Are there such cuboids that can be cut in half into two cuboids similar to the original cuboid?
Let the edges of the cuboid be denoted by $a \leq b \leq c$.

The division cannot occur by halving the edges of length $a$ or $b$, because similarity can only arise if the shortest edge is matched with the shortest edge, and so on, in order of size. For example, if $a: \frac{a}{2}=2: 1$, while the lengths of the other edges remain unchanged, there is no such correspondence. Therefore, the division can only be possible by halving the edge $c$, and in this case, the edges of the new cuboid are

$$
\frac{c}{2} \leq a \leq b
$$

due to the inequality among the original edges.

Since the volume of the original solid is twice the volume of the solid after the division, the ratio of their similarity is $1: \sqrt[3]{2}$.

Thus,

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

from which $a: b: c=1: \sqrt[3]{2}: \sqrt[3]{4}$ (approximately $1: 1.26: 1.587$).

Therefore, if the ratio of the edges of the cuboid is such, it can be cut in half into two cuboids similar to the original.

Kovács László (Bolyai J. E. Szki., IV. o. t.)