We need a way to identify the six pieces. 

One looks a bit like a house and another is sort of corkscrew shaped;  "House" and "Corkscrew."  Both contain five unit cubes.  One more piece is made of five unit cubes: we call it "Long."  All the above pieces, and two more, cannot lie flat: those other two are shaped like "Long" but have only four unit cubes: "Short.".  The last piece is four units and is flat (it also has no mirror image that differs from itself): "Flat."
We need to distinguish "right" from "left."

All except "Flat" have an axis that winds through the piece somewhat like the thread of a screw.  We identify each piece by the handedness of this screw-thread line.  Examine the above drawings, and imagine the mirror image of each piece.

Here are the twelve different puzzles, identified by the configurations
of Right and Left...and the number of (known) solutions for each puzzle.

HOUSE
SCREW
LONG
SHORT
SOLUTIONS
R
R
R
R/L
7
R
L
L
R/L
3
R
L
L
L/L
3
R
R
R
L/L
2+
R
R
L
R/R
2
R
R
L
R/L
2
R
L
R
L/L
2
R
R
R
R/R
1
R
R
L
L/L
1
R
L
R
R/L
1
R
L
L
R/R
1
R
L
R
R/R
0
The "2+" case has a third solution almost indistinguishable from one of the others.

We can note that the easiest puzzle has all the same handedness except for one of the Shorts.
And the impossible puzzle has all the same handedness except for the Corkscrew.

The number of solutions is somewhat tentative.  Please let us know
if you find a different number for any of the puzzles.

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