There is a rule that lets you transform between valid solutions, right? What happens when you use that rule to help impose additional constraints onto the problem such that you are solving for a particular solution rather than a solution? I feel like that should sometimes let you choose pathologically transformed variants. For example - lets say you were stuck with eight and nine as your valid answers in a bottom left corner, but the constraints don't forbid nine. We ought to be able to pick nine in this situation under the logic that there will be a transformation rule which makes nine a valid solution. Such a choice means we are now solving for a particular solution - not a solution - because eight might have worked just as well.
In other words can you use the "do you know a related problem technique" to make your problem more specific?
So I wrote a nice elegant program that crunched through every possibility (using 12 nested loops), eventually spitting out solution after solution after solution. The first time it produced a valid solution, I was thrilled -- and from then on I felt like I'd already somehow solved all the puzzles in advance.