Three iron cubes, each with an edge length of a positive integer, and the sum of their edge lengths is $38 \mathrm{~cm}$. After melting them, a cube with an edge length of $20 \mathrm{~cm}$ is obtained. What are the edge lengths of the original three iron cubes from largest to smallest?
Three cubes have edge lengths of $x \mathrm{~cm}, y \mathrm{~cm}$, and $z \mathrm{~cm}$, respectively, and $013^{3}$, then $z \geqslant 14$.
When $z=14$, if $y=14$, then $x=10$, but this does not satisfy equation (1), so $y<14$. Thus, $x \leqslant y \leqslant 13$.
Therefore, $x^{3}+y^{3}+z^{3} \leqslant 13^{3}+13^{3}+14^{3}<8000$, which is a contradiction.
So, $z \geqslant 15$.
Since $z^{3}<20^{3}$, it follows that $z<20$. Thus, $z \leqslant 19$.
Hence, $15 \leqslant z \leqslant 19$.
From equation (1), we have $x^{3}+y^{3}=8000-z^{3}$, which means
$(x+y)\left(x^{2}-x y+y^{2}\right)=8000-z^{3}$.
Thus, $(x+y) \mid\left(8000-z^{3}\right)$, and we have
$(38-z) \mid\left(8000-z^{3}\right)$.
Table 1
\begin{tabular}{|c|c|c|}
\hline$z$ & $38-z$ & $8000-z^{3}$ \\
\hline 15 & 23 & 4625 \\
\hline 16 & 22 & 3904 \\
\hline 17 & 21 & 3087 \\
\hline 18 & 20 & 2168 \\
\hline 19 & 19 & 1141 \\
\hline
\end{tabular}

From Table 1, we can see that only $z=17$ satisfies equation (2). In this case,
\[
\left\{\begin{array}{l}x+y=21, \\ x^{3}+y^{3}=3087 .\end{array}\right.
\]
Dividing (4) by (3) gives
\[x^{2}-x y+y^{2}=147.\]
From equations (3) and (5), we get $x=7, y=14$ or $y=7, x=14$.
Since $0<x \leqslant y \leqslant z$, we take $x=7, y=14$.
Thus, the edge lengths of the three iron cubes, from largest to smallest, are $17 \mathrm{~cm}, 14 \mathrm{~cm}, 7 \mathrm{~cm}$.
(Du Jiang, College of Mathematics and Computer Science, Chongqing Normal University, 400047; Jia Qingzhu, Jishan Middle School, Shili District, Nanchong City, Sichuan Province, 400047)