Class 6 Maths Ganita Prakash Chapter 1 Solutions (NCERT 2026–27) – Patterns in Mathematics

These Class 6 Maths Ganita Prakash Chapter 1 solutions cover Patterns in Mathematics from the new NCF-2023 textbook (Reprint 2026–27). Every Figure it Out, Math Talk and Try This question is solved step by step, with the number sequences, shape sequences and the surprising relations between them explained in full, so you can master the chapter and revise it quickly.

Class: 6 Subject: Mathematics Book: Ganita Prakash Chapter: 1 Exercises: Figure it Out (Sections 1.1–1.6) Session: 2026–27
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Chapter 1 Overview

Chapter 1 of Ganita Prakash, Patterns in Mathematics, introduces mathematics as the search for patterns and the explanations for why they exist. It begins with number sequences — counting numbers, odd and even numbers, triangular numbers, square numbers, cube numbers, Virahāṅka numbers and powers of 2 and 3 (Table 1). It then shows how these sequences can be visualised with pictures (Table 2) and how they relate to each other in beautiful ways — for example, adding consecutive odd numbers gives square numbers. Finally it explores shape sequences (Table 3) such as regular polygons, complete graphs, stacked squares and triangles and the Koch snowflake, and links them back to number sequences. The Class 6 Maths Ganita Prakash Chapter 1 solutions below work through every Figure it Out, Math Talk and Try This task step by step.

Key Concepts & Definitions

Pattern: a regularity in numbers or shapes that follows a rule, so we can predict what comes next and explain why.

Number theory: the branch of mathematics that studies patterns in whole numbers (0, 1, 2, 3, …).

Triangular numbers (1, 3, 6, 10, 15, …): numbers of dots that can be arranged in an equilateral triangle; each is the sum of counting numbers 1 + 2 + 3 + …

Square numbers (1, 4, 9, 16, 25, …): dots that fill a square grid; the n-th square number is n × n.

Cube numbers (1, 8, 27, 64, 125, …): dots that fill a cube; the n-th cube number is n × n × n.

Hexagonal numbers (1, 7, 19, 37, …): dots arranged in growing hexagons.

Virahāṅka (Fibonacci) numbers (1, 2, 3, 5, 8, 13, 21, …): each number is the sum of the previous two.

Geometry: the branch of mathematics that studies patterns in shapes (in 1D, 2D, 3D or more).

Regular polygon: a closed shape with all sides equal and all angles equal — triangle, square, pentagon, hexagon, and so on.

Important Sequences & Patterns (Chapter 1)

Counting numbers: 1, 2, 3, 4, 5, … (add 1 each time).

Odd numbers: 1, 3, 5, 7, 9, …  •  Even numbers: 2, 4, 6, 8, 10, … (add 2 each time).

Triangular numbers: 1, 3, 6, 10, 15, … (add 2, then 3, then 4, …).

Squares: 1, 4, 9, 16, 25, … = 1×1, 2×2, 3×3, …  •  Cubes: 1, 8, 27, 64, 125, … = 1×1×1, 2×2×2, …

Powers of 2: 1, 2, 4, 8, 16, … (double each time)  •  Powers of 3: 1, 3, 9, 27, 81, … (triple each time).

Virahāṅka numbers: 1, 2, 3, 5, 8, 13, 21, … (each = sum of previous two).

Key relation: 1 + 3 + 5 + … + (sum of first n odd numbers) = n2 (the sum of consecutive odd numbers starting at 1 is always a square).

Figure it Out — Sections 1.1 & 1.2

Questions are reproduced verbatim from the NCERT Ganita Prakash (Grade 6) textbook; the worked solutions are original and verified against the answers given for the chapter.

Figure it Out (page 2)

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

SOLUTION Mathematics is used everywhere in daily life. For example: paying for fruits, vegetables and groceries and getting back the correct change; calculating the speed of a vehicle or how long a journey will take; reading clocks and calendars; designs and patterns in buildings, tiles and rangoli; finding the area of a plot or a room; measuring ingredients while cooking; and keeping score in games. Many more such contexts exist — discuss other examples from your own day.

2. 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 Mathematics has driven almost every major advance. It lets scientists design and analyse experiments; it runs our economy (banking, prices, taxes) and democracy (counting and predicting votes); it lets engineers build safe bridges, tall buildings and other structures; and it powers all our technology — TVs, mobile phones, computers, vehicles, planes — as well as accurate calendars and clocks. (A discussion question — share your own examples with your teacher and classmates.)

Figure it Out (page 3)

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

SOLUTION Yes — each sequence follows a clear rule: All 1’s: every term is 1.   Counting numbers: add 1 each time.   Odd numbers: start at 1, add 2.   Even numbers: start at 2, add 2. Triangular numbers (1, 3, 6, 10, …): add 2, then 3, then 4, …   Squares (1, 4, 9, 16, …): n × n.   Cubes (1, 8, 27, 64, …): n × n × n. Powers of 2 (1, 2, 4, 8, 16, …): double each time.   Powers of 3 (1, 3, 9, 27, 81, …): triple each time. Virahāṅka numbers (1, 2, 3, 5, 8, 13, 21, …): each number is the sum of the previous two (5 = 2 + 3, 8 = 3 + 5, 13 = 5 + 8).

2. 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 Continuing each sequence by its rule gives the next three numbers (shown in bold):
SequenceNext three numbersRule
All 1’s: 1, 1, 1, …1, 1, 1Every term is 1.
Counting: 1, 2, 3, 4, 5, …6, 7, 8Add 1 each time.
Odd: 1, 3, 5, 7, 9, …11, 13, 15Start at 1, add 2 each time.
Even: 2, 4, 6, 8, 10, …12, 14, 16Start at 2, add 2 each time.
Triangular: 1, 3, 6, 10, 15, …21, 28, 36Add 2, then 3, then 4, … (add the next counting number).
Squares: 1, 4, 9, 16, 25, …36, 49, 64The n-th term is n × n.
Cubes: 1, 8, 27, 64, 125, …216, 343, 512The n-th term is n × n × n.
Virahāṅka: 1, 2, 3, 5, 8, 13, 21, …34, 55, 89Each number is the sum of the previous two.
Powers of 2: 1, 2, 4, 8, 16, 32, 64, …128, 256, 512Double each time.
Powers of 3: 1, 3, 9, 27, 81, 243, 729, …2187, 6561, 19683Triple each time.

Figure it Out — Section 1.3 (Visualising Number Sequences, page 5)

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

SOLUTION Copy each picture from Table 2 and add one more step: All 1’s → one more single dot.   Counting numbers → a row of 6 dots.   Odd numbers → an “L” of 11 dots.   Even numbers → a 2-wide block of 12 dots. Triangular → a triangle of 21 dots (6 rows).   Squares → a 6 × 6 grid of 36 dots.   Cubes → a 6 × 6 × 6 cube of 216 dots. (A drawing task — the counts of dots in the next picture are given above.)

2. 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?

SOLUTION Triangular numbers: these many dots can be arranged neatly in the shape of an equilateral triangle (1; then 3 in two rows; then 6 in three rows; and so on). Square numbers: these many dots can be arranged in a perfect square grid — 1 = 1×1, 4 = 2×2, 9 = 3×3, 16 = 4×4, 25 = 5×5. Cubes: these many dots can be stacked to fill a perfect cube — 1 = 1×1×1, 8 = 2×2×2, 27 = 3×3×3, and so on. (Refer to Table 2 to see the pictures.)

3. 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!

SOLUTION 36 is the 8th triangular number: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, so 36 dots form a triangle with rows of 1, 2, 3, …, 8 dots. 36 is also the 6th square number: 6 × 6 = 36, so the same 36 dots fill a 6 × 6 square grid. Draw both arrangements in your notebook — the same number can play different roles depending on the context.

4. 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 These are hexagonal numbers. The differences are 6, 12, 18, … (each step adds the next multiple of 6): 1 → 7 (+6), 7 → 19 (+12), 19 → 37 (+18). The next difference is +24, so the next number is 37 + 24 = 61.

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

SOLUTION Powers of 2 (1, 2, 4, 8, 16, …): start with 1 dot, then keep doubling — show each step as a block that is twice as big as the previous one (1, then 2, then 2×2 = 4, then 2×2×2 = 8, …), as on page 6 of the book. Powers of 3 (1, 3, 9, 27, …): start with 1 dot; replace each dot by a group of 3, then each of those by a group of 3 again. This grows 1 → 3 → 9 → 27, which can be drawn as a branching tree or as squares of side 1, 3, 9, …

Figure it Out — Section 1.4 (Relations among Sequences, pages 8–9)

1. 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?

SOLUTION Yes. Arrange the dots as a square that has been “tilted” into diagonals. Each diagonal of a square grid holds 1, 2, 3, … up to the middle, then 3, 2, 1 again. For example, a 3 × 3 square (9 dots) has diagonals of sizes 1, 2, 3, 2, 1, and 1 + 2 + 3 + 2 + 1 = 9 = 32. Since the diagonals of any n × n square count up to n and back down, the sum 1 + 2 + … + n + … + 2 + 1 always equals n2.

2. 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 + … + 99 + 100 + 99 + … + 3 + 2 + 1?

SOLUTION The sum counts up to 100 and back down, so it equals 1002. ∴ 1 + 2 + … + 99 + 100 + 99 + … + 2 + 1 = 100 × 100 = 10,000.

3. 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 Adding up: 1, 1 + 1, 1 + 1 + 1, … = 1, 2, 3, 4, … — you get the counting numbers. Adding up and down: 1; 1 + (1 + 1) + 1 = 4; 1 + (1 + 1) + (1 + 1 + 1) + (1 + 1) + 1… — this builds 1, 4, 9, 16, …, the square numbers, because adding a constant 1 up and down across an n-wide block fills an n × n square.

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

SOLUTION 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, … = 1, 3, 6, 10, … — you get the triangular numbers. Picture: place rows of 1, 2, 3, … dots one above another; they stack into a triangle (see Table 2, page 4). Two such triangles fit together to make a rectangle, which is why each triangular number can be drawn neatly.

5. 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?

SOLUTION 1 + 3 = 4,  3 + 6 = 9,  6 + 10 = 16,  10 + 15 = 25, … — you get the square numbers 4, 9, 16, 25, … Why: two consecutive triangles — one with rows 1, 2, …, n and a smaller one with rows 1, 2, …, (n − 1) — fit together exactly to form an n × n square. (See the triangle/square pictures in Table 2.)

6. 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?

SOLUTION The running totals are 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15, 1 + 2 + 4 + 8 + 16 = 31, … (each is one less than the next power of 2). Adding 1 to each gives 2, 4, 8, 16, 32, … — the powers of 2 again. Why: 1 + 2 + 4 + … + 2k = 2k+1 − 1, so adding 1 jumps you up to the next power of 2.

7. 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 × 6) + 1 = 7,  (3 × 6) + 1 = 19,  (6 × 6) + 1 = 37,  (10 × 6) + 1 = 61,  (15 × 6) + 1 = 91, … You get 7, 19, 37, 61, 91, … — these are the hexagonal numbers (from 1, 7, 19, 37, … onward). Why: a hexagon can be split into a central dot plus six identical triangular arrangements around it, i.e. 6 × (a triangular number) + 1.

8. 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?

SOLUTION 1,  1 + 7 = 8,  1 + 7 + 19 = 27,  1 + 7 + 19 + 37 = 64, … — you get the cube numbers 1, 8, 27, 64, … Why: a cube of side n can be built up in hexagonal “shells”; adding successive hexagonal numbers fills out the cube one layer at a time, so the running totals are 13, 23, 33, …

9. 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 Here are two of many possible patterns: (a) The difference between consecutive square numbers gives the odd numbers: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, … This is because the next square is built by adding an “L” of odd many dots around the previous square. (b) Eight times a triangular number plus 1 is a square: 8 × 1 + 1 = 9, 8 × 3 + 1 = 25, 8 × 6 + 1 = 49, … (An open exploratory task — record any patterns you discover and try to explain them.)

Figure it Out — Sections 1.5 & 1.6 (Patterns in Shapes, pages 11–12)

Figure it Out (page 11)

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

SOLUTION Yes — each shape sequence follows a rule: Regular Polygons: the number of sides increases by 1 each time — triangle, quadrilateral, pentagon, hexagon, … (3, 4, 5, 6, 7, 8, 9, 10, …). Complete Graphs (K2, K3, …): each adds one more point, and every point is joined to every other point. Stacked Squares: 1, 4, 9, 16, 25, … little squares (square numbers).   Stacked Triangles: 1, 4, 9, 16, 25, … little triangles (square numbers). Koch Snowflake: each line segment is replaced by a “speed bump”, multiplying the number of segments by 4: 3, 3×4, 3×4×4, …

2. 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 Regular Polygons: next shape after the decagon is an 11-sided regular polygon (hendecagon) — add one more equal side. This can always be drawn. Complete Graphs: next is K7 — add a 7th point and join it to all the others. This can be drawn, though it gets crowded. Stacked Squares / Triangles: next shape adds one more row, giving the next square number (36) of little shapes. Koch Snowflake: the next shape replaces every segment by a speed bump again, giving 4 times as many, much tinier, segments — harder to draw because the pieces become very small.

Figure it Out (page 11–12)

1. 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 Number of sides = 3, 4, 5, 6, 7, 8, 9, 10, … — the counting numbers starting from 3. Number of corners = 3, 4, 5, 6, 7, 8, 9, 10, … — the same sequence. Why: in any closed figure each side ends at a corner and each corner joins two sides, so the number of sides always equals the number of corners (vertices).

2. 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 The number of lines is 1, 3, 6, 10, 15, … — the triangular numbers. Why: with n points, each new point is joined to all the points already there, adding 1, 2, 3, … lines in turn. The running total 1 + 2 + 3 + … is exactly the triangular-number sequence.

3. 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, 4, 9, 16, 25, … — the square numbers. Why: the k-th stacked square is a k × k grid, which contains k × k = k2 little squares.

4. 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, 4, 9, 16, 25, … — again the square numbers. Why: the rows contain 1, 3, 5, 7, … little triangles (odd numbers), and 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, … The sum of the first k odd numbers is k2, so the totals are the square numbers. (This is the same as the “up and down” pattern 1, 1 + 2 + 1, …)

5. 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 answer is 3, 12, 48, …, i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)

SOLUTION Each step replaces every segment by 4 smaller segments, so the count is multiplied by 4 each time. Total line segments: 3, 12, 48, 192, 768, … Corresponding sequence: 3, 3×4, 3×4×4, 3×4×4×4, … — that is, 3 times the powers of 4.

Math Talk & In-text Tasks — Answered

These are the reflective Math Talk prompts and short in-text tasks inside the chapter; the determinate ones are answered, and the explanatory ones are explained.

In-text (Section 1.4) — Sum of the first 10 odd numbers By drawing a similar picture, can you say what is the sum of the first 10 odd numbers? Answer. The sum of the first n odd numbers is n2. So the sum of the first 10 odd numbers (1 + 3 + 5 + … + 19) = 102 = 100.
In-text (Section 1.4) — Sum of the first 100 odd numbers Now by imagining a similar picture, or by drawing it partially, can you say what is the sum of the first 100 odd numbers? Answer. The sum of the first 100 odd numbers = 1002 = 10,000. (The L-shaped layers of 1, 3, 5, … build a 100 × 100 square.)
Math Talk (Section 1.4) — Why odd numbers add to squares Why does adding up odd numbers give square numbers? Do you think it will happen forever? Answer. Yes, it happens forever. To grow an n × n square into an (n+1) × (n+1) square you add one new row and one new column — an L-shape of (2n + 1) dots, which is the next odd number. So each new odd number turns one square into the next, and 1 + 3 + 5 + … always lands on a square number.
Math Talk (Section 1.2) — The All 1’s and powers sequences Rewrite each sequence of Table 1 with the next three numbers and state its rule (Powers of 2 / Powers of 3 / Virahāṅka). Answer. Powers of 2 double each time: …, 32, 64, then 128, 256, 512. Powers of 3 triple each time: …, 243, 729, then 2187, 6561, 19683. Virahāṅka numbers add the previous two: …, 13, 21, then 34, 55, 89. (Full table given in the Figure it Out solution above.)
Math Talk (Section 1.3) — A number with two roles 36 is both a triangular and a square number. Can you find other numbers that can be represented in more than one way? Answer. Yes. 1 is a triangular, square and cube number all at once. The next number that is both triangular and square after 1 and 36 is 1225 (= 49th triangular number = 352). The same number can play different roles depending on how its dots are arranged.

Common Mistakes to Avoid

Watch out for these

  • Confusing the triangular numbers (1, 3, 6, 10, …, add the next counting number) with the odd numbers (1, 3, 5, 7, …, add 2) — check the differences.
  • Forgetting that the sum of consecutive odd numbers starting at 1 is a perfect square (n2), not a triangular number.
  • In the “up and down” sum 1 + 2 + … + n + … + 2 + 1, the answer is n2 — the peak value n is counted only once, not twice.
  • Writing the next power of 2 by adding 2 instead of doubling (and the next power of 3 by tripling, not adding 3).
  • Mixing up sides and lines: in regular polygons sides = corners (counting numbers from 3), but in complete graphs the lines follow the triangular numbers.
  • For the Koch snowflake, multiplying the count by 3 instead of 4 — each segment becomes 4 segments, so it is 3 × powers of 4.

Practice MCQs & Assertion–Reason

1. The branch of mathematics that studies patterns in whole numbers is called:

(a) geometry    (b) algebra    (c) number theory    (d) statistics

2. The next number in the triangular-number sequence 1, 3, 6, 10, 15, … is:

(a) 18    (b) 20    (c) 21    (d) 25

3. The sum of the first 10 odd numbers is:

(a) 55    (b) 90    (c) 100    (d) 110

4. In the Virahāṅka (Fibonacci) sequence 1, 2, 3, 5, 8, 13, …, the next number is:

(a) 18    (b) 21    (c) 24    (d) 26

5. Adding pairs of consecutive triangular numbers (1 + 3, 3 + 6, 6 + 10, …) gives the:

(a) cube numbers    (b) square numbers    (c) even numbers    (d) powers of 2

6. The next number in the hexagonal sequence 1, 7, 19, 37, … is:

(a) 55    (b) 58    (c) 61    (d) 64

7. The number of lines in the sequence of Complete Graphs (K2, K3, K4, …) follows the:

(a) square numbers    (b) triangular numbers    (c) cube numbers    (d) odd numbers

8. The value of 1 + 2 + 3 + … + 99 + 100 + 99 + … + 2 + 1 is:

(a) 5050    (b) 9900    (c) 10000    (d) 10100

9. Adding up hexagonal numbers (1, 1 + 7, 1 + 7 + 19, …) gives the:

(a) square numbers    (b) triangular numbers    (c) cube numbers    (d) powers of 3

10. The number of line segments in the Koch Snowflake follows the sequence 3, 12, 48, …, which is 3 times the:

(a) powers of 2    (b) powers of 3    (c) powers of 4    (d) square numbers

Answer key: 1-(c), 2-(c), 3-(c), 4-(b), 5-(b), 6-(c), 7-(b), 8-(c), 9-(c), 10-(c).

For each Assertion–Reason question, choose: (A) Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion; (B) Both are true but the Reason is not the correct explanation; (C) Assertion is true but Reason is false; (D) Assertion is false but Reason is true.

A-R 1. Assertion: Adding consecutive odd numbers starting from 1 always gives a square number.

Reason: To grow an n × n square into the next square you add an L-shape of (2n + 1) dots, which is the next odd number.

A-R 2. Assertion: 36 is both a triangular number and a square number.

Reason: 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 and 36 = 6 × 6.

A-R 3. Assertion: In a complete graph the number of lines follows the powers of 2.

Reason: With n points, each new point is joined to all earlier points, adding 1, 2, 3, … lines.

A-R 4. Assertion: In any regular polygon the number of sides equals the number of corners.

Reason: In a closed figure every side ends at a corner and every corner joins two sides.

A-R 5. Assertion: Multiplying the triangular numbers by 6 and adding 1 gives the hexagonal numbers 7, 19, 37, …

Reason: A hexagon can be split into a central dot plus six identical triangular arrangements.

Answer key: 1-(A), 2-(B), 3-(D), 4-(A), 5-(A).

Quick Revision Summary

  • Mathematics is the search for patterns and the explanations for why those patterns exist.
  • Key number sequences: counting, odd, even, triangular (1, 3, 6, 10, …), squares (1, 4, 9, …), cubes (1, 8, 27, …), Virahāṅka (1, 2, 3, 5, 8, …), powers of 2 and powers of 3.
  • Adding the first n odd numbers gives n2; adding counting numbers 1 + 2 + 3 + … gives the triangular numbers.
  • Adding two consecutive triangular numbers gives a square; adding hexagonal numbers gives cubes; 6 × (triangular) + 1 gives hexagonal numbers.
  • Adding powers of 2 (1, 1+2, 1+2+4, …) gives one less than the next power of 2.
  • Shape sequences: regular polygons (sides = corners = 3, 4, 5, …), complete graphs (lines = triangular numbers), stacked squares/triangles (square numbers), Koch snowflake (3 × powers of 4).
  • Visualising sequences with dot pictures helps explain why these relationships are true.

How to score full marks in this chapter

Always identify the rule of a sequence first (what is added or multiplied each step), then continue it — double for powers of 2, triple for powers of 3, add the next counting number for triangular numbers. When a question asks “why”, give the dot-picture reason (e.g. an L-shape of odd dots builds the next square). State the resulting sequence by name (square, triangular, cube) and show one or two numerical checks so each step earns its mark.

Frequently Asked Questions

What is Class 6 Maths Ganita Prakash Chapter 1 about?

Chapter 1, Patterns in Mathematics, introduces number sequences (counting, odd, even, triangular, square, cube, Virahāṅka and powers of 2 and 3), shows how to visualise them with dot pictures, and explores the surprising relations between number sequences and shape sequences such as regular polygons, complete graphs and the Koch snowflake.

What sequence do you get by adding consecutive odd numbers?

You get the square numbers. 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, and so on — the sum of the first n odd numbers is always n2, because each new odd number adds an L-shape of dots that grows one square into the next.

Why is 36 both a triangular number and a square number?

36 is the 8th triangular number (1 + 2 + 3 + … + 8 = 36), so 36 dots form a triangle; it is also the 6th square number (6 × 6 = 36), so the same 36 dots fill a square grid. A number can play different roles depending on how its dots are arranged.

Are these Class 6 Maths Ganita Prakash Chapter 1 solutions free?

Yes. All solutions are free and follow the official NCERT Ganita Prakash textbook for the 2026–27 session, with every Figure it Out, Math Talk and Try This task solved and verified.

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