ALS Merged Cell Forcing Chains

This post demonstrates the merger of cell forcing chains by means of ALS nodes.  I  checkpoint  three triple cell forcing chains from the challenge of the previous post, and argue that comprehensive search for cell forcing chains beyond the dual level is humanly impractical. However, these chains from the Unsolvable 40 puzzle suggest that triple cell forcing chains can be discovered by merger into dual forcing chains during a scan for ALS aided AIC.

triple cell fc 2In the sequence of triple cell forcing chains previewed in the last post, the starter cell for triple cell forcing chain #2 is r3c1. The presence of an x34-ALS in the box of the starting cell allows chains from 3 and 4 to merge, effectively making this a dual cell forcing chain.

The sysudokie doing ALS aided AIC, and noticing the 34 merging AIC would have the incentive to dog out the 6-AIC as far as it needs to go to connect with the merged chain from 3 and 4. No problem to find the 346-ALS node to do it.

Is the dual cell forcing chain resulting from the merger something other than an ALS aided AIC almost nice loop?  No, there is effectively a strong link in the starting cell between the 6-candidate and the 34 combination of the merged chain.  It’s an ANL, although a fancy one.

triple cell fc 1The Stuart solver’s triple cell forcing chain #3 re-uses much of the #1 chain, giving an appearance of human intelligence.  We know better, of course.  But did you by chance hit on the alteration from #1 to #3?  The technique would be to delete only the curves attached to removed candidates, then to extend the chains to a new connection.

Not at all implausible,  but it depends on finding triple cell forcing chain #1 or #3 from scratch.

triple cell fc 1In the solver’s triple cell forcing chain #4,  the 3 and 6 candidates in starting cell r3c8 merge, which is incentive enough to find the 4-chain.  But how would you plausibly find this without exhaustive search?  Here is a scenario:  Seeing this ALS floating on a sea of its numbers, you might trace forcing chains from it to any prospective triple cells. That would uncover the 36 pair in r3c8, and suggest a search from the 4-candidate,  finding the triple cell forcing chain.

An interesting possibility here is to find that the 4 is AIC connected to the ALS as well.  Then any candidate seen by the ALS can be removed.

Unsolvable 40 has another lesson for us: The series of forcing chains runs dry.  The human solver on a comprehensive search for triple cell chains would have no hint that #4 is the last one,  and would now waste an enormous amount of time searching for another.

To sum up,  I have demonstrated that cell forcing chains with two starting candidates are redundant, and those with more than two cannot be comprehensively searched by human solvers.  That is to say, it is not humanly practical to systematically cover possibilities and guarantee that no effective cell forcing chain exists.

However, the examples from Unsolvable 40 suggest that working from a comprehensive set of ALS , AIC hinges and bv, sysudokies may be able to trace back from ALS to triple starting cells with merging chains, to discover new AIC ANL’s.

usair gridNext post I get into unit forcing chains, as defined by Andrew Stuart.  As we approach the end of this series on forcing chains, and anticipating the next topic, you might want to start basic solving on a primary example.  It’s the U.S.  Air Hemisphere puzzle of July 9, 2009, that provided examples of APE and of Death Blossom via ER in posts of last January.

Look for the box marking checkpoint at the end of the next post.  Small world that it is, my favorite Sudoku genius Alex de Abreu and I each encountered this puzzle on different flights.  He solves without pencil marks, in ink.  Alex counts this puzzle as his only true defeat, and I can now explain why that happened.

About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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