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?

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

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

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?

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?

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?

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?

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

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?

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

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

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?

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

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.

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

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

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

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

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

Triangular 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.

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

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

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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?

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 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.

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

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

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

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

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?

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

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

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

What happens when you round these numbers to the nearest whole number?

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.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?