A Square and A Cube Class 8: Easy Guide to Squares, Cubes & Roots
Learning about squares, cubes, and roots becomes much easier when you understand the patterns behind them. A Square and A Cube Class 8 introduces students to mathematical patterns, perfect squares, square roots, perfect cubes, cube roots, factors, and interesting number relationships.
This easy guide is designed for Class 8 students, teachers, and parents who want a quick way to understand and revise the chapter.
⭐ What Will You Learn?
In A Square and A Cube Class 8, you will explore:
🔐 The 100-locker puzzle
🔢 Factors and prime numbers
🟦 Perfect squares
√ Square roots
➕ Odd-number patterns
🔺 Triangular-number relationships
🧊 Perfect cubes
∛ Cube roots
🚕 Taxicab numbers
🧠 Mathematical patterns and problem-solving
The chapter connects number patterns with logical thinking and real mathematical problems.
🔐 The 100-Locker Puzzle
One of the most engaging parts of A Square and A Cube Class 8 is the locker puzzle.
Imagine 100 lockers, all initially closed, and 100 people numbered from 1 to 100.
The first person opens every locker. The second person changes every second locker. The third changes every third locker, and this continues until all 100 people have taken their turns.
💡 What Is the Mathematical Trick?
A locker is changed only by people whose numbers are factors of that locker number.
For example, locker 6 is affected by people 1, 2, 3, and 6.
If a locker is changed an odd number of times, it remains open. If it is changed an even number of times, it closes.
🟦 Why Do Square Numbers Stay Open?
Most numbers have factors that occur in pairs.
For example:
6 = 1 × 6 = 2 × 3
But a square number has one factor that pairs with itself:
9 = 1 × 9 = 3 × 3
Therefore, square numbers have an odd number of factors.
The lockers that remain open are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
## 🔢 Prime Numbers and the Locker Clue
The locker puzzle also introduces **prime numbers**.
A number with exactly two factors—1 and itself—is called a prime number.
The first five prime numbers are:
**2, 3, 5, 7, 11**
These numbers provide the final clue in the puzzle.
🟦 Perfect Squares: Spot the Pattern
A perfect square is obtained when a whole number is multiplied by itself.
Examples include:
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
So the sequence is
1, 4, 9, 16, 25, 36, …
🔎 Check a Number Quickly
A useful observation from **A Square and A Cube Class 8** is that a perfect square can end in:
0, 1, 4, 5, 6, or 9
It cannot end in:
2, 3, 7, or 8.
⚠️ Remember: ending in 0, 1, 4, 5, 6, or 9 does **not** automatically prove that a number is a square. It only helps you rule out some numbers.
➕ The Odd-Number Square Pattern
There is a beautiful relationship between odd numbers and squares:
1 = 1²
1 + 3 = 2²
1 + 3 + 5 = 3²
1 + 3 + 5 + 7 = 4²
Therefore:
🧠 Key Rule
The sum of the first n odd numbers is n².
This pattern provides a simple way to understand how consecutive square numbers are formed.
📈 Finding the Next Square
Suppose:
35² = 1225
To find 36², add the 36th odd number.
The 36th odd number is
2 × 36 − 1 = 71
Therefore:
36² = 1225 + 71
36² = 1296
🚀 Try This
If you know:
125² = 15,625
You can use the same pattern to work out 126² without starting the calculation from the beginning.
🔺 Square Numbers and Triangular Numbers
The chapter also shows a surprising connection between triangular numbers and squares.
Triangular numbers include:
1, 3, 6, 10, 15, …
Now add consecutive pairs:
1 + 3 = 4 = 2²
3 + 6 = 9 = 3²
6 + 10 = 16 = 4²
10 + 15 = 25 = 5²
So, the sum of two consecutive triangular numbers is a square number.
√ Square Roots Made Simple
A square root tells us which number was multiplied by itself to produce a given number.
For example:
8² = 64
Therefore:
√64 = 8
The chapter also explains that both 8 and −8 have a square of 64, although the positive root is normally used in school calculations.
✏️ Three Ways to Find Square Roots
1. Successive subtraction
Subtract consecutive odd numbers until you reach zero.
For 81:
81 − 1 − 3 − 5 − 7 − 9 − 11 − 13 − 15 − 17 = 0
There are nine steps, so:
√81 = 9.
2. Prime factorisation
Break the number into prime factors and make pairs.
For example:
324 = 2 × 2 × 3 × 3 × 3 × 3
Hence:
√324 = 18.
3. Estimation
For a large number, first locate it between two nearby perfect squares.
For example:
40² = 1600
50² = 2500
So √1936 lies between 40 and 50. Further checking gives:
√1936 = 44.
🧊 Cubic Numbers: Think in Three Dimensions
A cube is a solid shape with equal edges meeting at right angles.
To find the number of unit cubes inside a larger cube, multiply its side length three times.
For a cube with side 2:
2 × 2 × 2 = 8
For a cube with side 3:
3 × 3 × 3 = 27.
🧊 Perfect Cubes
Numbers such as:
1, 8, 27, 64, 125, …
are perfect cubes.
For example:
1³ = 1
2³ = 8
3³ = 27
A number such as 9 is not a perfect cube because it lies between:
2³ = 8
and
3³ = 27.
🚕 The Amazing Story of 1729
One memorable section of A Square and A Cube Class 8 discusses the famous number 1729, associated with mathematicians G. H. Hardy and Srinivasa Ramanujan.
The number can be written as the sum of two cubes in two different ways:
1729 = 1³ + 12³
and
1729 = 9³ + 10³.
This type of number is known as a Taxicab number.
∛ Understanding Cube Roots
A cube root reverses the process of cubing.
If:
x³ = y
then:
∛y = x
For example:
∛8 = 2
∛27 = 3
∛1000 = 10.
🔢 Prime Factorization and Cube Roots
For a perfect cube, every prime factor occurs in groups of three in its prime factorization.
For example:
∛64 = 4
∛512 = 8
∛729 = 9.
🧠 Easy Revision Tips for Class 8
To revise A Square and A Cube Class 8 effectively:
1. Learn the Basic Squares
Practice squares from 1² onward.
2. Remember the Odd-Number Rule
The sum of the first n odd numbers is n².
3. Understand Factors
Factor pairs explain why square numbers have an odd number of factors.
4. Practice Roots
Use subtraction, prime factorization, and estimation.
5. Learn Common Cubes
Memorize basic cubes such as
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6. Solve “Figure It Out.” Problems
The chapter includes questions involving square numbers, square roots, cube roots, factors, patterns, and logical reasoning.
🎯 Quick Practice Questions
Test yourself:
Q1. Which lockers remain open after all 100 people take their turns?
Q2. Is 327 a perfect square? Why?
Q3. Find √441.
Q4. What is the next square after 35²?
Q5. Find ∛512.
Q6. Is 9 a perfect cube?
Q7. Write 1729 as the sum of two cubes in two different ways.
❓ Frequently Asked Questions
What is the main topic of the A Square and A Cube class in Class 8?
The chapter focuses on mathematical patterns involving square numbers, square roots, cubes, cube roots, factors, primes, and relationships between numbers.
Why do only square-numbered lockers remain open?
Square numbers have an odd number of factors because one factor pair contains the same number twice, such as 3 × 3 = 9.
What is a perfect square?
A perfect square is a number obtained by multiplying an integer by itself, such as 25 = 5².
What is a square root?
The square root of a number is a value that, when multiplied by itself, produces that number. For example, √64 = 8.
What is a perfect cube?
A perfect cube is produced by multiplying a number by itself three times. Examples include 8 = 2³ and 27 = 3³.
What is a cube root?
A cube root is the number that, when multiplied by itself three times, gives the original number. For example, ∛27 = 3.
Why is 1729 famous?
1729 is a taxicab number because it can be represented as the sum of two positive cubes in two different ways: 1³ + 12³ and 9³ + 10³.
🌟 Final Takeaway
A Square and a Cube Class 8 is not just about memorizing squares and cubes. It teaches students to discover relationships, use factors, recognize patterns, estimate answers, and solve mathematical puzzles.
The easiest way to remember the chapter is
Find the pattern → Understand the rule → Apply the rule → Check your answer.
With regular practice, squares, cubes, and roots become much easier to understand and solve.
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