Chains and loops
(Note : This chapter may contradict the views and ideas that other authors have about chains and loops More subtle considerations of the topic are possible.)
Chains means chains of causality. This approach is not similar to any other methodology that has already been discussed here. They are considered the most difficult method.
A chain of causality is a loop which coincidentally ends at the point where it began.
The basic idea of a chain of causality is very simple. Event A leads to B leads to C leads to D , etc. leads to result X. The result of X is either a solution or a contradiction. If X is a solution A was a good idea; if X is a contradiction , A was a wrong decision.
How to find such a chain of causality? Either you simply stare at the puzzle until you discover it, or you try something randomly.
If you consider trying something , then it is advisable to look for a field that has the fewest possible candidates. Typically, you'll usually find a field that has only two candidates left. Then you make as many copies of the puzzle as there are candidates - usually two. In the one copy you solve the field for the first candidate, and in the other for the second . Then one works separately on further variants . At some point, possibly after making further copies, the variants lead either to a solution or to a contradiction. From this we can then draw conclusions about what earlier decisions were good or bad.
One can assume that most people aren't interested in putting in that much effort just to solve a puzzle. This approach is probably more suitable for computer-based solution methods.
In this context, it seems necessary to say a few words about the three possible outcomes which this procedure can produce:
Result I: One variant leads to a solution, while every other one ultimately leads to a contradiction. Very nice, that's how it should be!
Result II: Several variants lead to a solution. That would be neither unusual nor false . However, it is considered undesirable because it is expected of a puzzle that it has only one correct answer.
Result III: All variants ultimately lead to a contradiction. This we can justifiably call false. A puzzle without a solution is not a puzzle - well, at least not a meaningful one.
This approach could be called the brute-force method. Although it is annoying and time-consuming, it offers two distinct advantages. It always works, and it can make a definitive statement about how many solutions the puzzle has - regardless of the initial situation (even if the initial state is a completely empty grid).
Speaking of definitive statements. The previously described conventional methods are, in their logic, all inescapable. If a puzzle can be solved with the conventional methods, then it automatically has only one possible solution.
Taken together as a sequence, the conventional methods make a definitive statement.
The picture is only intended to illustrate the idea. An XY-wing leads to the same result.
If you decide to put a 2 in the yellow box, that sets a chain of events in motion that leads very quickly to the conclusion that the neighboring field has no more candidates.
Since this is suboptimal, the 2 was obviously a wrong decision. In the absence of other options, only 9 can be correct.