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?

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

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

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

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

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

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?

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.

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

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?

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.

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.

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 explain the strategy for winning this game with any target?

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

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

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

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

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

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

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?

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

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

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

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.

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.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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

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

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

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

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?

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

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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.

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

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

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

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