What relationship must hold between the edges of a rectangular parallelepiped so that it can be cut into two identical rectangular parallelepipeds similar to the original?
Let the edges of the rectangular prism be $a \leq b \leq c$. Suppose we cut the rectangular prism into two identical, similar rectangular prisms. The congruence of the parts implies that the cutting plane must perpendicularly bisect one of the edges of the original prism.

![](https://cdn.mathpix.com/cropped/2024_05_02_91a49eab4ffd375bb3feg-1.jpg?height=279&width=403&top_left_y=229&top_left_x=861)

We know that the ratio of the volumes of similar solids - in our case 2 - is equal to the cube of the similarity ratio. Thus, the lengths of the edges of the half-prisms are:

$$
\frac{a}{\sqrt[3]{2}} \leq \frac{b}{\sqrt[3]{2}} \leq \frac{c}{\sqrt[3]{2}}
$$

The longest edge of the original prism cannot appear among the edges of the half-prisms, so we must have cut the prism perpendicularly to this edge. The $\frac{a}{\sqrt[3]{2}}$ cannot appear among the edges of the original prism either, since it is smaller than the smallest edge of the original prism, so only $\frac{a}{\sqrt[3]{2}}=\frac{c}{2}$ is possible.

For the other edges, we have $a=\frac{b}{\sqrt[3]{2}}$ and $b=\frac{c}{\sqrt[3]{2}}$. The ratio of the edges is thus:

$$
a: b: c=a: \sqrt[3]{2} a:(\sqrt[3]{2})^{2} a=1: \sqrt[3]{2}: \sqrt[3]{4}
$$

Accordingly, a rectangular prism can be exactly cut into two identical, similar rectangular prisms if and only if the ratio of its edges is $1: \sqrt[3]{2}: \sqrt[3]{4}$.