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This proof continues on, and on, and on some more. You may need to refer to some
previous pages of this proof to understand this page properly:

- First Page - Almost Y wing styles as Chain links
- Second Page - Almost Hidden Pairs as Chain links
- Third Page - Almost coloring, Almost AIC, Introduction to Appending chains
- Fourth Page - An introduction to Super Cells

Two steps detailed on this page attack similar areas of the puzzle. Perhaps the general
theme of this page is best described as generalized Almost AIC. In many ways, this is the
most difficult page of this proof. Hopefully, I am equal to the task of detailing these two
steps. One very important concept is theoretically detailed. It may well be the most important concept!

### Appending Chains

The concept of appending chains is a very simple one. Suppose one has a theoretical chain:

Suppose that F -- E. Then:

Now, there is little value in this observation unless we have neither F for certain, nor {A == B -- C == D}
for certain. In other words, if {A==B -- C==D} is an almost AIC, and appending F makes the entire
chain valid, we clearly have also proven that E is false.

Perhaps this concept is best illustrated by example. Many of the techniques already illustrated
in this proof can be viewed as appending chains. However, the following one is clearly best viewed
that way.

Note: *Forbidding matrices* help derive not only this manner of chain appending, but also
many others.

### AIC == Single

Illustrated above is a very deep, but not very complex chain:

- b2=1 ==
_{1} {a2=9 == c2=9 -- c5=9 == c5=8 -- c1=8 == c1=4 -- b2=4 ==_{1} b2=3 --
ab1=3 == d1=3 -- d6=3 == d6=9 -- d9=9 == d9=1 -- d3=1 == b3=1}
- => b2=1 == {a2=9 == b3=1} => b23=1 == a2=9 => a2≠1

Suppose for a moment that cell c1 could still be 1. Then, it should be obvious that a2=1
would still be impossible, as c1=1 -- a2=1. In a similar fashion, if c3=1 were possible then
a2=1 would still be impossible. In this way, most complicated AIC can be built from the concept
of appending simpler chains. The value in this concept is that one can build theoretical chains
of practically unlimited complexity from simple chains of little complexity.

Suppose again for a moment that c1 is limited to 148 rather than 48. One could write the
following:

- {pair 14 at b2,c1} ==
_{1} {a2=9 == c2=9 -- c5=9 == c5=8 -- c1=8 ==_{1} b2=3 -- ab1=3 == d1=3 -- d6=3 == d6=9 -- d9=9 == d9=1 -- d4=1 == b4=1}
- => {pair 14 at b2,c1} == {a2=9 == b3=1} => a2≠1
- Note: AALS can often be used in chains in a manner analgous to
HAALS

A much simpler logical presentation (

and thus easier on the mind) is possible:

- c1=1 ==
_{2} {b2=1 ==_{1} {a2=9 == c2=9 -- c5=9 == c5=8 -- c1=8 ==_{2} c1=4 -- b2=4 ==_{1} b2=3 --
ab1=3 == d1=3 -- d6=3 == d6=9 -- d9=9 == d9=1 -- d3=1 == b3=1}}
- => c1=1 == {b2=1 == {a2=9 == b3=1}} => b23,c1=1 == a2=9 => a2≠1

In a similar fashion, the chain can be further appended! One can, for instance, add 3 to d9. Clearly,
d1=3 -- d69=3, thus the important conclusions of the macro chain will not be affected. In this manner,
many ALS type configurations can be rewritten as nothing but simple bivalue, bilocation chains
that are appended with information that is superfluous to the eliminations.

The Super Cell idea presented on the previous page can also be derived from merely appending
theoretical chains. I am almost certain that, in fact, all techniques that do not use Uniqueness of Solution that
can be derived from appending simple chains.

The idea of appending chains is simpler than the concept of branching. Start with a theoretical simple chain,
see that it can be appended in various ways, derive theoretical complex chains. Knowing what
these theoretically complex chains will look like, find them in puzzles. I find this a very simple,
straightforward, albeit mathematical approach to unlocking very tough sudoku puzzles.

Enough of this heady? digression. Time for more examples!

### A chain snippet leading to a certain elimination

To the left, the newly created stronger 1's in column a are a certain part of a larger chain.
I say this is certain, as all the additional information that this elimination requires was included
in the previous step. However, I will present it in a slightly different fashion.

### AIC == Chain snippet

Two ways to view this graphic will be presented. One is perhaps more typical, the other
more intuitive. More typically, one might write:

- a78=1 == a4=1 -- a4=4 == b46=4 -- b2=4 ==
_{1} {b2=1 ==_{1} b2=3 -- ab1=3 == d1=3 -- d6=3 == d6=9 -- d9=9 == d9=1}
- => a8=1 == {b2=1 == d9=1} => b9≠1

### AIC == Single

More intuitively the appended chain representation of the same elimination is:

- b2=1 ==
_{1} {a78=1 == a4=1 -- a4=4 == b46=4 -- b2=4 ==_{1} b2=3 -- ab1=3 == d1=3 -- d6=3 == d6=9 -- d9=9 == d9=1}
- => b2=1 == {a78=1 == d9=1} => b9≠1

Lately, I have grown to much prefer the latter presentation. Previously, I would have chosen
the former. As time has gone by, I have come to regard appending of simple chains as the clearest
way to find complex chains. In fact, since I have changed my thinking towards this concept, many
chains that previously were difficult for me to find have become relatively easy. Not quite easy, but

*relatively* easy.

Frustratingly enough, I have always had in my possession the tools to view complex chains this
way. Unfortunately, the logic escaped me. A careful analysis of my own writings on topics such
as Y Wing Styles practically begs this type of analysis. I can only plead ignorance for not
previously, a long time ago, viewing complex AIC in this manner.

### Back to the puzzle

After b9 ≠1, two cells finally solve. b9 = 6 %cell and c3 = 6 %grid.

### Some locked candidate eliminations

Illustrated above are two very easy steps.

- b2=1 == b3=1 => c2≠1
- a2=2 == c2=2 => i2≠2

Unfortunately, this puzzle is still not trivial!

This concludes the *fifth* page of this *proof*. It may feel like time to
drink! However, if you are still sober minded, there remains a few more interesting steps
that I would like to uncover. Find some of them on
page 6 (viewing a strong
link as an appendage through the use of a super cell).

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