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?
We have a set of four very innocent-looking cubes - each face coloured red, blue, green or white - and they have to be arranged in a row so that all of the four colours appear on each of the four long sides of the resulting cuboid.
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?
In the cube illustrated the point $P$ is the midpoint of $AB$
and the point $Q$ is one quarter of the way along the edge
The plane through $PDQ$ cuts the cube into two. Find the ratio
of the volumes of the two pieces.