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Squaring the Square -> Cubing the Rectangular Prism

 Quote by TheFerruccio I mean, given a rectangular prism of dimensions $$x_1, x_2, x_3$$, where $$x_1 \ge x_2 \ge x_3$$ can it be perfectly and finitely filled with unique cubes of edge length $$x_n$$, where $$x_{n+1} < x_{n}$$ and $$x_n < x_3 \forall n > 3$$
Are there any other restrictions? Do the values have to be integer or rational numbers of edge size or can they be real numbers of any kind?