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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)


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assassination-classroom
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.

[![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.


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Here's a perhaps simpler explanation. I just got to this problem in the anime
and wanted to solve it before they revealed the answer.

Ok. You have a cube with an atom in the middle. What you'll want to do is
break the cube into eight pieces. Take one of these pieces. There are two
atoms on opposite corners (vertexes). One of these atoms is the central atom
[A]0 and the other is the atom on the edge. It goes by simple logic that
exactly half of this eighth of a cube is closer to one atom than the other.
This goes for all the other 8 pieces, so the volume of the cube closer to the
central atom is simply half the total volume.

The problem is much simpler than it appears and I admit computing the volume
by geometric pieces before I realized I was being dumb. It doesn't require
very much math. Just a solid grasp of logic and a bit of cleverness.


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Supposedly, the answer itself in the manga isn't even correct. If the atoms
occupy the same amount of space within each cube, then they would be one half
of (a) - the invisible lines). You could essentially plug it into the formula
for solving the volume of a sphere which is four over three times pi times r
to the power 3. In this case, your r (radius) is a/2 (because the atoms take
up the same amount of space, therefore one atom would take up one half of the
invisible line)

Basically, you would do V = 4/3*π*(a/2)^3

V = 4/3*π*a^3/8

V = 4 (in 4/3) cancels out the 8 (In a^3/8) so you get 1/3*π*a^3/2

**

V = π*a^3/6

**

Misc: * = times, ^ = to the power of, π = pi.


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