Question

good problem:

show that four non consecutive vertices of a cube form the vertices of a regular tetrahedron

hence calculate the volume of a tetrahedron with unit sides

hence calculate the volume of an octahedron with unit sides

show that the centre points of each cube face are the vertices of a regular octahedron inside the tetrahedron

prove that the volume of this octahedron is half the volume of the tetrahedron