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.

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?

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

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?

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

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

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?

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

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

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

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

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

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.

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

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

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

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

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

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?

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

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?

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

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

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?

Explore the effect of combining enlargements.

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

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

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.

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

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

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

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?

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

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

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

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

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

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?

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

Explore the effect of reflecting in two intersecting mirror lines.

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?

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.