Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Can you find a way of counting the spheres in these arrangements?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

Watch this animation. What do you see? Can you explain why this happens?

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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?

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

Delight your friends with this cunning trick! Can you explain how it works?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

How many centimetres of rope will I need to make another mat just like the one I have here?

Find out what a "fault-free" rectangle is and try to make some of your own.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Can you describe this route to infinity? Where will the arrows take you next?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Are these statements always true, sometimes true or never true?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

This challenge asks you to imagine a snake coiling on itself.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?