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

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

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 animation. What do you see? Can you explain why this happens?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. 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?

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?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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?

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

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

This challenge is about finding the difference between numbers which have the same tens digit.

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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?

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

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 ...?

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

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

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?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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

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

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

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

What happens when you round these three-digit numbers to the nearest 100?

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

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

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 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.

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

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

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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?

Try out this number trick. What happens with different starting numbers? What do you notice?

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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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

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.

Find the sum of all three-digit numbers each of whose digits is odd.

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

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