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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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

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

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

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

What is the shape of wrapping paper that you would need to completely wrap this model?

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?

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?

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?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

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

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?

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

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

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?

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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

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

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?

Which of the following cubes can be made from these nets?

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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

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.

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

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

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

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.

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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?

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

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?

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?

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

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .