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

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

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

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?

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?

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

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

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?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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.

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?

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

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

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

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

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

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?

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

Try out this number trick. What happens with different starting numbers? What do you notice?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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?

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

An investigation that gives you the opportunity to make and justify predictions.

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

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

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

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

What happens when you round these three-digit numbers to the nearest 100?

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

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

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?

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

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?

Find the sum of all three-digit numbers each of whose digits is odd.

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.

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?

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

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

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?

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?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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