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 find a way of counting the spheres in these arrangements?

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

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?

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

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

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?

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

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?

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?

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

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?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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.

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

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

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?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

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

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.

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

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 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.

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

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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

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?

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

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?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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

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?

Explore the effect of combining enlargements.

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Can all unit fractions be written as the sum of two unit fractions?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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