Twisting Puzzles
3-Layer Pentahedron
The 3-Layer Pentahedron has 3 faces that have 9 cubies and look, for all the world, like they are the faces of a normal 3x3x3 cube. The top and bottom faces have only 6 cubies, 3 corners and 3 edge pieces.
If you look at the picture carefully, you can see how the puzzle is going to turn. You can turn the U, D and E layers in increments of 120°. Then, if you are holding it as you see in the image, you can do 180° turns of the R, L and F faces. You lose the B face if you hold it as shown in the image.
I've solved this puzzle a few times pretty much intuitively, although I haven't got as far as writing down a reliable method. I start by getting the corners into the correct places and then swap edges between the layers. Then I work on the middle layer.