Chapter 1 — Patterns in Mathematics (Ganita Prakash, Grade 6)

1.1 What is Mathematics?

Mathematics is, to a large extent, the hunt for patterns — and the hunt for reasons behind those patterns. Finding a pattern is only half the job. The other half is answering: why does it work?

Table of Contents

Patterns are everywhere around us — in leaves and flowers, in the design of our homes and classrooms, and in the way the sun, moon and stars move. We meet them while shopping, while cooking, while throwing a ball, while playing a game, while checking the weather and while using any gadget.

Because searching for patterns is both playful and creative, mathematicians happily call mathematics an art as well as a science.

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Why the "why" matters: once we explain a pattern, that explanation can be used far away from where it was first noticed.

Patterns in the motion of stars, planets and satellites → the theory of gravitation → satellites, rockets to the Moon and Mars.
Patterns in genomes → diagnosing and curing diseases.

How mathematics pushes humanity forward

Figure it Out (Page 2)

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

Ans. Yes — plenty:

Paying for fruits, vegetables and groceries; checking the change we get back.
Working out the speed of a bus or a train, and the time a journey will take.
Reading the designs and repeating patterns on buildings, floor tiles and sarees.
Measuring the area of a plot of land or of the rooms in our own house.
Doubling or halving a recipe while cooking; sharing food equally.
Reading a clock, a calendar, a bus timetable or a price tag with a discount.

Many more such situations exist — discuss more examples in class.

Q2. How has mathematics helped propel humanity forward?

Ans. This is a teacher–student discussion question. Some points to talk about:

Science experiments: measuring, recording readings and drawing conclusions all need numbers.
Economy and democracy: banking, budgets, taxes, and counting votes in an election.
Buildings and bridges: engineers calculate loads, lengths and angles so structures stay safe.
Machines and gadgets: TVs, mobile phones and computers run on numbers and logic.
Travel: bicycles, trains, cars and planes are designed and their routes planned using maths.
Timekeeping: calendars and clocks are built on careful counting of patterns in the sky.

1.2 Patterns in Numbers

The simplest patterns in mathematics are patterns of numbers, especially the whole numbers:

0, 1, 2, 3, 4, …

The branch of Mathematics that studies patterns in whole numbers is called number theory.
Number sequences are the most basic — and among the most interesting — patterns that mathematicians study.

Table 1: Examples of Number Sequences

Sequence	Name	Rule for forming it	Next three numbers
1, 1, 1, 1, 1, 1, 1, …	All 1's	Every term is 1	1, 1, 1
1, 2, 3, 4, 5, 6, 7, …	Counting numbers	Add 1 to the previous term	8, 9, 10
1, 3, 5, 7, 9, 11, 13, …	Odd numbers	Start at 1, add 2 each time	15, 17, 19
2, 4, 6, 8, 10, 12, 14, …	Even numbers	Start at 2, add 2 each time	16, 18, 20
1, 3, 6, 10, 15, 21, 28, …	Triangular numbers	Add the next counting number: +2, +3, +4, …	36, 45, 55
1, 4, 9, 16, 25, 36, 49, …	Squares	Multiply a number by itself (n × n)	64, 81, 100
1, 8, 27, 64, 125, 216, …	Cubes	Multiply a number by itself thrice (n × n × n)	343, 512, 729
1, 2, 3, 5, 8, 13, 21, …	Virahānka numbers	Add the two terms just before it (5 = 2+3, 8 = 3+5, 13 = 5+8)	34, 55, 89
1, 2, 4, 8, 16, 32, 64, …	Powers of 2	Double the previous term	128, 256, 512
1, 3, 9, 27, 81, 243, 729, …	Powers of 3	Multiply the previous term by 3	2187, 6561, 19683

Figure it Out (Page 3)

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

Ans. Yes, each one has a clear rule (see the "Rule" column above). In short:

Powers of 2: 1, 2, 4 = 2 × 2, 8 = 2 × 2 × 2, 16 = 2 × 2 × 2 × 2, …
Powers of 3: 1, 3, 9 = 3 × 3, 27 = 3 × 3 × 3, …
Virahānka numbers: 1, 2, 3, 5 = 2 + 3, 8 = 3 + 5, 13 = 5 + 8, …
The remaining patterns are shown as pictures in Table 2.

Q2. Rewrite each sequence of Table 1 with the next three numbers, and write the rule in your own words.

Ans. The full answer is given in the Table 1 chart above — the last two columns give the rule and the next three numbers for every sequence. Copy them into your notebook.

1.3 Visualising Number Sequences

Many number sequences can be drawn as pictures. Seeing a number as a picture of dots often makes its rule obvious, and it helps us understand ideas that look hard in words.

Table 2: Pictorial Representation of Some Number Sequences

Dot pictures: All 1's, Counting, Odd, Even, Triangular, Squares

Cube numbers: 1, 8, 27, 64, 125 …

Figure it Out (Page 5)

Q1. Copy the pictures of Table 2 and draw the next picture in each sequence.

Ans. Draw one more step of each dot picture:

Sequence	Next picture	Next number
All 1's	One single dot again	1
Counting numbers	A row of 6 dots	6
Odd numbers	A row of 6 dots above a row of 5 dots	11
Even numbers	Two rows of 6 dots each	12
Triangular numbers	A triangle with 6 dots in the bottom row	21
Squares	A 6 × 6 grid of dots	36
Cubes	A 6 × 6 × 6 block	216

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

Ans. Refer to Table 2 above and check the pictures yourself. The names come straight from the shapes the dots make:

Triangular numbers: that many dots can be packed neatly into a triangle (rows of 1, 2, 3, 4 … dots).
Square numbers: that many dots fill a perfect square grid (n rows of n dots).
Cubes: that many small blocks build a solid cube (n × n × n).

Q3. 36 is both a triangular number and a square number! Make pictures showing this.

Ans. Refer to Table 2 (page 4) and draw. 36 dots can be arranged in both ways:

36 is triangular AND square

This shows that one number can be pictured in more than one way, and can play different roles depending on the situation. Try picturing other numbers in different ways too.

Q4. What would you call the sequence 1, 7, 19, 37, … ? What is the next number?

Ans. These are hexagonal numbers — the dots form rings around a centre dot, making a hexagon shape.

The rings add 6, 12, 18, 24, … dots: 1, 1+6 = 7, 7+12 = 19, 19+18 = 37, 37+24 = 61.
Next number = 61.

Hexagonal numbers: 1, 7, 19, 37, 61 …

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

Ans.

Powers of 2: start with a dot. Copy the whole figure and join each old dot to its new copy. Dot → line (2) → square (4) → cube (8) → double-cube (16) → 32, and so on. Every step doubles the dots.
Powers of 3: use the same idea, but make two extra copies each time and join them. Dot (1) → triangle (3) → 3 joined triangles (9) → 27 → 81, and so on. Every step triples the dots.

Powers of 2: each picture is the earlier one plus a joined copy

1.4 Relations among Number Sequences

Different sequences are often linked to each other in surprising ways.

Example: What happens when we add up odd numbers?

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

Why does this happen? Will it go on forever? Yes — it goes on forever, and a picture tells us why.

A picture can explain it

Recall that a square number is the count of dots in a square grid. Now cut the square grid into L-shaped bands (elbows) starting from one corner. The bands hold 1, 3, 5, 7, … dots — exactly the odd numbers!

Each red elbow adds the next odd number of dots

Since such a picture can be drawn for a square of any size, this explains why adding up odd numbers starting from 1 always gives a square number.

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Rule: the sum of the first n odd numbers = n × n.

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

Ans. A 10 × 10 square → 100.

By imagining a similar picture, what is the sum of the first 100 odd numbers?

Ans. A 100 × 100 square → 10000.

Another relation: Adding up and down

1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36

So here is yet another road to the square numbers — count the counting numbers up and then back down.

Figure it Out (Page 8–9)

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

Ans. One neat way: tilt the square grid so it stands like a diamond. Now read the dots along the slanting rows. The rows contain 1, 2, 3, …, n, …, 3, 2, 1 dots — that is exactly "up and down". Since all these dots together form a square, the total must be a square number.

Slanting rows give 1, 2, 3, 2, 1 — and the whole shape is a square

Q2. What is the value of 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1?

Ans. The top number is 100, so the picture is a 100 × 100 square. Value = 100 × 100 = 10,000.

Q3. Which sequence do you get when you add the All 1's sequence up? And up and down?

Ans.

Adding up: 1, 1 + 1, 1 + 1 + 1, 1 + 1 + 1 + 1, … = 1, 2, 3, 4, … → the counting numbers.
Adding up and down: 1; 1 + (1+1) + 1; 1 + (1+1) + (1+1+1) + (1+1) + 1; … Written with 1's only, we simply count how many 1's there are: 1, 3, 5, 7, 9, … → the odd numbers.

Q4. Which sequence do you get when you add the Counting numbers up? Give a pictorial explanation.

Ans. 1, 1+2, 1+2+3, 1+2+3+4, … = 1, 3, 6, 10, … → the triangular number sequence.
Picture: place a row of 1 dot, then 2 dots below it, then 3, then 4 … The dots stack into a triangle. So each sum is a triangular number (refer Table 2). Try it with an isosceles right triangle arrangement too.

Q5. What happens when you add up pairs of consecutive triangular numbers (1+3, 3+6, 6+10, 10+15, …)?

Ans. We get 4, 9, 16, 25, … → the square numbers.
Why: take a triangle of dots and a slightly smaller triangle. Turn the smaller one upside down and slide it against the bigger one — together they fit exactly into a square. (Refer Table 2, page 4 for the dot picture.)

6 + 10 = 16 — the two triangles interlock into a 4 × 4 square

Q6. What happens when you add up powers of 2 starting with 1 (1, 1+2, 1+2+4, 1+2+4+8, …)? Now add 1 to each. What do you get? Why?

Ans.

The sums are 1, 3, 7, 15, 31, …
Adding 1 to each gives 2, 4, 8, 16, 32, … → the powers of 2 again!
Why: every one of these sums is exactly 1 less than the next power of 2. Picture two equal towers of blocks: 1 + 2 + 4 + 8 is one block short of 16, because the missing single block would complete the doubling. So sum + 1 = the next power of 2. (Refer the picture on page 6.)

Q7. What happens when you multiply triangular numbers by 6 and add 1?

Ans.

(1 × 6) + 1 = 7 | (3 × 6) + 1 = 19 | (6 × 6) + 1 = 37 | (10 × 6) + 1 = 61 | (15 × 6) + 1 = 91

We get 7, 19, 37, 61, 91, … → the hexagonal numbers (from the second one onward).
Why: a hexagon has 6 identical triangular slices around one centre dot. So 6 triangles + 1 centre dot = a hexagon. (For the picture, refer Q4 on page 5.)

Q8. What happens when you add up hexagonal numbers (1, 1+7, 1+7+19, 1+7+19+37, …)? Explain using a picture of a cube.

Ans. We get 1, 8, 27, 64, … → the cube numbers.
Why: look at a cube from one corner. The blocks nearest that corner form a hexagon-shaped layer, and each further layer is the next hexagonal number. Peeling a cube layer by layer from a corner gives exactly 1, 7, 19, 37, … So adding them back builds the whole cube. (For the picture, refer Table 2, page 4.)

Hexagonal layers of a cube

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

Ans. Some patterns you can find and check:

Triangular number × 2 = a rectangle: 2 × 6 = 3 × 4, 2 × 10 = 4 × 5. Two equal triangles slide together into a rectangle.
Square − previous square = an odd number: 9 − 4 = 5, 16 − 9 = 7. This is just the extra L-shaped band.
Adding even numbers: 2, 2+4 = 6, 2+4+6 = 12, 2+4+6+8 = 20 → these are twice the triangular numbers.
Cube − previous cube: 8 − 1 = 7, 27 − 8 = 19, 64 − 27 = 37 → the hexagonal numbers, matching Q8.
Powers of 2 and 3: each new term is the previous one doubled or tripled, so the pictures keep copying themselves.

Draw a picture for each and check that the reason is clear.

1.5 Patterns in Shapes

Besides numbers, mathematics also studies patterns of shapes. These shapes may be in one, two or three dimensions (1D, 2D, 3D) — or even more. The branch that studies patterns in shapes is called geometry.

Shape sequences are one important kind of shape pattern.

Table 3: Examples of Shape Sequences

Shape sequence	The members	Rule for the next shape
Regular Polygons	Triangle, Quadrilateral, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon	Add one more side (and one more corner) each time
Complete Graphs	K2, K3, K4, K5, K6	Add one more point, then join it to every existing point
Stacked Squares	1, 2 × 2, 3 × 3, 4 × 4, 5 × 5 grids	Add one more row and one more column of little squares
Stacked Triangles	Triangles built from 1, 4, 9, 16, 25 little triangles	Add one more row of little triangles at the bottom
Koch Snowflake	Triangle → star → frillier star …	Replace every straight line '—' by a "speed bump" (a middle bump)

Regular Polygons — sides go 3, 4, 5, 6, 7, 8, 9, 10 …

Complete Graphs: every point joined to every other point

Stacked Squares and Stacked Triangles

Koch Snowflake: 3, 12, 48, 192, 768 …

Figure it Out (Page 11)

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

Ans. Yes.

Regular Polygons: 3, 4, 5, 6, 7, 8, 9, 10 — one way to read this is the number of sides of each shape, growing one at a time.
Complete Graphs: 1, 3, 6, 10, 15 (the number of lines).
Stacked Squares: 1, 4, 9, 16, 25 (little squares).
Stacked Triangles: 1, 4, 9, 16, 25 (little triangles).
Koch Snowflake: 3, 3×4, 3×4×4, 3×4×4×4, 3×4×4×4×4 (line segments).

Q2. Redraw each sequence in Table 3. Can you draw the next shape in each? Why or why not? Describe the rule.

Ans.

Regular Polygons: yes — draw an 11-sided shape with equal sides and equal corners.
Complete Graphs: yes — draw K7: seven points, each joined to every other.
Stacked Squares: yes — a 6 × 6 grid.
Stacked Triangles: yes — add a sixth row of little triangles.
Koch Snowflake: the rule is easy, but drawing gets harder each time, because the new bumps become smaller and smaller. We can draw only a few steps neatly by hand.

1.6 Relation to Number Sequences

Shape sequences and number sequences are often linked in surprising ways. These links help us understand both the shapes and the numbers better.

Example: Regular Polygons

The number of sides in the Regular Polygon sequence is given by the counting numbers starting at 3: 3, 4, 5, 6, 7, 8, 9, 10, … That is why the shapes are named regular triangle, quadrilateral (square), pentagon, hexagon, heptagon, octagon, nonagon, decagon, and so on.

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The word 'regular' means the shape has equal-length sides and equal angles — all sides look the same and all corners look the same. (Angles are studied in the next chapter.)

Figure it Out (Page 11–12)

Q1. Count the sides in each Regular Polygon. Which number sequence do you get? What about the corners? Are they the same? Why?

Ans.

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 … → yes, the same sequence.
Why: in any closed figure, number of sides = number of corners (vertices). Each side ends at a corner, and each corner is shared by exactly two sides — so they always match one-to-one.

Q2. Count the lines in each Complete Graph. Which number sequence do you get? Why?

Ans. 1, 3, 6, 10, 15 → the triangular number sequence.
Why: when a new point is added, it must be joined to every point already there. So we add 1, then 2, then 3, then 4 new lines … and adding the counting numbers up gives triangular numbers.

Graph	Points	New lines added	Total lines
K2	2	1	1
K3	3	2	3
K4	4	3	6
K5	5	4	10
K6	6	5	15

Q3. How many little squares are there in each Stacked Squares shape? Which number sequence does this give? Why?

Ans. 1, 4, 9, 16, 25 → the square number sequence.
Why: the n^th shape has n rows with n little squares in each row, so the total is n × n. Squares can be drawn using these numbers of dots as well.

Q4. How many little triangles are there in each Stacked Triangles shape? Which number sequence does this give? Why? (Hint: how many triangles in each row?)

Ans. 1, 4, 9, 16, 25 → the square number sequence.
Why: count row by row. The rows contain 1; then 1 + 2 + 1; then 1 + 2 + 3 + 2 + 1; and so on. This is the "adding up and down" pattern, which always gives square numbers.

1 | 1+2+1 = 4 | 1+2+3+2+1 = 9 | 1+2+3+4+3+2+1 = 16 …

Q5. To get the next Koch Snowflake, each line segment '—' is replaced by a 'speed bump'. How many line segments are in each shape? What is the corresponding number sequence?

Ans.

Total line segments in each shape: 3, 12, 48, 192, 768
Corresponding sequence: 3, 3 × 4, 3 × 4 × 4, 3 × 4 × 4 × 4, 3 × 4 × 4 × 4 × 4, … that is, 3 times the Powers of 4. (This sequence is not shown in Table 1.)
Why: each straight segment is replaced by a speed bump made of 4 smaller segments, so the count is multiplied by 4 at every step.

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Shape → Number links to remember for exams:

Shape sequence	What we count	Number sequence
Regular Polygons	Sides / Corners	Counting numbers from 3
Complete Graphs	Lines	Triangular numbers
Stacked Squares	Little squares	Square numbers
Stacked Triangles	Little triangles	Square numbers
Koch Snowflake	Line segments	3 × Powers of 4