On this post, X-panel chain analysis is illustrated on a Hodoku example. Hobiger’s coverage of X-chains is shallow on fundamentals and distracted with named X-chains. Several X-chain highlights of the blog are recalled.
Hodoku starts the chain types with X-chains. Hobiger over-defines the type in the same manner as XY-chains, saying they must end in slinks. This overlooks, for one thing, their use as forcing chains.
X-chains may look simple, but that is deceptive. They are the basis of grouping and ER forcing chains, demanding very thorough analysis at times. X-chain loops can also get complicated. Unlike XY-chains, the links do not carry strong and weak badges on their sleeves. Sometimes X-chains work because they can cut through whatever is going on in the grid around them. With all that, it is absolutely not useful to name different lengths and shapes of X-chains as if different methods are involved.
Have your Sudoku student list and locate the slinks not noted in the Hodoku grid. The fill strings of line marking are left on. Two naked singles turned up in line marking, but the going was tough.
When I discovered some of the complexities of X-chains, I made a decision that analysis of fish and X-chains require that candidates of a number be examined absent the noise of clues and other candidates. So I constructed a template for that, the X-panel. Puzzles that survive the bv scan have their surviving candidates copied to an X-panel. Each panel is scanned for X-chains and fish. Later, the X-panel becomes an aid in coloring and and pattern analysis.
At left is the 7-panel containing the 7-chain. Chains are suggested by the line slinks. For analysis it is convenient to mark the alternating inference path on the panel with a single curve or freeform, which is translated into alternating curves on the grid. It is a prime example of constructing a chain, then seeing its results – the opposite of starting a chain with a premise.
It’s not always that easy. Here is a 1-panel on which I thought I had found a grouped 1-ANL in the same puzzle. But when I add the 1-candidate that I missed in r4c1, it doesn’t work. Fill in the slink/wink alternation along the path and picture how it would have looked on the grid.
The Hodoku Techniques page has a link to Single Digit Patterns, where the popularly named X-chain types are illustrated, including Skyscraper, 2-String Kite, and Turbot Fish. I regard these categories more of a hindrance than a help, since the human solver should be constructing short X-chains instead. You can even have Hodoku generate 2-String Kites for you to find, one after another, but that’s fantasy baseball, my friend.
Bernhard’s inclusion of ER, the Empty Rectangle of this section, is a mistake. Although his description does clarify it somewhat, this inclusion itself suggests that ER is a grouped X-chain. It is actually a form of weak link, a way of ”seeing”. I prefer to identify it as an aid to spotting helpful grouped forcing chains.
When I realized that Hodoku was leaving X-chains at that, going to XY-chains and then into AIC, I was disappointed. They were a lot more fun when I was exploring them in early 2012. Yes, that’s when my abc diagram for eliminating (a) confirming (b) ANL and nice loops (c) was drawn up. I liked it so much I used it for awhile as the blog header.
Hodoku explains (b), the confirming almost nice loop, as an X-chain with the slink terminals ending on the same candidate. I use the ANL designation of my preferred author, Andrew Stuart, adding confirming to distinguish its result. Starting around in either direction, you assert the candidate is false, and arrive back proving, in that case, it’s true. So it can’t be false, can it? Another example of proper use of chain tracing from a premise; namely, in the proof of a chain property. Only unschooled beginners are allowed to use it to solve puzzles
So what fun stuff did Hobiger leave out? There’s the business of good and bad patterns of winks, and even but not odd length slink loops, McCollum’s rule, and Andrew’s guardians, and my own prolonged dance with grouping, going on into Stephen’s Nishio turned Grouped ANL Test. In fairness, everyone ought to get to play with these as X-chain entities on the X-panel before encountering them on AIC grids.
But it is what it is, and we must move on with Hodoku to AIC loops. And unfortunately, here we must deal with the confusing legacy of Paul Stephens once more.