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

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?

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"?

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

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?

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

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

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

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

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

Surprise your friends with this magic square trick.

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?

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

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

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

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

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?

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?

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

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

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

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.

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

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

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

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

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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

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?

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

Explore the effect of reflecting in two intersecting mirror lines.

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

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

Explore the effect of combining enlargements.

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

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?

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

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

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.