A “slinks only” X-loop of an even number of slinks consists of alternating true and false candidates. That we know by the logic of slinks. We just don’t know if it’s the arbitrary one we pick, or the next one, that is true. This phenomenon is the basis for coloring. We use two colors, and remain uncommitted on truth until one candidate of the chain proves false. It’s then like today’s politics. All candidates of the removed color are false and should be removed, and all of the opposite color are true.
What about an odd number of slinks without winks in a loop? The alternating true/false pattern of a slink chain is a problem. It forces two true candidates to see each other. I had that in mind when I ran across the site “Sudoku for Everyone” by Phillip McCollum, and these diagrams under his link for Strong X-Chains. I’ve translated them into the now familiar X-panels:
For such even numbered slink chains ending in the same unit, Phillip states that both ends can be removed, and moves on. I’m thinking “Not so fast! Does ending in the same unit determine where the double T’s go?“ Going back over Phillip’s argument – that two candidates wind up in the same unit – did not satisfy me as a reason the breakdown occurs there. You can just rotate the “if true” starting point, and with it, the offended unit, around the loop. That won’t do, because it leaves the coloring art in big trouble. You might be unlucky enough to start coloring on the wrong candidate, and remove the wrong one on the other end of the slink chain.
Fortunately, the matter is resolved if we reverse T’s and F’s in the McCollum diagrams. They would then reflect Phillip’s accurate conclusion that both ends in the same unit can be removed, for the now more evident reason that additional candidates X must be present in the terminating unit, making the loop closing link a wink, not a slink. That is the answer for odd slink loops. They do not exist, because termination of an even linked chain in a unit cannot be avoided, and a wink will be forced there.
To fill in the McCollum argument, T to T diagrams show that one of the end candidates must be removed. F to F diagrams then show that if one of them is false, they are both false. And don’t forget, slink inference determines that every loop candidate odd slinks away from the wink is true, and every candidate even slinks away is false. No need to memorize that, just go around the loop in both directions, saying False, True, False, True, . . .
By way of blog news, I went back and revised the post of 10/13/11 on Line marking. As the blog has grown to be more informative for all levels of solvers, it is apparent that it should be more accessible to less experienced solvers. The revision illustrates the mechanics of line marking in more detail, and puts the argument for efficiency in more practical terms. A page on box marking has been added also, to help beginners get started.
It became apparent in doing this revision that the blog is beginning to need forward links, to subjects that are mentioned for completeness in the current topic, but are presented comprehensively in later posts.
I am thinking that a reference page is also in order, to highlight grid location conventions and to contain a glossary giving definitions and alternative terminology used in other sources.
Of course, your suggestions on improving the blog for beginners and for all readers are welcome.