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?

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?

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

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

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

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?

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?

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

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

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?

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

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

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

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?

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

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

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

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

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

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

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?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

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.

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 find all the ways to get 15 at the top of this triangle of numbers?

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

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

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

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?

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

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?

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 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.

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

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

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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?

Explore the effect of reflecting in two intersecting mirror lines.

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

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?

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.

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?