It starts quite simple but great opportunities for number discoveries and patterns!

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Can you find examples of magic crosses? Can you find all the possibilities?

"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Here are some ideas to try in the classroom for using counters to investigate number patterns.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

An environment which simulates working with Cuisenaire rods.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

These lower primary activities offer opportunities for children to create, recognise and extend number patterns.

Here are some more lower primary number pattern tasks for you to try.

This activity creates an opportunity to explore all kinds of number-related patterns.

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Can you go from A to Z right through the alphabet in the hexagonal maze?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

In this article for teachers, Bernard Bagnall describes how to find digital roots and suggests that they can be worth exploring when confronted by a sequence of numbers.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Investigate these hexagons drawn from different sized equilateral triangles.

How many different sets of numbers with at least four members can you find in the numbers in this box?

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

These upper primary activities offer opportunities for children to recognise, extend and explain number patterns.

This article for primary teachers outlines how we can encourage children to create, identify, extend and explain number patterns and why being able to do so is useful.

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my square?

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Investigate the successive areas of light blue in these diagrams.

This number has 903 digits. What is the sum of all 903 digits?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Find the next number in this pattern: 3, 7, 19, 55 ...

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When did July 1st fall on a Monday again?