A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

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.

Plane to See

Age 16 to 18 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.