Problem source

\begin{questionparts}
\item
Show that four vertices of a cube, no two of which are adjacent,
form the vertices of a regular tetrahedron. 
Hence, or otherwise, find the volume of a regular 
tetrahedron whose edges are of unit length.
\item
Find the volume of a regular octahedron whose edges are of unit length.
\item
Show that the centres of the faces of a cube form the vertices
of a regular octahedron. Show that its volume is half that of the 
tetrahedron whose vertices are the vertices of the cube.
\end{questionparts}
\noindent
[{\em A regular tetrahedron (octahedron) 
has four (eight) faces, all equilateral triangles.}]