---
title: "NCERT Solutions Class 6 Maths Chapter 1 Patterns in Mathematics"
url: https://www.swavid.com/maths/class/6/chapter/patterns-in-mathematics/ncert-solutions
dateModified: 2026-10-07T14:49:08+00:00
---

# NCERT Solutions Class 6 Maths Chapter 1 Patterns in Mathematics

This chapter's questions cover pattern recognition in number sequences, geometric figures, and relationships between numbers and shapes. Students explore visual proofs and sequences such as squares, triangular numbers, and powers.

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## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Mathematics is used in various daily activities such as paying for fruits, vegetables, and groceries.
2. It is applied in calculating the speed of vehicles and understanding designs or patterns in buildings.
3. It is also used to find the area of any plot or our own home.

**Answer:** Mathematics helps us in everyday tasks like shopping, calculating speed, and measuring areas.

> Common mistake: Giving vague answers without specific daily life examples.

### Question 2

*3 marks · Short answer*

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.)

**Solution**

1. Mathematics helps in carrying out scientific experiments and running our economy and democracy.
2. It is essential for building complex structures like bridges and houses.
3. It enables the manufacturing of technology such as computers, mobile phones, clocks, and vehicles.

**Answer:** Mathematics drives progress through science, technology, and engineering.

> Common mistake: Listing examples without explaining how mathematics supports them.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Observe the given sequences in Table 1 of the textbook.
2. Identify the rule for each sequence, such as adding a fixed value or multiplying by a constant.
3. Recognize standard sequences like counting numbers, odd numbers, even numbers, squares, cubes, and Virahānka numbers.

**Answer:** The patterns in Table 1 include all 1's, counting numbers, odd numbers, even numbers, triangular numbers, squares, cubes, Virahānka numbers, and powers of 2 and 3.

> Common mistake: Failing to identify the recursive rule in Virahānka numbers where each term is the sum of the preceding two terms.

### Question 2

*3 marks · Short answer*

Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

**Solution**

1. Take each sequence from Table 1 and write down the next three terms using the identified rule.
2. For counting numbers ($1, 2, 3, 4, 5, 6, 7$), the next three terms are $8, 9, 10$.
3. For powers of 2 ($1, 2, 4, 8, 16, 32, 64$), the next three terms are $128, 256, 512$.

**Answer:** Each sequence is extended by three terms, and the formation rule is stated as described in Table 1.

> Common mistake: Applying addition instead of multiplication for geometric sequences like powers of 2 or 3.

## Figure it Out

### Question 1

*Activity*

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

**Solution**

1. Refer to Table 2 on page 4 of the textbook for the pictorial representations of sequences such as All 1s, counting numbers, odd numbers, even numbers, triangular numbers, squares, and cubes.
2. Draw the next picture for each sequence by following the established pattern of dots or objects for the next term in the sequence.

**Answer:** Completed the drawings of the next pictures for each sequence in Table 2.

### Question 2

*3 marks · Short answer*

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

**Solution**

1. They are called triangular numbers because $1, 3, 6, 10, 15, \dots$ dots can be arranged perfectly in the shape of triangles.
2. They are called square numbers or squares because $1, 4, 9, 16, 25, \dots$ dots can be arranged perfectly in the shape of squares in a grid.
3. They are called cubes because $1, 8, 27, 64, 125, \dots$ unit cubes can be stacked together to form larger solid cubes.

**Answer:** They are named after the geometric shapes ($1, 3, 6, \dots$ for triangles, $1, 4, 9, \dots$ for squares, and $1, 8, 27, \dots$ for cubes) that can be formed using those numbers of dots or blocks.

> Common mistake: Confusing the types of shapes or writing definitions without referring to the pictorial representation.

### Question 3

*Activity*

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! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!

**Solution**

1. Recognise that $36$ is the sixth triangular number ($1 + 2 + 3 + 4 + 5 + 6 = 36$) and also the sixth square number ($6 \times 6 = 36$).
2. Draw a triangular arrangement of 36 dots and a square grid arrangement of $6 \times 6$ dots in the notebook to illustrate both properties.

**Answer:** Drew both the triangular and square pictorial arrangements for 36 dots.

### Question 4

*3 marks · Short answer*

What would you call the following sequence of numbers?
$1, 7, 19, 37$
That's right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?

**Solution**

1. The given sequence of hexagonal numbers is 1, 7, 19, 37, ...
2. The differences between consecutive terms are 6, 12, 18, which increase by 6 each time.
3. The next difference is $18 + 6 = 24$, so the next number is $37 + 24 = 61$.

**Answer:** 61

> Common mistake: Adding a constant difference instead of an increasing difference of multiples of 6.

### Question 5

*3 marks · Short answer*

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

**Solution**

1. Powers of 2 can be visualised using line segments, squares, cubes, and hypercubes in higher dimensions as shown on page 6.
2. Powers of 3 can be visualised using points, triangles, 3D block structures, and their higher-dimensional extensions as shown on page 14.

**Answer:** Powers of 2 and 3 can be visualised pictorially using geometric shapes and dimensional growth.

> Common mistake: Not relating the powers to geometric scaling.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Recall that square numbers are made by counting the number of dots in a square grid.
2. We can arrange dots in a diamond-like square grid corresponding to the sums $1, 1+2+1, 1+2+3+2+1, \dots$, where each row or diagonal segment grows and then shrinks symmetrically.
3. This pictorial representation shows that adding counting numbers up and down forms square numbers of dots ($1, 4, 9, \dots$).

**Answer:** Adding counting numbers up and down gives square numbers, as shown by the symmetric diamond dot grids in the textbook.

> Common mistake: Confusing the counting numbers up and down pattern with the consecutive odd numbers pattern.

### Question 2

*3 marks · Short answer*

By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of $1 + 2 + 3 + \dots + 99 + 100 + 99 + \dots + 3 + 2 + 1$?

**Solution**

1. Given: The expression $1 + 2 + 3 + \dots + 99 + 100 + 99 + \dots + 3 + 2 + 1$.
2. Formula: Adding counting numbers up and down to $n$ gives the square number $n^2$.
3. Substitution: Here $n = 100$, so the sum is $100^2$.
4. Result: $10,000$

**Answer:** 10,000

> Common mistake: Forgetting to square the peak number or incorrectly taking $n = 99$.

### Question 3

*3 marks · Short answer*

Which sequence do you get when you start to add the All 1's sequence up? What sequence do you get when you add the All 1's sequence up and down?

**Solution**

1. When you start to add the All 1's sequence up ($1, 1+1, 1+1+1, \dots$), you get the counting numbers sequence ($1, 2, 3, 4, \dots$).
2. When you add the All 1's sequence up and down, you get sums of the form $1 + (1+1) + 1$, etc., which form a sequence of symmetric sums.
3. Thus, adding All 1's up gives counting numbers, and adding them up and down gives square numbers.

**Answer:** Adding All 1's up gives counting numbers, and adding them up and down gives square numbers.

> Common mistake: Listing the intermediate additions instead of identifying the resulting number sequence.

### Question 4

*3 marks · Short answer*

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

**Solution**

1. When you start to add the counting numbers up ($1, 1+2, 1+2+3, 1+2+3+4, \dots$), you get the triangular numbers sequence ($1, 3, 6, 10, \dots$).
2. A smaller pictorial explanation can be given by arranging dots in right-angled triangles where each new row adds $2, 3, 4, \dots$ dots.
3. Counting the total dots in each triangular arrangement gives the triangular numbers.

**Answer:** We get the triangular number sequence ($1, 3, 6, 10, \dots$), which can be visualised using right-angled triangular grids.

> Common mistake: Confusing triangular numbers with square numbers when adding counting numbers.

### Question 5

*3 marks · Short answer*

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

**Solution**

1. Calculate the sum of pairs of consecutive triangular numbers: $1 + 3 = 4$, $3 + 6 = 9$, $6 + 10 = 16$, $10 + 15 = 25$.
2. Observe that the resulting sequence of numbers is $4, 9, 16, 25, \dots$
3. Identify this sequence as the square numbers, which can be explained pictorially by combining two triangular arrangements to form a square grid.

**Answer:** We get the square numbers ($4, 9, 16, 25, \dots$).

> Common mistake: Confusing square numbers with triangular numbers by simply adding the terms without finding their sums.

### Question 6

*3 marks · Short answer*

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

**Solution**

1. Taking sums of powers of 2 starting with 1 gives: $1, 1+2=3, 1+2+4=7, 1+2+4+8=15, \dots$, which is the sequence $1, 3, 7, 15, 31, \dots$.
2. Now add 1 to each of these numbers to get: $1+1=2, 3+1=4, 7+1=8, 15+1=16, \dots$, which gives the powers of 2 sequence ($2, 4, 8, 16, 32, \dots$).
3. This happens because the sum of powers of 2 up to $2^n$ is always $2^{n+1} - 1$, so adding 1 gives $2^{n+1}$.

**Answer:** We get the sequence $1, 3, 7, 15, 31, \dots$, and adding 1 to each gives the powers of 2 sequence ($2, 4, 8, 16, 32, \dots$).

> Common mistake: Forgetting to add 1 to every term in the sequence as instructed.

### Question 7

*3 marks · Short answer*

What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?

**Solution**

1. Multiply each triangular number by 6 and add 1.
2. The calculations give: $(1 \times 6) + 1 = 7$, $(3 \times 6) + 1 = 19$, $(6 \times 6) + 1 = 37$, $(10 \times 6) + 1 = 61$, and $(15 \times 6) + 1 = 91$.
3. We get the sequence of hexagonal numbers: $7, 19, 37, 61, 91, \dots$

**Answer:** We get the sequence of hexagonal numbers: $7, 19, 37, 61, 91, \dots$

> Common mistake: Arithmetic errors while multiplying triangular numbers by 6 before adding 1.

### Question 8

*3 marks · Short answer*

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

**Solution**

1. Add up the hexagonal numbers progressively: $1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64, \dots$
2. The sums obtained are $1, 8, 27, 64, \dots$
3. This gives the sequence of cube numbers, which can be visualised by growing concentric shells of dots forming cubes as shown in Table 2.

**Answer:** We get the sequence of cube numbers: $1, 8, 27, 64, \dots$

> Common mistake: Confusing hexagonal numbers with triangular numbers when taking cumulative sums.

### Question 9

*3 marks · Short answer*

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?

**Solution**

1. Examine sequences in Table 1 such as square numbers and triangular numbers.
2. A known relation is that adding up consecutive odd numbers starting from 1 always gives square numbers.
3. Similarly, adding up two consecutive triangular numbers gives a square number.

**Answer:** Adding up consecutive odd numbers starting from 1 yields the sequence of square numbers.

> Common mistake: Not providing a clear relation or explanation connecting two sequences from Table 1.

## Figure it Out

### Question 1

*3 marks · Short answer*

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

**Solution**

1. Examine the shape sequences in Table 3 as shown in the textbook.
2. The sequences follow rules based on the number of sides, vertices, or building blocks such as triangles and squares.
3. For example, the regular polygons sequence increases by one side at each step, starting from a triangle with 3 sides.

**Answer:** The pattern in each sequence of Table 3 is formed by regularly increasing the number of sides, vertices, or geometric components.

> Common mistake: Confusing the number of sides with the number of corners or lines.

### Question 2

*Activity*

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.

**Solution**

1. Redraw the given shape sequences from Table 3 in the notebook.
2. Analyze the rule for forming shapes in each sequence.
3. Draw the next shape by extending the pattern, such as adding a side to a polygon or adding a row of elements.

**Answer:** Shapes redrawn in the notebook with the next shape added according to the observed rule.

### Question 1

*3 marks · Short answer*

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?

**Solution**

1. Count the number of sides for each shape in the sequence of Regular Polygons: 3, 4, 5, 6, 7, 8, 9, 10, which gives the counting number sequence starting with 3.
2. Count the number of corners for each shape in the sequence: 3, 4, 5, 6, 7, 8, 9, 10, which gives the exact same number sequence.
3. This happens because in any closed polygon, the number of sides is always equal to the number of corners (vertices).

**Answer:** We get the counting number sequence starting with 3 for both sides and corners, because in any closed figure, the number of sides equals the number of corners.

> Common mistake: Stating that the number of sides and corners are different.

### Question 2

*3 marks · Short answer*

Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

**Solution**

1. Count the number of lines in each shape of the sequence of Complete Graphs ($K_2, K_3, K_4, K_5, K_6$).
2. The number of lines obtained is 1, 3, 6, 10, 15.
3. This is the triangular number sequence because connecting all pairs of vertices in a complete graph forms the triangular numbers.

**Answer:** We get the triangular number sequence (1, 3, 6, 10, 15), as every pair of vertices is connected by a line segment.

> Common mistake: Miscounting the lines connecting the vertices in complete graphs.

### Question 3

*3 marks · Short answer*

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?

**Solution**

1. Count the number of little squares in each shape of the sequence of Stacked Squares.
2. The number of little squares in each shape is 1, 4, 9, 16, 25, ...
3. This gives the square number sequence because the stacked squares form larger square grids of size $1 \times 1$, $2 \times 2$, $3 \times 3$, and so on.

**Answer:** We get the square number sequence (1, 4, 9, 16, 25) because the shapes are arranged as square grids.

> Common mistake: Counting the total perimeter instead of the little squares inside.

### Question 4

*3 marks · Short answer*

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?)

**Solution**

1. Count the number of little triangles in each shape of the sequence of Stacked Triangles from Table 3.
2. The number of little triangles in each shape are $1$, $4$, $9$, $16$, $25$, and so on.
3. This gives the square number sequence, which can also be seen by adding counting numbers up and down per row ($1$, $1+2+1$, $1+2+3+2+1$, etc.).

**Answer:** Square number sequence ($1, 4, 9, 16, 25, \dots$), obtained by counting the little triangles in each shape.

> Common mistake: Listing counting numbers instead of square numbers by forgetting to count all triangles in the rows.

### Question 5

*3 marks · Short answer*

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?

**Solution**

1. Count the total number of line segments in the initial shape of the Koch Snowflake to get 3.
2. Multiply the number of line segments by 4 for each subsequent stage as each segment is replaced by 4 smaller segments.
3. Write down the corresponding number sequence as $3, 12, 48, 192, 768, \dots$, which represents 3 times powers of 4.

**Answer:** The total number of line segments in each shape is $3, 12, 48, 192, 768, \dots$, corresponding to 3 times powers of 4.

> Common mistake: Counting only the added segments instead of multiplying the total previous number of segments by 4.

## Frequently asked questions

### How many total questions are there in NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics?

This chapter is divided into multiple 'Figure it Out' sections containing a total of 25 questions across various types like short answer and activities. You can find SwaVid's free PDF and step-by-step solutions for all these questions on this page only.

### Which topics do the questions in this Class 6 chapter cover?

The questions cover concepts such as number sequences, hexagonal numbers, geometric representation of numbers, and patterns in everyday life. They also explore relationships between triangular numbers, cube numbers, and powers of 2 and 3.

### Which question types are considered challenging in Class 6 Maths Chapter 1 and how should we approach them?

Questions involving visualising powers of 2 and 3, or finding relations between hexagonal and cube numbers, can be tricky. To approach them, break down the pictorial representations and look for a recurring rule before applying the formula.

### How can I write answers for full marks in this Class 6 chapter?

To secure full marks, clearly show your working steps for number sequences or shape patterns instead of writing just the final number. Referring to SwaVid's free PDF and step-by-step solutions on this page only will help you structure your answers correctly.

### Is the free PDF available for Class 6 Maths Chapter 1 Patterns in Mathematics?

Yes, complete solutions aligned with the new NCERT book for the 2026-27 session are ready for students. You can access SwaVid's free PDF and step-by-step solutions for this chapter on this page only.

## Related pages

- [Class 6 Maths chapters](https://www.swavid.com/maths/class/6)

Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student