Cubes and cuboids

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?

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?

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 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

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?

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

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