My original problem is:

1) Let $XYZT.X'Y'Z'T'$ be a cube. Given $A\in XYY'X',B\in XYZT,C\in Y'Z'$ and $D\in TT'$. Is there a way to dissect the cube into Trirectangular Tetrahedra and $ABCD$?

I have no had a clear approach for this problem, just trying to answer easier questions.

2) Can a cube be dissected into trirectangular tetrahedra? I know it cannot be dissected into 6 such ones Equidecomposability of a Cube into 6 Trirectangular Tetrahedra. But I wanna generalize the question a little bit here. Clearly if $ZXY'T'$ can be dissected into trirectangular tetrahedra, the question is solved.

3) Can a 3-dimensional figure be dissected into cubes?

Those are my current questions and approaches, I hope u can help. Thank you

  • 3
    $\begingroup$ Equidecomposability of solids is the subject of Hilbert's Third Problem. There is much literature on it. Dehn provided a condition for determining when two solids might be equidecomposable. $\endgroup$ – Blue Nov 3 '18 at 2:52

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