Clear explanation of Karma-kun's answer to the final math problem in Assassination Classroom?


I've been trying to understand Karma-kun's answer to the math problem in the
Examination Finals.

I have searched for references and explanations for the problem but to no
avail. I still couldn't understand the horrible explanations for the problem,
and the answer assumes that you know their thought process is, which I don't.

[This](https://young-il-long-kiyoshi.tumblr.com/post/125623681248/on-the-
topic-of-the-final-maths-question-in-the) has an English translation of the
problem:

> The translated problem reads: “ _As shown in the diagram to the right, line
> segment A aligns itself periodically into a cube, and atoms are located at
> each vertex. Within, forming a uniform crystal lattice structure, Na, K, and
> other alkali metals attack the crystal lattice. Within the cube, focus on a
> certain atom A and within the points in the nearby space, let D represent
> the area that comprises the atoms closest to A. Find the volume of D._ ”
>
> This is a better worded question that I found on mangahere in the tsukkomis.
> Credits to the writer: “ _A bunch of atoms surround Atom A. Imaginary lines
> of length (a) from the centers of the atoms form a cube around A. Find the
> volume Atom A takes up, if it has the same volume as all the atoms. Hint:
> use the volume of the cube._ ”

[Another link](https://angryanimebitches.com/2016/03/assassination-classroom-
season-2-episode-12-the-principal-is-a-sore-loser/) I found mentioned the
problem but not explaining it.

There is also [a YouTube video](https://www.youtube.com/watch?v=itpDijhPlPQ)
that explains Karma-kun's answer but even with his explanation, I still can't
understand how he came to the conclusion that the area of the center atom is
equal to half the area of the cube divided by two. **a^3/2**.

* * *

The reason I'm posting this is that there is no other way for me to understand
the problem without asking in a thread. I just want an explanation of the
answer from a "high-school" level perspective, not a professional Ph.D. level
one. I don't want to hear "equidistant", "infinitesimal", "vertex" or any
other technical terms that I can't comprehend.

I know I demand too much, but please help me understand how Karma-kun got
**a^3/2**. The only perspective that I got from Karma is that all 8 atoms form
a cube of their own, and the middle point of that imaginary cube is equivalent
to all the atoms inside the 8 individual cubes.

And the video says so as well, but how the in the world has it gone from **A
1/8** to **2A**? and then **a^3/2**

as shown here:

![](https://i.stack.imgur.com/7oJPQm.png)




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.

[![a unit
cell](https://i.stack.imgur.com/cueyn.jpg)](https://i.stack.imgur.com/cueyn.jpg)  
(source: [sciencehq.com](https://www.sciencehq.com/wp-content/uploads/cubic-
unit-cell.jpg))

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.




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.

[![a unit
cell](https://i.stack.imgur.com/cueyn.jpg)](https://i.stack.imgur.com/cueyn.jpg)  
(source: [sciencehq.com](https://www.sciencehq.com/wp-content/uploads/cubic-
unit-cell.jpg))

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.

