Is it possible to count all possible Solutions of a Problem?

All right, thank you for the quick and detailed response. I appreciate the new insights into the solution finding process very much. I guess that leads to the conclusion, that it is not really implementable in my case (at least in a non tedious and non laborious way) since the explicitly forbidding 'method' seems to be the only way to get the job done. 

Perhaps it was not quite thought through to think there could be something like a 'block()' method to just iterate through until no more new solution is found.

Again, thanks for the effort.