# Equidecomposability of a Cube into Trirectangular Tetrahedra and a given tetrahedron

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

• 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. – Blue Nov 3 '18 at 2:52