fdh319 wrote:I didn't know that a chain may sometimes be misleading. So my next question is; Suppose a chain which confirms the contradiction chain isn't possible (2nd chain in my post), whom are you going to believe? And suppose the contradiction chain (first chain in my post) isn't possible, how are you going to decide which chain is true and the other is false? I know that the false candidate will finally result in a conflict, but this is not the case.

Hi fdh319. There are two ways to see and use forcing chains. One is to pick a binary condition (e.g. a certain candidate must be either true or false but not both or neither) and assume it one way. If you find two contradicting chains (or any other contradiction) for a single assumption, you know it must be false and its binary opposite true. If you don't find a contradiction, then you can't conclude anything. True candidates can't lie but false ones can, so basically you're trying to catch a liar.

The other way is to pick a group of candidates (or more generally boolean conditions) where you know that at least one of them must be true, and then find chains for each of them such that they all agree on a result. In that case you're proving the agreed result instead of anything about the assumptions themselves, which makes it "non-assumptive" (considered more elegant by most solvers). Why does it work? For the same reason as the other one. True candidates can't lie, and since our choice of a tested group must include at least one true candidate the only way to make everyone agree is on a true result. False candidates can be swayed either way but true ones are single-minded so they could never agree on a false result (as long as you don't make mistakes with your chains).

Both methods, especially the latter, are discussed in

this thread, though it also happens to deal with BUGs. In a BUG+n situation you know that at least one of the +n candidates must be true to prevent the illegal BUG pattern, so if you find agreeing chains for every one of them, then you know the agreed result must be true. The same principle works for any group of candidates if at least one of them has to be true (e.g. all candidates in a cell, all instances of a digit in a house). Important: with this method you're not trying to find out which of those tested candidates are true or false -- just a result they agree on, which is called a verity (as opposed to a contradiction). More on both contradiction and verity types of forcing chains

here. In most cases contradiction and verity types of forcing chains are just two ways to see the same thing, so they can be easily converted into each other (often just by reversing the chains).

Concerning the BUG+1. Of course I have heard about it, but haven't yet mastered it.

I highly recommend you look into BUGs. They're quite common and fun ways to solve a puzzle. Besides, they teach you about the principles laid out above. BUG+1 is the simplest case where there's only one extra candidate which must be true. In any other BUG+n situation you have to find an agreed result for every extra candidate.

Right now I'm exploring the possibilities of chains and less interested in a puzzle's solution

That's the right attitude if you really want to learn!