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

Can you explain the strategy for winning this game with any target?

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?

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

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

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

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?

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

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?

Got It game for an adult and child. How can you play so that you know you will always win?

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

This activity involves rounding four-digit numbers to the nearest thousand.

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?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Can you find sets of sloping lines that enclose a square?

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.

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

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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

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?

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

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.

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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

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

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

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.

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

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'.

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

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

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.

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

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?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?