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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 :)

Evgeny Kuznetsov
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    With the staircase dissection of a rectangle into two pieces you can only change the aspect ratio by a rational factor, which won't be enough in general. You can instead cut a rectangle into more pieces using straight lines for any factor. See for example here for a rectangle into a square, but the same works for any rectangle into any rectangle. For a parallelipiped you just have to do that twice, along different axes, to set two of the parallelipiped's dimensions. – Jaap Scherphuis Sep 01 '21 at 10:22

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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.)

PiGuy314
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    The resulting figure looks to me not like a cube, but an octahedron. – Lubin Sep 24 '21 at 17:37
  • Remember that the pyramids will have a height of 1/2 and a base side length of 1, so they will have a slant angle of 45 degrees. This is turn means that two of these pyramids at a right angle to each other will connect seamlessly (90/2=45). This should create a cube. – PiGuy314 Sep 26 '21 at 12:59
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    I’m sorry, forget the angles and count vertices. Each face of the original cube gives rise to a vertex, thus six vertices, not eight. And at each new vertex, four faces come together, not three. Still looks like an octahedron to me. – Lubin Sep 26 '21 at 17:54
  • You do have a point. My logic was likely wrong. I can take down the answer if you like. Still trying to find a solution, though. – PiGuy314 Sep 26 '21 at 18:21
  • Up to you. I think wrong answers can be illuminating. And I have to confess that I never thought of this kind of dissection of the cube. – Lubin Sep 27 '21 at 14:34
  • Then I will leave it up, along with a note that it is not the correct answer. – PiGuy314 Sep 27 '21 at 16:59