Clear explanation of karma-kun's answer to the final math problem in Assassination Classroom?
All the atoms in the crystal lattice are identical. Let's say the volume of an atom at the vertex (not corner, if it is a corner, you will need a slightly different reasoning) is B. Each atom at the vertex takes up (1/8)B of the cube.  See diagram.



For a cube with length a, it consists of the central atom A with volume D and  8 atoms at the vertices each taking up (1/8)B of the cube. Thus,

a3 = D + 8 × (1/8)B

However, since we  know that the atoms at the vertices are actually identical to the central atom, The volume B is equal to D and hence,

a3 = B + 8 × (1/8)B ==> B = (1/2)a3

QED

All the technical terms that you have mentioned are what I was learning in my high school. This question is not hard, as Karma has said. It is a combined question of maths and chemistry.