Rubik1x3x3 Scramble Cube

Designed by Katsuhiko Okamoto, made by Gentosha, Japan, 2010.
(plastic, 2.25 by 2.25 by 3/4 inches)

A generalization of the Rubik 1x3x3 Floppy Cube Rubik 1x3x3 Floppy Cube where a 1x1x2 portion can rotate (see also the Rubik 1x3x3 Floppy Mirror Cube); here are a sequence of three moves to make an edge piece sit up:

Jaap's Page presents an analysis, but by just playing it is relative easy to solve, and it is easy to make interesting shapes, like the one on the left below, and then return the puzzle to flat.

Notation: L (left), R (right), F (front), B (back) denote flip that side, L90 to turn the left side 90 degrees clockwise, B90 to turn the back side 90 degrees clockwise.

Once flat, moves for the 1x3x3 Floppy Cube can be used to solve or to be left with one edge flipped as shown on the right above. To fix this, make the edge opposite to this edge sit up, then use the edge pair flipping transformation from the 1x3x3 Floppy Cube (LFL RBR), and then put the opposite edge back down; all 90 rotations are in pairs, so all movement can be clockwise:
L90 B90 L90   L F L   R B R   L90 B90 L90

Further Reading
Jaap's Page, from: