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

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?

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

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 find a way of counting the spheres in these arrangements?

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.

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?

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?

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

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

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 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 you’re left with odd or even? How do the images help to explain this?

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

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

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

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'.

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

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

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

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!

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.

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

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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

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.

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.

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

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

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?

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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.

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

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?

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

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

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.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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

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

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?

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?

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?

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

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?