The Sets

Let's have a look at the tetracubes, which consist of four cubes joined together at the faces. There are eight different pieces and now we will cut one cube diagonally. If you can separate a triangular prism with one cut, you will get 36 different pieces, called sliced or clipped tetracubes.
On the other side you can start with tricubes and add a half cube joining the square faces. But there are pieces in the set of sliced tetracubes, that can't be made this way as the following example shows.

Only 32 pieces can be derived from tricubes. I call this subset extended tricubes. Here is the complete set of clipped tetracubes with the pieces not being extended tricubes red colored.

For other numbers of cubes the table shows how many pieces are possible.

cubes12 3456

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