NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 01 A Square and A Cube PDF Download

Chapter 1: A Square and a Cube

Queen Ratnamanjuri had a will written that described her fortune of ratnas (precious stones) and also included a puzzle. Her son Khoisnam and their 99 relatives were invited to the reading of her will. She wanted to leave all of her ratnas to her son, but she knew that if she did so, all their relatives would pester Khoisnam forever. She hoped that she had taught him everything he needed to know about solving puzzles. She left the following note in her will -

"I have created a puzzle. If all 100 of you answer it at the same time, you will share the ratnas equally. However, if you are the first one to solve the problem, you will get to keep the entire inheritance to yourself. Good luck."

The minister took Khoisnam and his 99 relatives to a secret room in the mansion containing 100 lockers.

The minister explained - "Each person is assigned a number from 1 to 100.

• Person 1 opens every locker.
• Person 2 toggles every 2nd locker (i.e., closes it if it is open, opens it if it is closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, ... and so on).
• Person 4 toggles every 4th locker (4th, 8th, 12th, ... and so on).

This continues until all 100 get their turn.

In the end, only some lockers remain open. The open lockers reveal the code to the fortune in the safe."

Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?

Hint: Find out how many times each locker is toggled.

If a locker is toggled an odd number of times, it will be open. Otherwise, it will be closed. The number of times a locker is toggled is the same as the number of factors of the locker number. For example, for locker #6, Person 1 opens it, Person 2 closes it, Person 3 opens it and Person 6 closes it. The numbers 1, 2, 3, and 6 are factors of 6. If the number of factors is even, the locker will be toggled by an even number of people and it will eventually be closed.

Note that each factor of a number has a 'partner factor' so that the product of the pair of factors yields the given number. Here, 1 and 6 form a pair of partner factors of 6, and 2 and 3 form another pair.

Does every number have an even number of factors?

We see in some cases, like \( 2 \times 2 \), that the numbers in the pair are the same.

Can you use this insight to find more numbers with an odd number of factors?

For instance, 36 has a factor pair \( 6 \times 6 \) where both numbers are 6. Does this number have an odd number of factors? If every factor of 36 other than 6 has a different factor as its partner, then we can be sure that 36 has an odd number of factors. Check if this is true.

Hence all the following numbers have an odd number of factors -

\( 1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4, \ldots \)

A number that can be expressed as the product of a number with itself is called a square number, or simply a square. The only numbers that have an odd number of factors are the squares, because they each have one factor which, when multiplied by itself, equals the number. Therefore, every locker whose number is a square will remain open.

Teacher's Note

When a number has an odd number of factors, one factor pairs with itself. For example, in 16, the factor 4 pairs with itself: \( 4 \times 4 = 16 \). This is why only perfect squares have an odd number of factors. Try listing the factors of 12 and 25 - you'll see that 12 has an even number but 25 has an odd one.

Write the locker numbers that remain open.

1.1 Square Numbers

Why are the numbers, 1, 4, 9, 16, ..., called squares? We know that the number of unit squares in a square (the area of a square) is the product of its sides. The table below gives the areas of squares with different sides.

We use the following notation for squares.

\( 1 \times 1 = 1^2 = 1 \)

\( 2 \times 2 = 2^2 = 4 \)

\( 3 \times 3 = 3^2 = 9 \)

\( 4 \times 4 = 4^2 = 16 \)

\( 5 \times 5 = 5^2 = 25 \)

\( \vdots \)

In general, for any number \( n \), we write \( n \times n = n^2 \), which is read as 'n squared'.

Can we have a square of sidelength \( \frac{3}{5} \) or 2.5 units?

Yes, there area in square units are \( \left(\frac{3}{5}\right)^2 = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} \), and \( (2.5)^2 = (2.5) \times (2.5) = 6.25 \).

Teacher's Note

The exponent 2 means you multiply the number by itself exactly once, not twice. So \( 5^2 = 5 \times 5 = 25 \), not \( 5 \times 2 \). A common mistake is to think \( 5^2 = 5 \times 2 = 10 \); remember that the 2 is the exponent (how many times), not a second number to multiply by.

Khoisnam immediately collects word clues from these 10 lockers and reads, "The passcode consists of the first five locker numbers that were touched exactly twice."

Which are these five lockers?

The lockers that are toggled twice are the prime numbers, since each prime number has 1 and the number itself as factors. So, the code is 2-3-5-7-11.

Key Points

• A square number is made by multiplying a number by itself: \( n \times n = n^2 \). The first few square numbers are 1, 4, 9, 16, 25, 36, and so on.
• Square numbers are the only numbers that have an odd number of factors, because one factor (the square root) pairs with itself.
• We can find square numbers for any positive number, including fractions and decimals: \( \left(\frac{3}{5}\right)^2 = \frac{9}{25} \) and \( (2.5)^2 = 6.25 \).

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