A study in Chain Forging: Page 7

Welcome back!

This proof page basically is more examples of using Almost Y wing styles. Please note that if this concept is confusing, it has an alternate interpretation as appending a simple chain. Try to look at both almost Y wing styles presented upon this page in mutliple fashions, including merely appending such simple chains.

Previous pages of this proof are practically a prerequisite for this page:

  1. First Page - Almost Y wing styles as Chain links
  2. Second Page - Almost Hidden Pairs as Chain links
  3. Third Page - Almost coloring, Almost AIC, Introduction to Appending chains
  4. Fourth Page - An introduction to Super Cells
  5. Fifth page - A further introduction to Appending Chains
  6. Sixth page - Overlap of concepts: Super Cells as Appendages

An Almost Very Common Y Wing Style

Almost VCYWS 17

To the left, one can find cell h8 again is involved in an almost Y wing Style. In this case, because of the recent reductions in cell g4, a different almost Y wing style is available:

  • {h8=1 *==*h8=7 -- h1=7 == g1=7 -- g4=7 == g4=1} -- h6=1

Very Common Y style roughly centered in a chain

Almost VCYWS chained up

Note above that not much needs to be added to either side of the very common Y Wing Style to complete the chain. Essentially, the very common Y wing style establishes: a8=2 -- h6=1. It is perhaps useful to be aware of such remote or non-native or derived weak links. My personal preference is to view complex AIC as the movement of strong links, but it maybe useful to others to use the equivalent notion of remote weak links. One AIC or forbidding chain representation of the above graph is:

  • c9=2 == a8=2 -- h8=2 ==1 {h8=1 ==1 h8=7 -- h1=7 == g1=7 -- g4=7 == g4=1} -- h6=1 == c6=1
    • => c9=2 == c6=1 => c9≠1
This view basically uses a Y Wing Style as an internal argument. However, the same set of strong inferences can be viewed differently, with a Y wing style as the base, and some strong inferences added as appendages:
  • {c9=2 == a8=2 -- h8=2 ==1 h8=1 -- h6=1 == c6=1} ==1
  • h8=7 -- h1=7 == g1=7 -- g4=7 == g4=1 -- h6=1 == c6=1
    • => c9=2 == {c6=1 == c6=1} => c9=2 == c6=1 => c9≠1
There is yet at least one other way to view the same set of inferences:
  • {c9=2 == a8=2 -- h8=2 == 1 h8=7 -- h1=7 == g1=7 -- g4=7 == g4=1 -- h6=1 == c6=1} ==1 h8=1
    • => c9=2 == c6=1 with h8=1 as just an internal appendage => c9=2 == c6=1 => c9≠1
The latter presentation is for me less visual, but is precisely in line with viewing some complex chains as being merely bivalue/bilocation chains with superfluous appendages that do not effect the outcome. A forbidding matrix representation of this chain most clearly identifies h8=1 as precisely such a superfluous appendage. Thinking in terms of such superfluous appendages has some significant value in finding such chains.

I have chosen to present such multiple views of the same elimination to illustrate the flexibility of the underlying logical presentations.

Single candidate chain on 1s

fc on 1s

The newly stronger set of 1s in column c allow the following chain on only candidate 1:

  • c7 == c6 -- a4 == g4 => g7≠1

This newly stronger set of 1s in column c also allow the following chain that has the same general matrix form as the first step illustrated on this page.

An Almost Y Wing Style using candidates 147

Almost YWS 147

The fact that this Almost Y Wing Style may be useful is keyed by the strong 3s in row 7. The possibly useful chain can be written:

  • {c7=1 == c6=1 -- c6=7 == b4=7 -- b7=7 *==* b7=4} -- c7=4
All that is needed to complete the possible usefulness of this chain is:
  1. Restricted 3s emanating from box b8
  2. Restricted 4s emanating from box b8
  3. A common intersection for these restrictions
It should be fairly simple to find these links!

A Y Wing Style roughly centered in a chain again

Almost YWS 147 chained up

Note here how this elimination is fairly symmetrical upon the Almost Y Wing Style. One possible chain representation of this elimination is:

  • a1=3 == a7=3 -- b7=3 ==1 {b7=4 ==1 b7=7 -- b4=7 == c6=7 -- c6=1 == c7=1} -- c7=4 == c1=4
    • => a1=3 == c1=4 => a1≠4
As in the previous example on this page, there are many other ways to write this chain. Again, in a very pure appendage form:
  • b7=4 ==1 internal appendage {a1=3 == a7=3 -- b7=3 ==1 b7=7 -- b4=7 == c6=7 -- c6=1 == c7=1 -- c7=4 == c1=4}
    • => a1=3 == c1=4 => a1≠4
Note that I have added the detail, internal appendage, as for the purposes of this chain, the possible existence of 4 at b7 is superfluous. I am finding this type of chain surgery to be a fairly useful and powerful candido-suction technique.

This concludes the seventh page of this exercise in Advanced Forbidding Chains . I may finally finish this puzzle proof with one additional page. (I may also elect to add an addendum type page to review not only the proof, but also concepts used). Eventually, the link to page 8 (Almost single candidate chains or Almost Coloring) will become active.

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13/Jun/07 6:53 PM
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18/Jun/07 12:09 AM
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