times (EVEN)

Prime factor 3 appears 1 time (ODD)

Prime factor 13 appears 1 time (ODD)

NOT all exponents are even → 156 is NOT a perfect square

**Worked Example 3: Is 1156 a Perfect Square?**

Prime factorisation of 1156:

1156 = 4 × 289

= 2² × 289

= 2² × 17²

**Analysis:**

Prime factor 2 appears 2 times (EVEN)

Prime factor 17 appears 2 times (EVEN)

All exponents are even → 1156 is a perfect square

**Finding the Square Root:**

√1156 = 2¹ × 17¹ = 34

**Verification:** 34 × 34 = 1156 ✓

**Worked Example 4: Is 2800 a Perfect Square?**

Prime factorisation of 2800:

2800 = 28 × 100

= 4 × 7 × 100

= 2² × 7 × 10²

= 2² × 7 × (2 × 5)²

= 2² × 7 × 2² × 5²

= 2⁴ × 5² × 7¹

**Analysis:**

Prime factor 2 appears 4 times (EVEN)

Prime factor 5 appears 2 times (EVEN)

Prime factor 7 appears 1 time (ODD)

NOT all exponents are even → 2800 is NOT a perfect square

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ESTIMATING SQUARE ROOTS

Strategy for Estimating Large Square Roots

**Goal:** Find √1936 where 1936 is a perfect square, but it's too large to calculate directly.

**Step-by-Step Method:**

**Step 1: Find an approximate range**

Identify nearby perfect squares you know well

Since 40² = 1600 and 50² = 2500, we have 40 < √1936 < 50

**Step 2: Use units digit to narrow possibilities**

1936 ends in 6

Numbers that, when squared, give units digit 6 are those ending in 4 or 6

So √1936 must end in either 4 or 6

Possibilities: 44 or 46

**Step 3: Test the midpoint**

Calculate 45²:

45² = (40 + 5)²

= 40² + 2(40)(5) + 5²

= 1600 + 400 + 25

= 2025

**Step 4: Compare and refine range**

Since 2025 > 1936, we know √1936 < 45

Combined with our units digit analysis, the answer must be 44

**Step 5: Verify**

44² = (40 + 4)² = 1600 + 320 + 16 = 1936 ✓

**Answer:** √1936 = 44

Estimating Non-Perfect Squares

**Worked Example: Find √250**

**Step 1: Find the range**

100 = 10², 400 = 20²

So 10 < √250 < 20 (still a large range!)

**Step 2: Narrow down**

Calculate 15² = 225 and 16² = 256

Since 225 < 250 < 256

We have 15 < √250 < 16

**Step 3: Estimate more precisely**

256 is much closer to 250 than 225 is

250 is only 6 away from 256

√250 is approximately 15.8 or 15.9

Most accurate estimate: √250 ≈ 15.8 (but definitely less than 16)

Practical Application: Handkerchief Problem

**Problem:** Akhil has a square cloth with area 125 cm². He wants to know if he can cut a square handkerchief of side 15 cm. If not, what's the maximum integer side length?

**Solution:**

Side of original cloth: √125

Perfect squares near 125: 11² = 121 and 12² = 144

Since 121 < 125 < 144, we have 11 < √125 < 12

For a handkerchief with integer side length 15 cm, we would need area 15² = 225 cm²

But 225 > 125, so it's impossible

The largest square with integer sides has side = 11 cm (area = 121 cm²)

This handkerchief will fit in the cloth with area remaining

**Answer:** Maximum handkerchief side = 11 cm

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WORKED EXAMPLES FROM "FIGURE IT OUT" (Square Numbers)

Problem 1: Identifying Non-Perfect Squares

**Question:** Which of the following are NOT perfect squares?

(i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

**Solution:**

(i) 2032: Ends in 2 → NOT a perfect square (Perfect squares can only end in 0, 1, 4, 5, 6, 9)

(ii) 2048: Ends in 8 → NOT a perfect square

(iii) 1027: Ends in 7 → NOT a perfect square

(iv) 1089: Ends in 9 → Could be a perfect square

Prime factorisation: 1089 = 3² × 121 = 3² × 11²

All exponents are even → IS a perfect square (33²)

**Answer:** 2032, 2048, and 1027 are NOT perfect squares

Problem 2: Finding Which Square Has Last Digit 4

**Question:** Which one among 64², 108², 292², 36² has last digit 4?

**Solution:**

Numbers whose squares end in 4:

Numbers ending in 2: 2² = 4 (ends in 4) ✓

Numbers ending in 8: 8² = 64 (ends in 4) ✓

Now check each:

64²: 64 ends in 4 → 64² ends in 6 (NO)

108²: 108 ends in 8 → 108² ends in 4 ✓ (YES)

292²: 292 ends in 2 → 292² ends in 4 ✓ (YES)

36²: 36 ends in 6 → 36² ends in 6 (NO)

**Answer:** Both 108² and 292² have last digit 4

Problem 3: Finding 126² Given 125²

**Question:** Given 125² = 15625, what is 126²?

**Solution:**

Using the relationship: (n+1)² = n² + 2n + 1

126² = 125² + 2(125) + 1

= 15625 + 250 + 1

= 15876

Alternatively, looking at the options:

126² = 125² + (something)

The difference is:

(n+1)² - n² = 2n + 1 = 2(125) + 1 = 251

So 126² = 15625 + 251

**Answer:** (iv) 15625 + 251 = 15876

Problem 4: Finding Side Length from Area

**Question:** Find the length of the side of a square whose area is 441 m².

**Solution:**

If area = 441 m², then side = √