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

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

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?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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

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

Investigate the totals you get when adding numbers on the diagonal of this pattern in threes.

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

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

A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

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?

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?

Can you sort numbers into sets? Can you give each set a name?

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

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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

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

Show that all pentagonal numbers are one third of a triangular number.

Can you find any perfect numbers? Read this article to find out more...

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

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?

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

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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

What is the remainder when 2^{164}is divided by 7?

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.

An account of some magic squares and their properties and and how to construct them for yourself.

What are the coordinates of this shape after it has been transformed in the ways described? Compare these with the original coordinates. What do you notice about the numbers?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

What is the total number of squares that can be made on a 5 by 5 geoboard?