NCERT Solutions Class 6 Maths Chapter 1 Patterns in Mathematics

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 6 · M AT H S

NCERT Solutions

Chapter 1: Patterns in
Mathematics

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

1 – 12 7 27 English

Solutions, notes, sample papers & more at 36 pages

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 1: Patterns in Mathematics
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics from the
NCERT textbook Ganita Prakash. Every Figure it Out question from pages 2, 3, 5, 8, 11 and 12 is solved with
reasoning and diagrams.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 1 – 12

SECTIONS QUESTIONS

7 27

MEDIUM

English

Figure it Out — Page 2
Section 1.1 What is Mathematics?

MATH TALK

Q1 Can you think of other examples where mathematics helps us in our everyday lives?

Yes — mathematics is hidden in almost everything we do in a day. Here are examples you can
discuss in class:

Shopping — adding up a bill, working out a discount, checking the change the shopkeeper
returns.
Cooking — measuring flour and water, doubling a recipe for twice the number of guests.
Travel — reading a railway timetable, calculating speed, distance and time, estimating petrol
needed.
Money — saving pocket money, making a monthly budget, understanding interest in a bank
passbook.
Sports — run rate in cricket, batting average, points table, angles while playing carrom or
billiards.
At home — measuring a room, counting tiles for a floor, finding the area of a plot of land.
Art and design — rangoli and kolam patterns, symmetry in temple architecture, repeating
designs on a saree border.
Technology — barcodes and QR codes, digital photographs, weather forecasting, passwords
and secret codes.

Page 1 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Math Talk: Make a list of every place you used a number today — from the time you
woke up to the price of your snack. Compare your list with a friend's.

Q2 How has mathematics helped propel humanity forward? (You might think of
examples involving: carrying out scientific experiments; running our economy and
democracy; building bridges, houses or other complex structures; making TVs,
mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)

Mathematics does not only describe patterns — its explanations become tools, and those tools
have changed how human beings live. A few of the biggest ways:

Science and space — patterns in the motion of stars and planets led to the theory of
gravitation. That single explanation lets us launch satellites, run GPS on a phone and send
rockets to the Moon and Mars.
Economy and democracy — banking, taxes, budgets, the census and the counting of crores
of votes in an election all rest on arithmetic and statistics.
Engineering and construction — bridges, dams and tall buildings are designed with
geometry and careful calculation of loads, so that they stand safely for decades.
Machines and devices — computers, mobile phones and TVs work on binary numbers (0
and 1) and coded signals; a bicycle's gears, a car's engine and a plane's wing are all designed
mathematically.
Time — calendars, clocks and the prediction of eclipses come from patterns in the motion of
the Sun and Moon.
Medicine — statistics decides whether a new medicine really works, and reading patterns in
genomes helps doctors diagnose and cure diseases.

Math Talk: This question is meant for discussion. Pick one item from the list, find
out one real Indian example of it (say, the Chandrayaan mission or the Statue of
Unity), and tell the class what mathematics was needed for it.

Figure it Out — Page 3

Page 2 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Section 1.2 Patterns in Numbers

MATH TALK

Q1 Can you recognise the pattern in each of the sequences in Table 1?

1, 1, 1, 1, 1, 1, 1, … (All 1’s)

1, 2, 3, 4, 5, 6, 7, … (Counting numbers)

1, 3, 5, 7, 9, 11, 13, … (Odd numbers)

2, 4, 6, 8, 10, 12, 14, … (Even numbers)

1, 3, 6, 10, 15, 21, 28, … (Triangular numbers)

1, 4, 9, 16, 25, 36, 49, … (Squares)

1, 8, 27, 64, 125, 216, … (Cubes)

1, 2, 3, 5, 8, 13, 21, … (Virahānka numbers)

1, 2, 4, 8, 16, 32, 64, … (Powers of 2)

1, 3, 9, 27, 81, 243, 729, … (Powers of 3)

Table 1 of the chapter — ten number sequences with their names.

Yes. Each sequence in Table 1 is built by one simple rule, repeated again and again:

All 1's — the same number 1 is written every time.
Counting numbers — add 1 to the previous number.
Odd numbers — start at 1 and keep adding 2.
Even numbers — start at 2 and keep adding 2.
Triangular numbers — add 2, then 3, then 4, then 5, … (the amount added grows by one
each time).
Squares — multiply a counting number by itself: 1×1, 2×2, 3×3, …
Cubes — multiply a counting number by itself three times: 1×1×1, 2×2×2, 3×3×3, …
Virahānka numbers — each number is the sum of the two numbers before it (2+3 = 5, 3+5 =
8, 5+8 = 13).
Powers of 2 — double the previous number.
Powers of 3 — multiply the previous number by 3.

Page 3 of 36

as e
Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

co m
m.
Why it matters: once you can say the rule in words, you can continue the sequence

m as e
l
as far as you like — you no longer need to remember the numbers.

m .co a g
l a se
a g
Q2 Rewrite each sequence of Table 1 in your notebook, along with the next three

com
ag
numbers in each sequence! After each sequence, write in your own words what is

m .
e
the rule for forming the numbers in the sequence.

g l as
a
m
1, 1, 1, 1, 1, 1, 1, … (All 1’s)

m .co
se
1, 2, 3, 4, 5, 6, 7, … (Counting numbers)

c1,om l a
e
.
m 3, 5, 7, 9, 11, 13, … ag
(Odd numbers)

g l as
a 2, 4, 6, 8, 10, 12, 14, … (Even numbers)

a s
com agl
1, 3, 6, 10, 15, 21, 28, … (Triangular numbers)

m .
ase
1, 4, 9, 16, 25, 36, 49, … (Squares)

1, 8, 27, 64, 125, 216, … agl (Cubes)

1, 2, 3, 5, 8, 13, 21, … (Virahānka numbers)
co m
m .
m ase
.co l
1, 2, 4, 8, 16, 32, 64, … (Powers of 2)

a g
a s em 1, 3, 9, 27, 81, 243, 729, … (Powers of 3)

agl
Table 1 of the chapter — ten number sequences with their names.

se m
com g l a
m . a
gl ase
a

The next three numbers of every sequence are shown in orange, and the rule is written beside
it.
co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
em . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 4 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

NUMBER SEQUENCE GIVEN NEXT THREE RULE IN YOUR OWN WORDS
SEQUENCE IN TABLE 1 NUMBERS

All 1's 1, 1, 1, 1, 1, 1, 1, … 1, 1, 1 Every number is 1. Nothing ever
changes.

Counting 1, 2, 3, 4, 5, 6, 7, … 8, 9, 10 Add 1 to the previous number.
numbers

Odd numbers 1, 3, 5, 7, 9, 11, 13, … 15, 17, 19 Start at 1 and keep adding 2. These
are the numbers not divisible by 2.

Even numbers 2, 4, 6, 8, 10, 12, 14, … 16, 18, 20 Start at 2 and keep adding 2. These
are the multiples of 2.

Triangular 1, 3, 6, 10, 15, 21, 28, … 36, 45, 55 Add 2, then 3, then 4, … The nth
numbers number is 1 + 2 + 3 + … + n.

Squares 1, 4, 9, 16, 25, 36, 49, … 64, 81, 100 Multiply a counting number by
itself: n × n.

Cubes 1, 8, 27, 64, 125, 216, … 343, 512, 729 Multiply a counting number by
itself three times: n × n × n.

Virahānka 1, 2, 3, 5, 8, 13, 21, … 34, 55, 89 Each number is the sum of the two
numbers numbers just before it.

Powers of 2 1, 2, 4, 8, 16, 32, 64, … 128, 256, 512 Double the previous number.

Powers of 3 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683 Multiply the previous number by 3.
…

Tip: To check a triangular number quickly, use Tn = n × (n + 1) ÷ 2. For example the
10th triangular number is 10 × 11 ÷ 2 = 55 — exactly what the pattern gives.

Figure it Out — Page 5

Page 5 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Section 1.3 Visualising Number Sequences

MATH TALK

Q1 Copy the pictorial representations of the number sequences in Table 2 in your
notebook, and draw the next picture for each sequence!

All 1’s
1 1 1 1 1

Counting
1 2 3 4 5 numbers

Odd numbers
1 3 5 7 9

Even numbers
2 4 6 8 10

Triangular
numbers

1 3 6 10 15

Squares

1 4 9 16 25

Cubes

1 8 27 64 125

Table 2 of the chapter — the first seven sequences of Table 1 drawn as pictures.

The sixth picture of each sequence is described below — draw it just after the fifth picture of
Table 2.

Page 6 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

SEQUENCE PICTURE ALREADY DRAWN NEXT PICTURE TO DRAW

All 1's 1, 1, 1, 1, 1 one more single dot → 1

Counting numbers 1, 2, 3, 4, 5 a line of 6 dots → 6

Odd numbers 1, 3, 5, 7, 9 an “L / T” shape of 11 dots → 11

Even numbers 2, 4, 6, 8, 10 two rows of 6 dots → 12

Triangular numbers 1, 3, 6, 10, 15 a triangle of 6 rows → 21

Squares 1, 4, 9, 16, 25 a 6 × 6 square of dots → 36

Cubes 1, 8, 27, 64, 125 a 6 × 6 × 6 cube of dots → 216

1
3
6
10
15
Triangular numbers 1, 3, 6, 10, 15 — each new picture adds one more row at the bottom. The next
picture has a 6-dot row, giving 21.

1
4
9
16
25
Square numbers 1, 4, 9, 16, 25 — each new picture adds one more row and one more column. The
next picture is 6 × 6 = 36.

Page 7 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q2 Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called
square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?

Because of the shape the dots make when you arrange them neatly:

Triangular numbers — that many dots can be packed into a perfect triangle: rows of 1, 2, 3,
4, … dots stacked one below the other. So 1 + 2 + 3 = 6 dots make a triangle of 3 rows, and 6
is a triangular number.
Square numbers — that many dots can be packed into a perfect square grid with the same
number of rows and columns. 4 = 2 × 2, 9 = 3 × 3, 25 = 5 × 5.
Cubes — that many dots (or small unit cubes) can be packed into a solid cube with equal
length, breadth and height. 8 = 2 × 2 × 2, 27 = 3 × 3 × 3, 125 = 5 × 5 × 5.

The idea: a number gets its name from the shape it can be arranged into. The same
number can therefore have more than one name — see the very next question.

Q3 You will have noticed that 36 is both a triangular number and a square number!
That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make
pictures in your notebook illustrating this!

Yes — 36 wears two hats at once:

As a triangle: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 (8 rows)

As a square: 6 × 6 = 36 (6 rows of 6)

36 = 1+2+ ⋯+8 (triangle) 36 = 6 × 6 (square)

Page 8 of 36

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Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

The same 36 dots — arranged as a triangle of 8 rows, and as a 6 × 6 square.
co m
e m.
m l as
.co a g
This shows that a number can play different roles depending on the context. Other numbers

se m
a
can also be shown in more than one way:

g l
a 1 is a triangular number, a square number and a cube.
16 is a square (4 × 4) and also a rectangle (2 × 8).

co m
. ag
64 is both a square (8 × 8) and a cube (4 × 4 × 4).

se m
l a
agthe next number that is both triangular and square is
Did you know? After 1 and 36,
1225 (= 35 × 35, and also 1 + 2 + 3 + … + 49). Such numbers are very rare.

co m
se m.
o m l a
Q4 m
.c would you call the following sequence of numbers? a1,g7, 19, 37 … That’s right,
se they are called hexagonal numbers! Draw these in your notebook. What is the next
What
l a
ag number in the sequence?

m a s
m.co agl
l a se
ag

co m
m .
m as e
.co a g l
se m
g l a
a 1 7 19 37
se m
com g l a
m . a
ase
The sequence of dot pictures printed with this question, labelled 1, 7, 19 and 37.

agl

c om
.
mfigure) with one dot in
s e
These are hexagonal numbers — the dots form a hexagon (a six-sided

. com and rings of dots around it.
the centre
a gla
a sem
agl c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 9 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1

7

19

37

Hexagonal numbers 1, 7, 19, 37. Each new picture adds one more ring around the outside; the rings
hold 6, 12 and 18 dots.

Look at how much is added each time:

1 +6→ 7 +12→ 19 +18→ 37 +24→ 61

The additions are 6, 12, 18, 24, … (multiples of 6). So the next ring holds 24 dots and

37 + 24 = 61

Answer: the next hexagonal number is 61. The sequence continues 1, 7, 19, 37, 61, 91, 127, …

Q5 Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of
3?

Yes. The trick is to show that each picture is made of copies of the picture before it.
Powers of 2 — every picture is two copies of the previous one, so the number of dots doubles: 1,
2, 4, 8, 16, …

Page 10 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1 2
4 8

16
Powers of 2: 1, 2, 4, 8, 16. Each step is simply the previous picture, taken twice.

Powers of 3 — every picture is three copies of the previous one, so the number of dots triples: 1,
3, 9, 27, …

1
3
9

27

Powers of 3: 1, 3, 9, 27. A single dot; three dots in a triangle; three of those triangles; three of those
groups again.

Why this picture is useful: it shows at a glance why the numbers grow so fast — at
every step the whole picture is copied 2 (or 3) times. This is called doubling and
tripling.

In-text Questions — Page 7
Section 1.4 Relations among Number Sequences

Q1 By drawing a similar picture, can you say what is the sum of the first 10 odd
numbers?

Adding odd numbers starting from 1 always gives a square number, and the answer is the
square of how many odd numbers you added.

Page 11 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1 + 3 + 5 + 7 + 9 + 11 = 36
A 6 × 6 square of dots split into bent “L” layers of 1, 3, 5, 7, 9 and 11 dots. This is why 1 + 3 + 5 + 7 + 9 +
11 = 36.

Here we are adding the first 10 odd numbers, so the picture is a 10 × 10 square:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 10 × 10 = 100

Answer: 100.

Q2 Now by imagining a similar picture, or by drawing it partially, as needed, can you
say what is the sum of the first 100 odd numbers?

We do not have to draw all the dots — the rule is now clear. The sum of the first n odd numbers
is n × n. So for 100 odd numbers we imagine a 100 × 100 square:

1 + 3 + 5 + … + 197 + 199 = 100 × 100 = 10 000

Page 12 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Answer: 10,000.

The big idea: the picture is not just decoration. Because such a square can be drawn
for any size, the picture proves the pattern goes on forever — for 10, for 100, for a
million odd numbers.

Figure it Out — Pages 8 & 9
Section 1.4 Relations among Number Sequences

TRY THIS

Q1 Can you find a similar pictorial explanation for why adding counting numbers up
and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …, gives square numbers?

Yes. Take a square grid of dots and colour it along the slanting diagonals instead of along L-
shapes.

1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36
A 6 × 6 square cut along its diagonals. The diagonals contain 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots —
counting up and then down.

Count the dots on each slanting line of a 6 × 6 square. Starting from the top-left corner they
contain

1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 dots

Every dot of the square lies on exactly one of these diagonals, so

Page 13 of 36

as e
Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

co m
e m.
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 6 × 6 = 36
m l as
.co a g
a s em
The same picture can be drawn for any size of square, which explains why adding counting

agl
numbers up and then down always gives a square number.

. c om ag
a s em picture, or drawing it partially, as needed, can
By imagining a large version of your

agl of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
Q2
you see what will be the value

. c om
In the picture above, the largest number in the middle tells us the side of them
s e square. Here the

. com is 100, so imagine a 100 × 100 square of dots.
middle number
a gla
a s em
agl 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1 = 100 × 100 = 10 000
m a s
m.co agl
se
Answer: 10,000.

g l a
a
Q3 Which sequence do you get when you start to add the All 1’s sequence up? What
co m
sequence do you get when you add the All 1’s sequence up and down?
m .
m as e
. c o a g l
a s em

a gl (a) Adding the All 1's sequence up:
se m
com g l a
. a
1=1
m
ase
agl
1+1=2

1+1+1=3

co m
.
1+1+1+1=4 …
em
om1, 2, 3, 4, 5, … — the counting numbers.
cget g l as
. a
sem(b) Adding the All 1's sequence up and down:
We

a
agl c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 14 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1=1

1+1+1=3

1+1+1+1+1=5

1+1+1+1+1+1+1=7 …

We get 1, 3, 5, 7, 9, … — the odd numbers.

Why: going “up and down” to a height of n uses n ones on the way up and n − 1 ones
on the way down, that is n + (n − 1) = 2n − 1 ones in all — and 2n − 1 is exactly the nth
odd number.

Q4 Which sequence do you get when you start to add the counting numbers up? Can
you give a smaller pictorial explanation?

1=1

1+2=3

1+2+3=6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15 …

We get 1, 3, 6, 10, 15, … — the triangular numbers.

Page 15 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1 + 2 + 3 + 4 + 5 = 15
Rows of 1, 2, 3, 4 and 5 dots stacked one below the other make a triangle of 15 dots. That is why the
sums are called triangular numbers.

The picture explains it immediately: putting a row of 1 dot, then 2 dots, then 3 dots, … one
below the other builds the triangle itself. So the running totals of the counting numbers are
exactly the triangular numbers.

Q5 What happens when you add up pairs of consecutive triangular numbers? That is,
take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … Which sequence do you get? Why? Can you explain
it with a picture?

1+3=4 3+6=9 6 + 10 = 16 10 + 15 = 25 15 + 21 = 36

We get 4, 9, 16, 25, 36, … — the square numbers.

Page 16 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

10 + 15 = 25
A 5 × 5 square split by its diagonal: the green part is a triangle of 15 dots, the orange part a triangle
of 10 dots. Together 10 + 15 = 25.

Why it happens: cut a square of dots along its main diagonal. One piece (including
the diagonal) is a triangle of dots, the other piece is the previous, slightly smaller
triangle. So two consecutive triangular numbers always fit together to fill a square
exactly.

Page 17 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q6 What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 +
2, 1 + 2 + 4, 1 + 2 + 4 + 8, …? Now add 1 to each of these numbers — what numbers do
you get? Why does this happen?

Step 1 — the running totals:

1=1

1+2=3

1+2+4=7

1 + 2 + 4 + 8 = 15

1 + 2 + 4 + 8 + 16 = 31 …

We get 1, 3, 7, 15, 31, … — each number is one less than a power of 2.
Step 2 — add 1 to each:

1+1=2 3+1=4 7 + 1 = 8 15 + 1 = 16 31 + 1 = 32

We get 2, 4, 8, 16, 32, … — the powers of 2 again!

Why it happens: think of the doubling picture. Take the pile of 1 + 2 + 4 + 8 dots and
add one more single dot. Now 1 + 1 = 2, and that 2 joins the next 2 to make 4, and
that 4 joins the next 4 to make 8, and so on — everything collapses into a single pile
of 16. In short:
1 + 2 + 4 + … + 2n−1 = 2n − 1, so adding 1 gives exactly 2n.

Page 18 of 36

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Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

co m
m.
What happens when you multiply the triangular numbers by 6 and add 1? Which
e
Q7

m l as
.co
sequence do you get? Can you explain it with a picture?

a g
se m
g l a
a

(1 × 6) + 1 = 7
co m
e m . ag
as
(3 × 6) + 1 = 19

(6 × 6) + 1 = 37 a g l
(10 × 6) + 1 = 61
co m
em.
as
(15 × 6) + 1 = 91
m l
m .co a g
l a se
We get 7, 19, 37, 61, 91, … — the hexagonal numbers (from question 4 of page 5).
a g
m a s
m .co agl
l a se
ag

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
m . co agl
l as 6
e × 6 + 1 = 37
a g

co m
m .
m ase
.co


a g l Page 19 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

The hexagonal number 37 = 6 × 6 + 1. The black dot is the centre; the six coloured wedges around it
are six identical triangles of 6 dots each.

Why it happens: a hexagon is made of six identical triangles meeting at one
central point. Remove the centre dot and the remaining dots split into 6 equal
triangular groups. So every hexagonal number is “6 × a triangular number, plus the 1
dot in the middle”.

Q8 What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7
+ 19, 1 + 7 + 19 + 37, …? Which sequence do you get? Can you explain it using a
picture of a cube?

1=1

1+7=8

1 + 7 + 19 = 27

1 + 7 + 19 + 37 = 64

1 + 7 + 19 + 37 + 61 = 125

We get 1, 8, 27, 64, 125, … — the cube numbers.

Why it happens (the cube picture): build a cube out of small unit cubes, one shell
at a time.

Start with 1 small cube — that is a 1 × 1 × 1 cube.
To turn it into a 2 × 2 × 2 cube you must add 7 more small cubes (8 − 1 = 7).
To turn that into a 3 × 3 × 3 cube you add 19 more (27 − 8 = 19).
To reach 4 × 4 × 4 you add 37 more (64 − 27 = 37).

The number of cubes in each new shell is exactly 1, 7, 19, 37, … — the hexagonal
numbers. So adding hexagonal numbers rebuilds the cube layer by layer, and the
totals are the cubes.

Check it yourself: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37, 125 − 64 = 61. The differences
of consecutive cubes are the hexagonal numbers.

Page 20 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q9 Find your own patterns or relations in and among the sequences in Table 1. Can you
explain why they happen with a picture or otherwise?

1, 1, 1, 1, 1, 1, 1, … (All 1’s)

1, 2, 3, 4, 5, 6, 7, … (Counting numbers)

1, 3, 5, 7, 9, 11, 13, … (Odd numbers)

2, 4, 6, 8, 10, 12, 14, … (Even numbers)

1, 3, 6, 10, 15, 21, 28, … (Triangular numbers)

1, 4, 9, 16, 25, 36, 49, … (Squares)

1, 8, 27, 64, 125, 216, … (Cubes)

1, 2, 3, 5, 8, 13, 21, … (Virahānka numbers)

1, 2, 4, 8, 16, 32, 64, … (Powers of 2)

1, 3, 9, 27, 81, 243, 729, … (Powers of 3)

Table 1 of the chapter — ten number sequences with their names.

Here are some patterns you can discover — try to find more of your own.

Adding even numbers: 2, 2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20 → 2, 6, 12, 20, 30 …
These are exactly double the triangular numbers, and each one is a rectangle of dots: 1 ×
2, 2 × 3, 3 × 4, 4 × 5, …
Differences of squares: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9 → the odd numbers. (This
is the same L-shape picture as on page 7, read backwards.)
Differences of cubes: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37 → the hexagonal numbers.
Adding cubes: 1 + 8 = 9, 1 + 8 + 27 = 36, 1 + 8 + 27 + 64 = 100 → squares of the triangular
numbers: 3², 6², 10².
Triangular × 8 + 1: (1×8)+1 = 9, (3×8)+1 = 25, (6×8)+1 = 49, (10×8)+1 = 81 → the squares of
odd numbers.
Virahānka numbers: add the first few — 1 + 2 + 3 + 5 = 11, which is 2 less than 13, the next-
but-one Virahānka number. Try it again: 1 + 2 + 3 + 5 + 8 = 19 = 21 − 2. The pattern holds!
Powers of 2 and odd/even: every power of 2 after 1 is even, and no power of 3 is ever even.

Page 21 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Try This: Choose any two sequences from Table 1 and add them term by term. For
example odd + even: 1 + 2 = 3, 3 + 4 = 7, 5 + 6 = 11 → 3, 7, 11, 15 … a new sequence
that goes up by 4 each time. Explain why.

Figure it Out — Page 11

Page 22 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Section 1.5 Patterns in Shapes

MATH TALK

Q1 Can you recognise the pattern in each of the sequences in Table 3?

Regular Polygons

Triangle Quadrilateral Pentagon Hexagon

Heptagon Octagon Nonagon Decagon

Complete Graphs

K2 K3 K4 K5 K6

Stacked Squares

Stacked Triangles

Koch Snowflake

Table 3 of the chapter — five shape sequences.

Yes. Each shape sequence follows one clear rule:

Page 23 of 36

as e
Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

SHAPE RULE FOR GOING TO THE NEXT SHAPE
. c om RELATED NUMBER
SEQUENCE
m a s
SEQUENCE
em
Regular m.
co Add one more side (and one more corner) each time — ag3,l 4, 5, 6, 7, 8, 9, 10 …
l a se
a g
Polygons triangle, quadrilateral, pentagon, hexagon, …

Complete Add one more dot, then join it to every dot already 1, 3, 6, 10, 15 …

m
.co ag
Graphs Kn there. (triangular)

a s em
agl
Stacked Add one more row and one more column of small 1, 4, 9, 16, 25 …
Squares squares. (squares)

Stacked Add one more row of small triangles at the bottom.
co m
m.
1, 4, 9, 16, 25 …

se
Triangles (squares)

com Replace every straight line segment by a “speed bump” g l a
. a 3, 12, 48, 192 … (3 ×
em
Koch Snowflake

a s
agl
of 4 smaller segments. powers of 4)

m a s
m.co agl
l a se
ag
1
4
co m
m .
m
9
as e
.co 16
a g l
se m
g l a 25
a
Stacked Squares: 1, 4, 9, 16, 25 small squares.
se m
com g l a
m . a
ase
agl

co m
1
m .
m 4
as e
.co a g l
m 9
ase
agl
16
25
.c
s e m
comyellow together). a
. agl
Stacked Triangles: 1, 4, 9, 16, 25 small triangles (upward-pointing green and downward-pointing

e m
g l as
a

co m
m .
m ase
.co


a g l Page 24 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q2 Try and redraw each sequence in Table 3 in your notebook. Can you draw the next
shape in each sequence? Why or why not? After each sequence, describe in your
own words what is the rule or pattern for forming the shapes in the sequence.

Regular Polygons

Triangle Quadrilateral Pentagon Hexagon

Heptagon Octagon Nonagon Decagon

Complete Graphs

K2 K3 K4 K5 K6

Stacked Squares

Stacked Triangles

Koch Snowflake

Table 3 of the chapter — five shape sequences.

Page 25 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Yes for four of them, and “only in principle” for the fifth.

Regular Polygons — the next shape after the decagon is an 11-sided regular polygon
(hendecagon). Easy to draw: keep all sides equal and all corners alike.
Complete Graphs — the next is K7: mark 7 dots on a circle and join every pair. It has 21 lines,
so the drawing gets crowded but it can certainly be done.
Stacked Squares — the next has 6 rows of 6 small squares = 36.
Stacked Triangles — the next has 6 rows, 36 small triangles in all.
Koch Snowflake — the next stage can be drawn, but each stage has 4 times as many line
segments (3 → 12 → 48 → 192 → 768 …) and each is 3 times shorter. After a few stages the
bumps become too tiny to draw by hand or even to see. So the rule is perfectly clear, but the
picture soon goes beyond what a pencil can manage.

The rule in one line each: polygons — add a side; complete graphs — add a dot
and join it to all the others; stacked squares/triangles — add a row; Koch snowflake
— replace every segment by a bump made of 4 segments.

Figure it Out — Pages 11 & 12

Page 26 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Section 1.6 Relation to Number Sequences

TRY THIS

Q1 Count the number of sides in each shape in the sequence of Regular Polygons.
Which number sequence do you get? What about the number of corners in each
shape in the sequence of Regular Polygons? Do you get the same number
sequence? Can you explain why this happens?

Regular Polygons

Triangle Quadrilateral Pentagon Hexagon

Heptagon Octagon Nonagon Decagon

Complete Graphs

K2 K3 K4 K5 K6

Stacked Squares

Stacked Triangles

Koch Snowflake

Page 27 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Table 3 of the chapter — five shape sequences.

SHAPE TRIANGLE QUADRI ‐ PENTAGON HEXAGON HEPTAGON OCTAGON NONAGON DECAGON
LATERAL

Sides 3 4 5 6 7 8 9 10

Corners 3 4 5 6 7 8 9 10

Both rows give the same sequence — the counting numbers starting from 3: 3, 4, 5, 6, 7, 8, 9,
10, …

Why the two counts are always equal: in a closed figure made of straight lines,
walk once around the boundary. Every side ends at a corner, and at every corner
exactly two sides meet. So sides and corners come in a perfect one-to-one pairing —
each side can be matched with the corner at its end. Therefore number of sides =
number of corners (vertices) in any closed figure, not just regular ones.

Page 28 of 36

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Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

co m
m.
Count the number of lines in each shape in the sequence of Complete Graphs.
e
Q2

m l as
.co
Which number sequence do you get? Can you explain why?

a g
a s em
a gl Regular Polygons

com
e m . ag
g l as
Triangle
a Quadrilateral Pentagon Hexagon

co m
em.
m l as
m .co a g
l a se
a g Heptagon Octagon Nonagon Decagon

m a s
Complete Graphs
m .co agl
l a se
ag

co m
K2 K3 K4 K5
m . K6

mStacked Squares as e
. c o a g l
se m
g l a
a
se m
com g l a
Stacked Triangles
m . a
ase
agl

co m
m .
m as e
.co
Koch Snowflake
a g l
se m
g l a
a c
m .
m a s e
co agl
Table 3 of the chapter — five shape sequences.

m .
as e
a g l

co m
m .
m ase
.co


a g l Page 29 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

COMPLETE GRAPH K2 K3 K4 K5 K6 K7

Dots 2 3 4 5 6 7

Lines 1 3 6 10 15 21

We get 1, 3, 6, 10, 15, 21, … — the triangular numbers.

Why it happens: go from one graph to the next. When you add a new dot to K2
(which has 2 dots), the new dot must be joined to those 2 dots, so 2 new lines
appear. Adding the next dot brings 3 new lines, then 4, and so on. So the totals are

1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10 …

which are exactly the running sums of the counting numbers — the triangular
numbers.

Another way to see it: in Kn each of the n dots is joined to the other (n − 1) dots.
That counts every line twice (once from each end), so the number of lines is n × (n −
1) ÷ 2. For K6: 6 × 5 ÷ 2 = 15. ✔

Page 30 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q3 How many little squares are there in each shape of the sequence of Stacked
Squares? Which number sequence does this give? Can you explain why?

Regular Polygons

Triangle Quadrilateral Pentagon Hexagon

Heptagon Octagon Nonagon Decagon

Complete Graphs

K2 K3 K4 K5 K6

Stacked Squares

Stacked Triangles

Koch Snowflake

Table 3 of the chapter — five shape sequences.

Page 31 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

1
4
9
16
25

Stacked Squares — 1, 4, 9, 16 and 25 little squares.

1, 4, 9, 16, 25, 36 …

We get the square numbers.

Why it happens: the nth shape is a big square made of n rows, and each row holds n
little squares. So the total is n × n = n². For example the 4th shape has 4 rows of 4
little squares = 16.

Page 32 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q4 How many little triangles are there in each shape of the sequence of Stacked
Triangles? Which number sequence does this give? Can you explain why? (Hint: In
each shape in the sequence, how many triangles are there in each row?)

Regular Polygons

Triangle Quadrilateral Pentagon Hexagon

Heptagon Octagon Nonagon Decagon

Complete Graphs

K2 K3 K4 K5 K6

Stacked Squares

Stacked Triangles

Koch Snowflake

Table 3 of the chapter — five shape sequences.

Page 33 of 36

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Class 6 Maths Chapter 1 Patterns in Mathematics
a g l AglaSem · NCERT Solutions

co m
m.

m as e
.co a g l
se m
g l a
a
1
co m
4
e m . ag
9
g l as
a 16
25
c o m
.
m together: 1, 4,
s e
Stacked Triangles — count the upward-pointing and downward-pointing little triangles

. com a gla
9, 16, 25.

a s em
a gl
1, 4, 9, 16, 25 …

m a s
m .co agl
se
Again we get the square numbers.

g l a
a
Why it happens (follow the hint): count the little triangles row by row.

co m
m .
as e
com2 → 3 triangles (2 pointing up, 1 pointing down)
Row 1 → 1 triangle
. a g l
sem
Row
a
agl Row 3 → 5 triangles (3 up, 2 down)
Row 4 → 7 triangles (4 up, 3 down)
se m
com g l a
m . a
e
9 s… — the odd numbers. And we already know from
The rows contain 1, 3, 5, 7, a
g l
page 7 that adding theaodd numbers gives the squares:

co m
m .
e
1 + 3 + 5 + 7 = 16 = 4 × 4
m l as
.co
m That is why a stacked triangle of n rows contains n² little triangles. a g
l a se
ag
.c
s e m
m a
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 34 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Q5 To get from one shape to the next shape in the Koch Snowflake sequence, one
replaces each line segment ‘—’ by a ‘speed bump’. As one does this more and more
times, the changes become tinier and tinier with very very small line segments.
How many total line segments are there in each shape of the Koch Snowflake? What
is the corresponding number sequence?

The Koch Snowflake sequence of Table 3 — the first five shapes.

The starting shape is an equilateral triangle with 3 line segments. Each step replaces every
single segment by a speed bump made of 4 smaller segments, so the count is multiplied by 4
every time.

STAGE 1ST 2ND 3RD 4TH 5TH

Line segments 3 12 48 192 768

As a product 3 3×4 3×4×4 3×4×4×4 3×4×4×4×4

The number sequence is 3, 12, 48, 192, 768, …, that is 3 × the powers of 4. (This sequence is not
in Table 1.)

Why: one segment → 4 segments. So if a stage has k segments, the next stage has
4k. Starting from 3, the counts are 3, 3×4, 3×4², 3×4³, …

Chapter at a glance
Mathematics is the search for patterns and for the explanations of why those patterns exist.
The most basic patterns in mathematics are number sequences — counting numbers, odd,
even, triangular, square, cube, Virahānka numbers and powers of 2.
Number sequences are often linked in beautiful ways. Adding odd numbers gives squares;
adding counting numbers gives triangular numbers; adding hexagonal numbers gives
cubes.
Drawing a sequence as a picture often explains why a pattern works — a proof you can see.

Page 35 of 36

Class 6 Maths Chapter 1 Patterns in Mathematics AglaSem · NCERT Solutions

Shape sequences — regular polygons, complete graphs, stacked squares and triangles, the
Koch snowflake — are the second big family of patterns, and each is tied to a number
sequence.

Quick revision

SEQUENCE FIRST FEW HOW TO GET THE NEXT PICTURE IT AS
TERMS TERM

Counting 1, 2, 3, 4, 5 +1 a line of dots
numbers

Odd numbers 1, 3, 5, 7, 9 +2 bent “L” layers of a square

Even numbers 2, 4, 6, 8, 10 +2 two equal rows of dots

Triangular 1, 3, 6, 10, 15 add the next counting number a triangle of dots

Squares 1, 4, 9, 16, 25 add the next odd number a square grid of dots

Cubes 1, 8, 27, 64, 125 add the next hexagonal number a solid cube

Hexagonal 1, 7, 19, 37, 61 add the next multiple of 6 rings around a centre dot

Virahānka 1, 2, 3, 5, 8, 13 add the previous two spirals and plant growth

Powers of 2 1, 2, 4, 8, 16 ×2 two copies of the last
picture

Powers of 3 1, 3, 9, 27, 81 ×3 three copies of the last
picture

Page 36 of 36