[scotty79]

If you start with all possible pixel arrangements, generate numbers from them then all unique sets of number are puzzles uniquely solvable only with logic. Those that were generated from multiple arrangements of pixels have multiple solutions.

Logic has two modes, one is going forward and the other is "assuming conversely" and finding if it leads to contradiction to eliminate that possibility. Both are equally valid and logical. There's no guessing involved. You simply pick any option to eliminate, assume it and try to lead to contradiction. If you don't find the contradiction you don't keep your assumption and the following steps that ended in a state that is not a contradiction, instead you try to eliminate another option instead in the same manner. Only when you successfully eliminate at least one you come back to trying forward logic again. The ultimate trick is not to search the contradiction using only the forward logic, but recursively using the entire solver to find if you landed in a contradiction.

Those two modes are basically the only logic that math uses for bulk of its proofs.