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

Question

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

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

This has an english translation of the problem:

young-il-long-kiyoshi.tumblr.com/post/125623681248/on-the-topic-of-the-final-maths-question-in-the

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Another link I found mentioning the problem but not explaining it.

angryanimebitches.com/2016/03/assassination-classroom-season-2-episode-12-the-principal-is-a-sore-loser/

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Their is also a youtube video that explains Karma-kun's answer but even his explanation I still can't understand how he to the conclusion that the area of the center atom is equal to the half the area of the cube divided by two. a^3/2

video link: Weird Geometry (Assassination Classroom S2 E12)

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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 in "high-school" level of perspective, not a professional PhD level one...

I don't want to hear "equidistant", "inifinitesimal", "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 the 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:

Answer 1

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.

Answer 2

I know I'm late to this question, but 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 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.

Answer 3

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.