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 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?

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?

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.

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?

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?

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?

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?

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

How many models can you find which obey these rules?

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

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

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

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

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 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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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