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

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?

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.

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?

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

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

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?

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

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

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!

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 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?

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

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

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

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

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?

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.

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

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.

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

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

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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.

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

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

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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.

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

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

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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?

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

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Can all unit fractions be written as the sum of two unit fractions?

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

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

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.

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

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

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?