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

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

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.

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

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

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?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

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

Investigate these hexagons drawn from different sized equilateral triangles.

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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

Can you figure out how sequences of beach huts are generated?

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?

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

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?

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?

What's the greatest number of sides a polygon on a dotty grid could have?

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.

Which of these pocket money systems would you rather have?

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

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Can you find a way to identify times tables after they have been shifted up or down?

Just because a problem is impossible doesn't mean it's difficult...

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?

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.

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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

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