Does the puzzle usually include repeated shapes on the higher difficulties, because if they do, you might be able to reduce the time needed to find a solution by removing a pair and the running the algorithm(s) to see if the puzzle can be solved and just place the repeated shapes on top of eachother. In your example, it could reduce the possibilites to a twelve-thousanth of the number of possibilities you would get if you ran your original algorithm. Though I suppose that would only be for a special case.
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