What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

What is the greatest number of squares you can make by overlapping three squares?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Exploring and predicting folding, cutting and punching holes and making spirals.

Can you find ways of joining cubes together so that 28 faces are visible?

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?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

What is the best way to shunt these carriages so that each train can continue its journey?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Can you visualise what shape this piece of paper will make when it is folded?

Make a flower design using the same shape made out of different sizes of paper.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

What shape is made when you fold using this crease pattern? Can you make a ring design?

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

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Can you cut up a square in the way shown and make the pieces into a triangle?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

Reasoning about the number of matches needed to build squares that share their sides.

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Make a cube out of straws and have a go at this practical challenge.

Can you make a 3x3 cube with these shapes made from small cubes?

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

Can you work out what is wrong with the cogs on a UK 2 pound coin?

Which of these dice are right-handed and which are left-handed?

Can you fit the tangram pieces into the outlines of the convex shapes?