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?

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?

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

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?

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?

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

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

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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.

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

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?

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

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

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?

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

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

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

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 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?

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

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

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

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?

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

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.

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 ...?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

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

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?

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?

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

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

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

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

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

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

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

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.