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.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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?

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?

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

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?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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?

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

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?

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

How many models can you find which obey these rules?

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 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?

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

This problem provides training in visualisation and representation of 3D shapes. You will need to imagine rotating cubes, squashing cubes and even superimposing cubes!

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

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

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

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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. . . .

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?

There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?

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

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?

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

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?

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

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?

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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. . . .

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?

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?

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

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

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