I have found this problem in a 5th grade problem-book :D, but ashaimingly to me I can not solve it for two days now :D
There are given two copies of a cube. We need to dissect the two initial copies of a cube into a cube twice as large volume (that is volume of which is equal to a sum of volumes of the two initial congruent cubes).
I tried to think of it in a generalized form. Suppose there is given a right paralellipiped. How to dissect the given body into a cube of the same volume.
Note that Dehn's invariant tells us, that it shall be possible to do that.
I guess I can see how to manage it in case if dimensions of a right parallelepiped are integers and a volume apears to be aperfect cube. But it seems to me irrelevant in general. Shortly speacking in this particular case you can cut a parallelipiped like stairs and shift, you then you can construct a cube in finitely many steps.
Mainly Im am interested in answering to a generalized question, but an answer to the initial one with two cubes is valuable for me to :)
The question with the two cubes can be solved by splitting the first cube into 6 rectangular pyramids, each with their vertex at the center of the cube and a face of the cube as their base. If you place those six pyramids on each face of the second cube, you can made a cube with twice the volume as desired.
(note: This answer is not correct, and is here only for information pusposes.)