This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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?

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

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?

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?

A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

When dice land edge-up, we usually roll again. But what if we didn't...?

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

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

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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?

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

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

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

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?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

The challenge for you is to make a string of six (or more!) graded cubes.

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?

If you had 36 cubes, what different cuboids could you make?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

What is the largest cuboid you can wrap in an A3 sheet of paper?

How many models can you find which obey these rules?

How can we as teachers begin to introduce 3D ideas to young children? Where do they start? How can we lay the foundations for a later enthusiasm for working in three dimensions?

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

Are these statements always true, sometimes true or never true?

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

Make a cube with three strips of paper. Colour three faces or use the numbers 1 to 6 to make a die.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Which faces are opposite each other when this net is folded into a cube?

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?

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

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

A description of how to make the five Platonic solids out of paper.

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

A task which depends on members of the group working collaboratively to reach a single goal.