Four Points on a Cube

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?

Instant Insanity

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.

Classic Cube

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?

Plane to See

Stage: 5 Challenge Level:

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 $EF$.

The plane through $PDQ$ cuts the cube into two. Find the ratio of the volumes of the two pieces.