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

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?

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 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?

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

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?

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?

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

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?

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

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.

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

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

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

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.

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

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

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?

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

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

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.

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?

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

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?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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

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

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?

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?

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

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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

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?

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?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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?

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

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?

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?

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

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?