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?

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?

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

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

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

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

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

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?

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

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

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?

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?

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

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

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?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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.

What's the largest volume of box you can make from a square of paper?

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?

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?

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

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

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

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

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

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

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.

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

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

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

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?

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

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Here are two kinds of spirals for you to explore. What do you notice?

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