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. . . .

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.

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

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?

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

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?

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?

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

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?

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?

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

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

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

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

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

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?

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

How many models can you find which obey these rules?

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

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

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

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?

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?

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

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.

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

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.

How many faces can you see when you arrange these three cubes in different ways?

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