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[alanr555]

Place holder while message is composed.

[alanr555]

It was good to get back to a VH after the gentle exercise of the
puzzles presented over the vacation period. The "Hard" one was
solved in about a quarter hour using just Mandatory Pairs.

This puzzle started with 28 cells completed. Using the basic techniques
a further 22 were resolved leaving 31 blank at the crunch point - with
the grade still shewing as Very Hard.

All but two cells of the middle block of 27 (rows 4,5,6) could be solved
relatively easily. This left a puzzle of "two halves" with a quite broad
waistband in between.

The key to solution was to note that the "contact" between the upper and
lower parts is limited and that the lower part has a much greater
preponderance of "strong links" (ie cells with only two values).

I found implication chains of length five and four stages which gave
the same value in the destination cell - starting from an origin which
had strong links, vertically for one value and horizontally for the
other. This seems to be the pattern for which to be looking.

In order for implication chains to work, there needs to be a "polarity
reverser" (and the same concept would probably apply to the method
known as colouring - although I find that much more complex than
implication chains).

In this case the "reversal" was effected by passing through (on just
ONE of the chains!) a triple of form xy/yz/xyz. The xy/yz cells
allowed the strong link to be maintained but the exit cell had three
values in it - allowing a continuation that would not have been possible
in a "closed triple" (xy/yz/zx). Coincidentally, the "other" chain goes
into a closed triple and (like a black hole!) such a triple does not
provide an escape route. Thus the destination cell for both routes
was a cell within the closed triple.

With the polarity reversal this destination cell had the same value
for each route. It was then a question of completing the puzzle
starting from that cell using strong links to give definitive values
to the remaining thirty cells. This process, also, was interesting as
three culs de sac were encountered - requiring a retracing of the
solution route to branch points. This is the phenomenon that provides
that "buzz" of rapid solution towards the end - and we sometimes
forget to see that we are really just following a number of chains or
subchains of a binary nature.

I have not identified the crunch-point cells as I understand that it is
"bad form" to reveal a solution too early in the day. However, this
post may give some clues or may even inspire someone with a
very logical mind to take up the challenge of formulating a set of
guidelines on how to find the "crunch point" cells that open up a
puzzle to the advanced techniques.

There is a lot of potency in the binary concepts and it may be that
they are a useful element in determining polarity reversers - although
it has still to be demonstrated that such reversers exist or are useful!

[storm_norm]

hmmmm, maybe I did something wrong, but I found no advanced moves were needed.

[andras]

[tlanglet]

A simple xy-wing was sufficient for me, but I am going to look for alternative solutions.

Ted

[sdq_pete]

From this position:

[alanr555]

[Marty R.]

I don't have any intermediate grids, but I removed two candidates from the 36 rectangle in boxes 7 and 9 to break up a DP, then an XY-Wing 56-25-26 pivoted in r8c6 finished it off.

[alanr555]

[quote="sdq_pete"]

[cgordon]

I used an xyz-wing <456> <45> <56> in R7C67 that took out the <5> in R7C5. Then an xy wing using the residual (good word that) <26> in R7C5 as the pivot with <25> R3C5 and <56> in R7C6 to take out the 5s in C6Box2. Would have posted the solution on a grid but I'm visiting Seattle and can't get the ASCII thing to work with a laptop.

[TKiel]

Alan,

[cgordon]

[keith]

This is a sticking point:

[Earl]

A more obvious triple (356) appears in R9, if your first eliminate by locked candiates from Box 1 and 2, reducing R9C2 to a single with a naked pair in Box 1.

Earl

[alanr555]

XYZ-wing

[alanr555]

[keith]

[TKiel]

Alan,

Your "Triple with a kink" understanding of the XYZ-wing is semi-appropriate (I almost used a reference to a naked/locked triple but decided against it to avoid confusion, for this very reason) but only if one realizes the kink applies not only to the location of the cells but also to the logic that applies, which makes it most unlike a triple.

In my previous post, I explained the logic (as I saw it) for the XYZ-wing: Basically, placing the common digit in the cell that sees the three cells of the wing will leave the pivot cell with no possible candidates. Placing a digit other than the common does not do the same thing. Let's look at what happens if we place (6) in r9c5.

Here's the grid:

[keith]

Not proposing to send people to other sites, but ...

http://www.brainbashers.com/sudokuxyzwing.asp

http://www.sudokuonline.us/xyz-wing.php

http://www.sudopedia.org/wiki/XYZ-Wing

Three of about 183,000 Google hits.

Keith

[alanr555]