The Floppy Cube is a 3x3x1 puzzle that was originally invented by Katsuhiko Okamoto.
It consists of 9 cubes arranged in a square, and each side of 3 cubes can be given a 180 degree turns as well 90 degree turns. Four edges are connected to a center. The edges does not move by rotate about the center piece. The other 4 corners can connect to any of the four sides of the edges including the 2 internal sides giving it a deformed shape. This is a very interesting and innovative puzzle which won an International Puzzle award.