Alternate Inference Chains (AIC)


Readers of this blog are familiar with an alternate inference chain in the form of an XY-chain on a bv map, or an xyt-chain on the grid, or as an X-chain or grouped X-chain on an X-panel.  Humans, as opposed to computers,  can best find these chains by construction, more than by searching. AIC, as they are called, generate two-candidate toxic sets – the best kind – very  effectively.   X-chain AIC ends  can also form candidate confirming loops.

The propagation of alternating true and false inferences along the chain depends only on the alternation of slinks and winks, and not on the underlying circumstances define the link type. “Alternate Inference Chains”, or “AIC” is also the name given to the method of assembling alternating links from all causes.

The flexibility of AIC makes it subject to the complexity of too many choices.  Systematic Sudoku controls the similar complexity  of the XY net by reducing bv interactions to a small, comprehendible set of curves.  This makes the XY-chain  a practical tool for the human solver, and enables XY chains to move into the SSOB immediately after line marking.  The corresponding,  but more severe,  complexity  of  the AIC had forced it back in the SSOB beyond more time consuming methods.  But fortunately, there is an organizing technique that enables human solvers to duplicate the impressive chains of computer solving engines, without exhaustive search.  This resource brings AIC to this spot in the blog, and in the order of battle, right after coloring.

To describe AIC analysis through example, let’s bring Maestro 47 of the previous post to the state of solving where it is needed.

A box marking trace is

1: Cm, SEm.  2: Wm.  3: Em=>NEm.  4: NWm=>NEm, Cm.  5: NEm.  6:  7: 8: Cm=>Nm, SEm.  9: SEm.

Line marking traces as

5f: r2, c3, c9  6f: r5, r6, r9, c6  7f:r1, r4, r7, r8  8f:r3  Close:c2, c5, c6, c7, c8.

Not a single clue so far.  In the bv scanI found no live XY-chain, SdC or APE.  One unproductive XY chain is shown below. On the 8-panel, there is there is an almost nice loop removing one indecisive candidate.

Coloring is so limited, we can see there are no results as yet.

Whew!  Now to develop alternate inference chains similar to the XY-chains in the bv map above,   we identify and mark cells similar to bv in that a chain enters with a slink and leaves with a slink.  This is the case when two or more candidates partner in two or more slinks.  We’ll call them AIC hinge cells.  We can also mark multiple candidate cells in which only one candidate partners  two or more slinks.  These are possible confirmation sites for almost nice (AN) loops. We do not mark cells with a single emerging slink. 

The AIC hinge cells are marked in green, and confirmation sites in light blue, with distinctive PowerPoint “scribble” marks, as  illustrated here with Maestro 47. To get the hang of it, justify each of these markings.

Now we leave the AIC marks in place, and superimpose promising XY and X chains on this grid, attempting to extend them via AIC.  ©PowerPoint makes this easy.

First up is the long XY-chain of the bv map above.  One end, the bv r3c5 extends immediately into AIC hinge cells.  We just extend it as far as we can, reaching the 3 candidate in r6c8.

As we do this, we can look back along the XY chain as we complete each slink extension, for matching toxic set partners along the chain. Two toxic sets result, on numbers 2 (blue) and 3(green) are seen in the next diagram. Both have no victims. 

Now we can seek an AIC  extension from another point along the XY chain.  A slink out from the 4-candidate in r6c6 goes into an AIC hinge and on to complete an almost nice (AN) loop confirming 3 in r5c8! This is shown in the diagram below.  Also, a slink out from 1 is possible, but the double slink requires the extension to return to the XY chain for an AN  loop, and it does not.

As a challenge and an insight, be aware that the long 2-chain of Maestro 47 can be extended similarly to make the very same confirmation.  Using our AIC hinges and X-chain node cells, see if you can construct the AN-loop. 

Then determine how the new clue

E3=>(W3m, E7m).

yields another AIC extension of the 2-chain to confirm a 2-clue in r5c3. 

Keep your AIC hinges until they are destroyed, and update your X-panels for more removals.   We will checkpoint your discoveries in the next post, and discover a few more AIC tricks, courtesy of Maestro 47.

Meanwhile, to get ready to fly AIC solo, you might like to do basic solving on Maestro 49.  But it won’t be easy.  Be prepared for one of the toughest line marks ever.  The puzzle maker of Will Shortz’s “Beware! Very Challenging” section of Ferocious Sudoku will love it!  I wonder if she’s reading my blog?  So far I have to conclude that no Sudoku expert is. Help me change that, if you know one.

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About Sudent

My real name is John Welch. I'm a happily married, retired professor (computer engineering), timeshare traveling, marathon running father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. The blog is about Sudoku solving. It covers how to start, basic solving to find candidates efficiently, and advanced solving methods in an efficient order of battle. It is about human solving methods, not computer solving.
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3 Responses to Alternate Inference Chains (AIC)

  1. Very nice blog post. I definitely appreciate this website. Keep it up!

    • Sudent says:

      Thanks, Sniper. You are the first acknowleged Sudoku writer to comment. What do you recommend I keep it up with, when all of the humanly practical techniques are covered? That’s going to be within a year of starting.

  2. Pingback: Revisiting Wex 435 | Systematic Sudoku

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