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

Here's a simple way to make a Tangram without any measuring or ruling lines.

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?

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

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

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

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

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

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

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

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?

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

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?

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?

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?

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

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

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

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

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

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

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 ...

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of this telephone?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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

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

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?

Exchange the positions of the two sets of counters in the least possible number of moves

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

Can you fit the tangram pieces into the outline of Granma T?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

Can you fit the tangram pieces into the outlines of these people?

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

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

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

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of this plaque design?

Can you fit the tangram pieces into the outline of the rocket?