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

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?

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?

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

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

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 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.

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

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

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

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?

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?

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

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

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

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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?

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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

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

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

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 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.

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

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

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

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

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

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

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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

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

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.

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

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?

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

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

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?

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.

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.