This project solves the guitar version of Scramble Squares although it can easily be modified to solve any of those puzzles of any size. Scrambe Squares has 4^9 * 9! combinations and far fewer solutions.
Each card side (Up, Left, Down, Right) has 1/2 of an image. These cards are classified in Cards.java as a Top (T) side and a Bottom (B) side. After this, the idea is to basically backtrack via graph traversal by grabbing cards, rotating them, and appling cards to the card board until a solution that "fits" is found. Some linear optimization is performed by giving up on a graph traversal that cannot be solved early.
If you would like to use this to solve your game, you will need to be able to build this project yourself. Change Cards.java to match the cards you have. Each Card expects 4 sides to exist.