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.

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?

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

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?

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?

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

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

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?

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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?

Triangle 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.

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

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?

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

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

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

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

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?

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

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

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

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.

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?

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

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?

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

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

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

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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 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.

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

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.