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So basically, my interpretation is that the problem is asking for the volume of the region inside a cube that is closer to its center than any of its vertices. 
 
So basically, my interpretation is that the problem is asking for the volume of the region inside a cube that is closer to its center than any of its vertices. 
   
Gakushu approached the problem by drawing the bisecting planes between the center of the cube and its vertices, then manually finding the volume of the resulting truncated octahedron. Amusingly, though, I'm pretty sure this solution could have worked nicely with a little more thought: if he divided the cube into eight smaller sub-cubes (in the obvious way), he would have noticed that the desired domain takes up exactly half of each sub-cube, and thus half the entire cube.
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Gakushu's approach for the problem consisted of drawing the bisecting planes between the center of the cube and its vertices, then manually finding the volume of the resulting truncated octahedron. Amusingly, though, I'm pretty sure this solution could have worked nicely with a little more thought: if he divided the cube into eight smaller sub-cubes (in the obvious way), he would have noticed that the desired domain takes up exactly half of each sub-cube, and thus half the entire cube.
   
 
Karma's solution was more indirect. Basically, he noted that if the center and vertices were all identical atoms, then they would all have an equally large domain in the space. The cube, then, consists of 1/8 the domain of each vertex atom and the full doman of the center atom, so the domain of the center atom takes up half the cube.
 
Karma's solution was more indirect. Basically, he noted that if the center and vertices were all identical atoms, then they would all have an equally large domain in the space. The cube, then, consists of 1/8 the domain of each vertex atom and the full doman of the center atom, so the domain of the center atom takes up half the cube.

Latest revision as of 02:14, February 21, 2020

Wait okay, so I think I finally understand the what's happening in the problem.

So basically, my interpretation is that the problem is asking for the volume of the region inside a cube that is closer to its center than any of its vertices. 

Gakushu's approach for the problem consisted of drawing the bisecting planes between the center of the cube and its vertices, then manually finding the volume of the resulting truncated octahedron. Amusingly, though, I'm pretty sure this solution could have worked nicely with a little more thought: if he divided the cube into eight smaller sub-cubes (in the obvious way), he would have noticed that the desired domain takes up exactly half of each sub-cube, and thus half the entire cube.

Karma's solution was more indirect. Basically, he noted that if the center and vertices were all identical atoms, then they would all have an equally large domain in the space. The cube, then, consists of 1/8 the domain of each vertex atom and the full doman of the center atom, so the domain of the center atom takes up half the cube.

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