NCERT Textbook: A Square and A Cube | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

 Page 1


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. 
A SQUARE AND A CUBE
1
Chapter 1.indd   1 Chapter 1.indd   1 10-07-2025   14:06:40 10-07-2025   14:06:40
Page 2


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. 
A SQUARE AND A CUBE
1
Chapter 1.indd   1 Chapter 1.indd   1 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
2
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 × 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 × 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 × 1, 2 × 2, 3 × 3, 4 × 4, ... 
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.
6:
1 × 6
2 × 3
Factors are
1, 2, 3 and 6.
1:
1 × 1
The only factor 
is 1.
4:
1 × 4
2 × 2
Factors are 
1, 2 and 4.
9:
1 × 9
3 × 3
Factors are 
1, 3 and 9.
Chapter 1.indd   2 Chapter 1.indd   2 10-07-2025   14:06:40 10-07-2025   14:06:40
Page 3


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. 
A SQUARE AND A CUBE
1
Chapter 1.indd   1 Chapter 1.indd   1 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
2
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 × 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 × 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 × 1, 2 × 2, 3 × 3, 4 × 4, ... 
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.
6:
1 × 6
2 × 3
Factors are
1, 2, 3 and 6.
1:
1 × 1
The only factor 
is 1.
4:
1 × 4
2 × 2
Factors are 
1, 2 and 4.
9:
1 × 9
3 × 3
Factors are 
1, 3 and 9.
Chapter 1.indd   2 Chapter 1.indd   2 10-07-2025   14:06:40 10-07-2025   14:06:40
A Square and A Cube
3
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 × 1 = 1
2
 = 1 
2 × 2 = 2
2
 = 4 
3 × 3 = 3
2
 = 9, 
4 × 4 = 4
2
 = 16 
5 × 5 = 5
2
 = 25.
.
.
.
In general, for any number n, we write n × n = n
2
, which is read as ?n squared’.
Can we have a square of sidelength 
3
5
 or 2.5 units? 
Yes, there area in square units are (
3
5
)
2
 = (
3
5
) × (
3
5
) = (
9
25
), 
and (2.5)
2
 = (2.5) × (2.5) = 6.25.
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.
Sidelength 
(in units)
Area 
(in sq units)
1 1 × 1 = 1 sq. unit
2 2 × 2 = 4 sq. units
3 3 × 3 = 9 sq. units
4 4 × 4 = 16 sq. units
5 5 × 5 = 25 sq. units
10 10 × 10 = 100 sq. units
Chapter 1.indd   3 Chapter 1.indd   3 10-07-2025   14:06:40 10-07-2025   14:06:40
Page 4


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. 
A SQUARE AND A CUBE
1
Chapter 1.indd   1 Chapter 1.indd   1 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
2
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 × 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 × 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 × 1, 2 × 2, 3 × 3, 4 × 4, ... 
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.
6:
1 × 6
2 × 3
Factors are
1, 2, 3 and 6.
1:
1 × 1
The only factor 
is 1.
4:
1 × 4
2 × 2
Factors are 
1, 2 and 4.
9:
1 × 9
3 × 3
Factors are 
1, 3 and 9.
Chapter 1.indd   2 Chapter 1.indd   2 10-07-2025   14:06:40 10-07-2025   14:06:40
A Square and A Cube
3
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 × 1 = 1
2
 = 1 
2 × 2 = 2
2
 = 4 
3 × 3 = 3
2
 = 9, 
4 × 4 = 4
2
 = 16 
5 × 5 = 5
2
 = 25.
.
.
.
In general, for any number n, we write n × n = n
2
, which is read as ?n squared’.
Can we have a square of sidelength 
3
5
 or 2.5 units? 
Yes, there area in square units are (
3
5
)
2
 = (
3
5
) × (
3
5
) = (
9
25
), 
and (2.5)
2
 = (2.5) × (2.5) = 6.25.
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.
Sidelength 
(in units)
Area 
(in sq units)
1 1 × 1 = 1 sq. unit
2 2 × 2 = 4 sq. units
3 3 × 3 = 9 sq. units
4 4 × 4 = 16 sq. units
5 5 × 5 = 25 sq. units
10 10 × 10 = 100 sq. units
Chapter 1.indd   3 Chapter 1.indd   3 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
4
The squares of natural numbers are called perfect squares. For 
example, 1, 4, 9, 16, 25, … are all perfect squares.
Patterns and Properties of Perfect Squares
Find the squares of the first 30 natural numbers and fill in the table below.
1
2
 = 1 11
2
 = 121 21
2
 = 441
2
2
 = 4 12
2
 = 22
2
 = 
3
2
 = 9 13
2
 =
4
2
 = 16 14
2
 =
5
2
 = 25 15
2
 =
6
2
 = 16
2
 =
7
2
 = 17
2
 =
8
2
 = 18
2
 =
9
2
 = 19
2
 =
10
2
 = 20
2
 =
What patterns do you notice? Share your observations and make 
conjectures.
Study the squares in the table above. What are the digits in the 
units places of these numbers? All these numbers end with 0, 1, 4, 5, 6 or 
9. None of them end with 2, 3, 7 or 8.
If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?
The numbers 16 and 36 are both squares with 6 in the units place. 
However, 26, whose units digit is also 6, is not a square. Therefore, we 
cannot determine if a number is a square just by looking at the digit in 
the units place. But, the units digit can tell us when a number is not a 
square. If a number ends with 2, 3, 7, or 8, then we can definitely say that 
it is not a square.
Write 5 numbers such that you can determine by looking at their units 
digit that they are not squares.
The squares, 1
2
, 9
2
, 11
2
, 19
2
, 21
2
, and 29
2
, all have 1 in their units place. 
Write the next two squares. Notice that if a number has 1 or 9 in the 
units place, then its square ends in 1.
Let us consider square numbers ending in 6: 16 = 4
2
, 36 = 6
2
, 196 = 14
2
,  
256 = 16
2
, 576 = 24
2
, and 676 = 26
2
. 
Math 
Talk
Math 
Talk
Chapter 1.indd   4 Chapter 1.indd   4 10-07-2025   14:06:40 10-07-2025   14:06:40
Page 5


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. 
A SQUARE AND A CUBE
1
Chapter 1.indd   1 Chapter 1.indd   1 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
2
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 × 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 × 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 × 1, 2 × 2, 3 × 3, 4 × 4, ... 
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.
6:
1 × 6
2 × 3
Factors are
1, 2, 3 and 6.
1:
1 × 1
The only factor 
is 1.
4:
1 × 4
2 × 2
Factors are 
1, 2 and 4.
9:
1 × 9
3 × 3
Factors are 
1, 3 and 9.
Chapter 1.indd   2 Chapter 1.indd   2 10-07-2025   14:06:40 10-07-2025   14:06:40
A Square and A Cube
3
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 × 1 = 1
2
 = 1 
2 × 2 = 2
2
 = 4 
3 × 3 = 3
2
 = 9, 
4 × 4 = 4
2
 = 16 
5 × 5 = 5
2
 = 25.
.
.
.
In general, for any number n, we write n × n = n
2
, which is read as ?n squared’.
Can we have a square of sidelength 
3
5
 or 2.5 units? 
Yes, there area in square units are (
3
5
)
2
 = (
3
5
) × (
3
5
) = (
9
25
), 
and (2.5)
2
 = (2.5) × (2.5) = 6.25.
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.
Sidelength 
(in units)
Area 
(in sq units)
1 1 × 1 = 1 sq. unit
2 2 × 2 = 4 sq. units
3 3 × 3 = 9 sq. units
4 4 × 4 = 16 sq. units
5 5 × 5 = 25 sq. units
10 10 × 10 = 100 sq. units
Chapter 1.indd   3 Chapter 1.indd   3 10-07-2025   14:06:40 10-07-2025   14:06:40
Ganita Prakash | Grade 8 
4
The squares of natural numbers are called perfect squares. For 
example, 1, 4, 9, 16, 25, … are all perfect squares.
Patterns and Properties of Perfect Squares
Find the squares of the first 30 natural numbers and fill in the table below.
1
2
 = 1 11
2
 = 121 21
2
 = 441
2
2
 = 4 12
2
 = 22
2
 = 
3
2
 = 9 13
2
 =
4
2
 = 16 14
2
 =
5
2
 = 25 15
2
 =
6
2
 = 16
2
 =
7
2
 = 17
2
 =
8
2
 = 18
2
 =
9
2
 = 19
2
 =
10
2
 = 20
2
 =
What patterns do you notice? Share your observations and make 
conjectures.
Study the squares in the table above. What are the digits in the 
units places of these numbers? All these numbers end with 0, 1, 4, 5, 6 or 
9. None of them end with 2, 3, 7 or 8.
If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?
The numbers 16 and 36 are both squares with 6 in the units place. 
However, 26, whose units digit is also 6, is not a square. Therefore, we 
cannot determine if a number is a square just by looking at the digit in 
the units place. But, the units digit can tell us when a number is not a 
square. If a number ends with 2, 3, 7, or 8, then we can definitely say that 
it is not a square.
Write 5 numbers such that you can determine by looking at their units 
digit that they are not squares.
The squares, 1
2
, 9
2
, 11
2
, 19
2
, 21
2
, and 29
2
, all have 1 in their units place. 
Write the next two squares. Notice that if a number has 1 or 9 in the 
units place, then its square ends in 1.
Let us consider square numbers ending in 6: 16 = 4
2
, 36 = 6
2
, 196 = 14
2
,  
256 = 16
2
, 576 = 24
2
, and 676 = 26
2
. 
Math 
Talk
Math 
Talk
Chapter 1.indd   4 Chapter 1.indd   4 10-07-2025   14:06:40 10-07-2025   14:06:40
A Square and A Cube
5
Which of the following numbers have the digit 6 in the units place?
 (i)  38
2
  (ii)  34
2
   (iii)  46
2
    (iv)  56
2
     (v)  74
2
    (vi)  82
2
Find more such patterns by observing the numbers and their squares 
from the table you filled earlier.
Consider the following numbers and their squares.
If a number contains 3 zeros at the end, how many zeros will its square 
have at the end?
What do you notice about the number of zeros at the end of a number 
and the number of zeros at the end of its square? Will this always 
happen? Can we say that squares can only have an even number of 
zeros at the end? 
What can you say about the parity of a number and its square?
Perfect Squares and Odd Numbers
Let us explore the differences between consecutive squares. What do 
you notice?
4 – 1 = 3  9 – 4 = 5  16 – 9 = 7  25 – 16 = 9
See if this pattern continues for the next few square numbers.
From this we observe that adding  consecutive odd numbers starting 
from 1 gives consecutive square numbers, as shown below.
10
2
 = 100
20
2
 = 400
40
2
 = 800
100
2
 = 10000
200
2
 = 40000
700
2
 = 49000
900
2
 = 81000 
We have 
one zero.
We have 
two zeroes.
But we have 
two zeroes.
But we have 
four zeroes.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36.
Chapter 1.indd   5 Chapter 1.indd   5 10-07-2025   14:06:40 10-07-2025   14:06:40