What comes next? Meeting patterns

Look at this row of numbers: 2, 4, 6, 8, … What comes next? Almost everyone says 10 straight away. Now try 1, 4, 9, 16, … That one takes a moment longer, but many people spot it: 25.

How did you know? Nobody told you the answer. You looked at the numbers you had, noticed something that stayed the same each time, and trusted it to carry on. That noticing is the heart of mathematics. A pattern is something that repeats or grows in a way you can describe, and the description is called the rule.

In this layer you will hunt for patterns in bangles and beads, in rangoli and kolam borders, in steps and stacks, in the hundred square and on the calendar, and in shapes made from matchsticks and dots. By the end you will be able to answer the question on the cover of this topic: how can you predict the 100th term without drawing 100 pictures?

Before you learned to read, you were already reading patterns. Day follows night follows day. Monday follows Sunday. In a song, the chorus comes back. On a train journey, the wheels go clickety-clack, clickety-clack over the rail joints.

Walk around your home or school and you will find more:

• Floor tiles that repeat square after square.
• Bangles stacked green, gold, green, gold.
• A rangoli or kolam at the door, where the same little shape is drawn again and again around a border.
• Steps on a staircase, each one the same height above the one before.
• Cricket overs: six balls, then a new over, then six more balls.
• The ticking of a clock: after 12 comes 1 again.

A pattern lets you predict. If you know the rule, you know what comes next without waiting to see it. That is why patterns matter so much: in music, weaving, building, computer programs, weather records and in all of mathematics.

Mathematicians sort patterns in many ways, but two big families are enough to start with.

In a repeating pattern, a small group of things, the repeating unit (sometimes called the core), comes round again and again. The pattern never really gets bigger; it just keeps going.

In a growing pattern, each step changes the one before by the same kind of rule: one more row, two more sticks, double the dots. The pattern keeps getting bigger (or, in a shrinking pattern, smaller).

Number patterns and shape patterns can be either kind. Most of this topic is about growing patterns, because they are where the surprises are.

Look at a string of beads: red, red, blue, red, red, blue, red, red, blue, …

The group red, red, blue repeats. That group is the repeating unit. It has 3 beads in it, so we can call it an AAB pattern: two of one thing, then one of another.

Other repeating units you will meet:

Name	Example	Unit length
AB	green, gold, green, gold, …	2
ABC	▲ ● ■ ▲ ● ■ …	3
AAB	red, red, blue, …	3
ABB	clap, stamp, stamp, …	3
ABCD	Mon, Tue, Wed, Thu, … (only part of the week)	4

The trick for any repeating pattern is to find the unit first, then count in units.

Repeating patterns are one of the oldest kinds of art in India.

A kolam, drawn every morning at the doorstep in many homes in Tamil Nadu and neighbouring states, starts with a grid of dots made from rice flour. Lines loop around the dots, and in a kolam border the same loop shape is drawn again and again all the way along. Muggulu in Andhra Pradesh and Telangana, rangoli in Maharashtra and Gujarat, alpana in Bengal and mandana in Rajasthan use the same idea.

Look closely and you will see two kinds of repetition:

• Along a line (a border): the unit is repeated by sliding it along. This is called a frieze or strip pattern.
• Around a centre (a flower or star shape): the unit is repeated by turning it. This makes the design look the same when you turn it by a quarter or a sixth of a turn.

Block-printed saris from Bagru and Sanganer, Kanchipuram silk borders, beadwork from the Toda and Banjara communities, the jali screens of old buildings, and the tiles on a railway station floor all use repeating units in the same way.

At a birthday party someone builds a pyramid of paper cups. The top row has 1 cup. The row under it has 2. Then 3, then 4, then 5.

How many cups altogether for a pyramid with 1 row, 2 rows, 3 rows, …?

Rows	Cups in the new bottom row	Total cups
1	1	1
2	2	1 + 2 = 3
3	3	3 + 3 = 6
4	4	6 + 4 = 10
5	5	10 + 5 = 15

The totals 1, 3, 6, 10, 15 make a growing pattern. Each time, you add one more than last time: add 2, then 3, then 4, then 5. These numbers are so famous they have a name, the triangular numbers, because you can arrange that many dots in a triangle.

Not every growing pattern grows by a changing amount. Some grow by the same amount every time:

• A tower of blocks, adding 2 blocks each day: 2, 4, 6, 8, …
• A staircase in a building where each step is 15 cm higher: 15, 30, 45, 60, … cm above the ground.
• Your piggy bank, if you put in ₹10 every Sunday and started with ₹50: ₹50, ₹60, ₹70, ₹80, …

A good first question for any growing pattern is: "By how much does it change each time?" Write the changes (the differences) underneath the pattern. If they are all the same, the pattern grows steadily. If they change, look for a pattern in the differences themselves.

The very first number patterns you learned were counting patterns:

• Counting on in ones: 1, 2, 3, 4, 5, …
• Counting in twos: 2, 4, 6, 8, … (pairs of socks, pairs of shoes)
• Counting in fives: 5, 10, 15, 20, … (fingers on hands, ₹5 coins)
• Counting in tens: 10, 20, 30, 40, … (₹10 notes)
• Counting in hundreds: 100, 200, 300, … (₹100 notes)

Counting in threes gives the 3 times table: 3, 6, 9, 12, 15, … Counting in any number gives that number's times table, and the numbers you land on are its multiples.

You can also count back: 50, 45, 40, 35, … (take away 5 each time) or 100, 90, 80, 70, … (take away 10).

Try to share some sweets between two friends so that both get the same number, with none left over.

• 6 sweets: 3 each. ✓
• 7 sweets: 3 each and 1 left over. ✗

Numbers that can be split into pairs with nothing left over are even: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, … Numbers that always leave one left over are odd: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …

If you line up counters in two rows, an even number makes two neat rows. An odd number always has one counter sticking out, like a little tail.

The last digit tells you straight away:

• Even numbers end in 0, 2, 4, 6 or 8.
• Odd numbers end in 1, 3, 5, 7 or 9.

So 3,578 is even and 1,00,001 is odd, however big they are. On a number line, odd and even take turns: odd, even, odd, even, … It is a repeating pattern with a unit of 2.

A hundred square is a grid of the numbers 1 to 100 in ten rows of ten. It is packed with patterns.

• Going across a row, each number is 1 more than the one before.
• Going down a column, each number is 10 more than the one above: 3, 13, 23, 33, … The last digit stays the same, and the tens digit goes up by one.
• The last column is 10, 20, 30, …, 100: counting in tens.
• The diagonal going down to the right goes up by 11: 1, 12, 23, 34, 45, …
• The diagonal going down to the left goes up by 9: 10, 19, 28, 37, 46, … These are the multiples of 9 (plus the column shift): 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 sit on a slanting line.

If you shade every multiple of 2, you get stripes going down (columns 2, 4, 6, 8, 10). Shade the multiples of 5 and you get two stripes (columns 5 and 10). Shade the multiples of 3 and you get slanting lines. Shade the multiples of 9 and you get one slanting line.

A calendar page is a number grid with 7 columns, one for each day of the week. That single fact creates lots of patterns.

In September 2026, the 1st is a Tuesday. The Mondays are the 7, 14, 21, 28th: they go up by 7, because a week has 7 days.

• Down a column (same weekday, next week): add 7.
• Across a row: add 1.
• Diagonal down-right: add 8 (one week plus one day).
• Diagonal down-left: add 6 (one week minus one day).

So if your birthday is on a Friday the 2nd, then the 9th, 16th, 23rd and 30th of that month are also Fridays. (In October 2026 the 2nd is a Friday; check a calendar.)

Take some matchsticks (or toothpicks, or ice-cream sticks) and make a square: that needs 4 sticks. Now add a second square joined to the first. You do not need 4 more sticks, because the two squares share a side. You only need 3 more: 7 sticks for 2 squares.

Squares in a row	1	2	3	4	5	6
Matchsticks	4	7	10	13	16	19

The matchstick pattern goes 4, 7, 10, 13, 16, 19: add 3 each time. Every new square needs one top, one bottom and one side, because the other side is already there.

Now try dots. Arrange dots in squares: a 1 by 1 square, a 2 by 2 square, a 3 by 3 square, and so on.

Size	1 × 1	2 × 2	3 × 3	4 × 4	5 × 5	6 × 6
Dots	1	4	9	16	25	36

These are the square numbers: 1, 4, 9, 16, 25, 36, … Each one is a number multiplied by itself.

Arrange dots in triangles instead (1 dot on top, 2 in the next row, 3 in the next, …) and you get 1, 3, 6, 10, 15, 21: the same triangular numbers as the cup pyramid. A staircase of blocks, one block in the first column, two in the next, three in the next, is the same pattern again.

Imagine a number machine. You feed a number in, the machine does something to it, and a new number comes out. Feed that number back in, and so on. The instruction inside the machine is the rule.

• Machine “add 3”, start at 1: 1 → 4 → 7 → 10 → 13 → …
• Machine “double” (multiply by 2), start at 1: 1 → 2 → 4 → 8 → 16 → …
• Machine “take away 4”, start at 30: 30 → 26 → 22 → 18 → …
• Machine “halve”, start at 64: 64 → 32 → 16 → 8 → 4 → …

Notice how different “add 2” and “double” are, even when they start in the same place:

Start at 2	1st	2nd	3rd	4th	5th	6th
Add 2	2	4	6	8	10	12
Double	2	4	8	16	32	64

They agree for the first two terms, then the doubling pattern races away. Two terms are never enough to be sure of a rule.