I am trying to tie up some loose ends.

This is a more detailed proof of the original problem:

任何一个长方形盒子不可能用不全等的立方体填满.

A more accurate statement of the problem is given by I94: For any (rectangular) box, there is no way to use a finite set of cubes of different sizes to completely fill it up.

Suppose there is a fillable box. Then there is a face of the box that touches more than one cube. We call this face the bottom of the box, and use the professor's idea of going up to find an infinite sequence of shrinking cubes.

Among the cubes that touches the bottom, there is a smallest one. Let's call it C1. C1 cannot touch any edge of the bottom. So it is surrounded on its four side faces by larger cubes that also touch the bottom. The top of C1, then, is lower than the tops of the surrounding cubes. The sides of the surrounding cubes and the top of C1 form a "well". Thus any cube that touches the top of C1 must have its bottom completely contained in the well. Let C2 be the smallest such cubes. Again C2 cannot touch any edge of C1. So it s surrounded on its four sides by larger cubes that also touch the top of C1. Continue this process we can form an infinite sequence of cubes C1, C2, ... such that C1 > C2 > ....

A question is whether we can fill a box with infinitely many cubes of different sizes. I think the answer is no. (Just a wild guess, no reason.)