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Dec 27th VH
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Thu Dec 27, 2007 4:21 am    Post subject: Dec 27th VH Reply with quote

Place holder while message is composed.
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Thu Dec 27, 2007 4:54 am    Post subject: Reply with quote

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!
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storm_norm



Joined: 18 Oct 2007
Posts: 1741

PostPosted: Thu Dec 27, 2007 7:26 am    Post subject: Reply with quote

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



Joined: 31 Oct 2007
Posts: 56
Location: Mid Wales

PostPosted: Thu Dec 27, 2007 10:53 am    Post subject: Reply with quote

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


My wife also solved it without using anything advanced, and beat me to it as well. Shocked

I couldn't see how she'd done it and had to use an x-y-z on 456, which broke it clean open.

John
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tlanglet



Joined: 17 Oct 2007
Posts: 2468
Location: Northern California Foothills

PostPosted: Thu Dec 27, 2007 1:34 pm    Post subject: Reply with quote

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

Ted
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sdq_pete



Joined: 30 Apr 2007
Posts: 119
Location: Rotterdam, NL

PostPosted: Thu Dec 27, 2007 1:39 pm    Post subject: Reply with quote

From this position:
Code:

+------------+--------------+-------------+
| 25  1  257 | 6   8   345  | 9   47  35  |
| 8   34 567 | 49  1   3459 | 2   47  356 |
| 9   34 56  | 24  25  7    | 1   8   356 |
+------------+--------------+-------------+
| 23  5  23  | 8   7   1    | 6   9   4   |
| 4   69 8   | 5   69  2    | 3   1   7   |
| 7   69 1   | 3   4   69   | 8   5   2   |
+------------+--------------+-------------+
| 356 7  9   | 1   256 456  | 45  236 8   |
| 1   8  4   | 27  3   56   | 57  26  9   |
| 356 2  35  | 479 569 8    | 457 36  1   |
+------------+--------------+-------------+

Play this puzzle online at the Daily Sudoku site
Code:
XYZ 256 R7C5
XY  257 R8C4


Peter
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Thu Dec 27, 2007 4:06 pm    Post subject: Reply with quote

Quote:

Maybe I did something wrong, but I found that no
advanced moves were needed.


That all depends upon one's definitions!!

I regard all wings, fish etc as "advanced" in that they are
"packets" of logic rather than transparently simple.

I include in "non-advanced" techniques such things as
recognising pairs/triples/quartets etc and the processes
of sole candidate (reduction of cell profile to a single
digit) or sole position (only appearance of a digit in a
row, column or region) plus "type one" unique rectangles
(ie same pair in all four corners) as these address the
basic rules.

Advanced techniques are those which depend on "seeing"
a particular PATTERN (as opposed to a grouping like pairs
and triples) from which implications can be drawn. Anything
which involves algebra (having worked back from z to w, are
there any techniques for v or u?) for me has a taint of
advanced-ness - and as for the fish!!

Perhaps we could agree some definitions as to what is advanced
by discussion in the the Methodology section - although I suspect
that SamGJ must have a working list in order to be able to get
the grader to assess the difficulty of puzzles submitted to it.
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Marty R.



Joined: 12 Feb 2006
Posts: 5731
Location: Rochester, NY, USA

PostPosted: Thu Dec 27, 2007 4:55 pm    Post subject: Reply with quote

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.
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Thu Dec 27, 2007 5:18 pm    Post subject: Reply with quote

[quote="sdq_pete"]
Code:

+------------+--------------+-------------+
| 25  1  257 | 6   8   345  | 9   47  35  |
| 8   34 567 | 49  1   3459 | 2   47  356 |
| 9   34 56  | 24  25  7    | 1   8   356 |
+------------+--------------+-------------+
| 23  5  23  | 8   7   1    | 6   9   4   |
| 4   69 8   | 5   69  2    | 3   1   7   |
| 7   69 1   | 3   4   69   | 8   5   2   |
+------------+--------------+-------------+
| 356 7  9   | 1   256 456  | 45  236 8   |
| 1   8  4   | 27  3   56   | 57  26  9   |
| 356 2  35  | 479 569 8    | 457 36  1   |
+------------+--------------+-------------+


Thank you for posting this.
My "crunch point" was later.
Recognising the triple (356) in row nine allows r9c5 to be resolved
as 9 and then r2c4 becomes the sole position for the 9 in row two.
This all resolves six more cells altogether and reduces the
profile on a further four cells.

Value 9 is set in r2c4, r5c2, r6c6, r9c5
and value 6 in r5c5 and r6c2.

r2c6 reduces to 345 and r7c5 to 25
while row nine gains a 47 remote pair in r9c4 and r9c7.

Then I had to resort to implication chains!

++
Quote:

Code:

XYZ 256 R7C5

XY  257 R8C4

Peter


I regret to say that the above is almost meaningless to me.
The XYZ refers to r7c5 and its value of 256.
As described above the 6 disappears as a consequence of
the triple in row nine. I presume that the XYZ is another
method of achieving the same result - but that is an utter
mystery to me as to how it does it.

The second line is more intriguing.
It refers to cell r7c5 (with profile 27).
This cell was the origin for my implication chains.

It is part of a closed quartet in row 8 and a closed triple in
column 4 - all strong links!

My implication chains led me to r7c7. This is part
of a closed triple in column 7 which intersects with
the quartet directly and indirectly with the triple
using the 47 mutual reception pair in row nine.

The chains are:
r8c4=2; r8c8=6; r9c8=3; r7c8=2; r7c5=5; r7c7=4
r8c4=2; r9c4=4; r9c7=7; r8c7=5; r7c7=4

I discovered that r7c7 MUST have value 4. Applying that
using sole candidate techniques etc transformed the puzzle
from VH to Easy.

+++

Clearly there is something very important here.
Both of us "homed in" on the same cell r8c4 - but
from different perspectives.

What does "XY 257 R8C4" mean?
How does it assist with the solution?

Can someone please explain to me - using this example -
in precisely detailed steps how this works, starting with
an indication of what is going to be achieved and then
the steps towards that achievement.

Maybe (?) I will then be able to emerge from the darkness
and start to understand a glimmer of what contributors to
this forum are describing when using such "shorthand".

I was a frequent contributor to this forum back in 2005/2006
and have only recently returned. Although a lot is the same,
there is a subtle change of culture (Tracy has posted a poll
on one aspect). I did once have the experience of arriving
at a foreign airport in a country where I speak the language
and suddenly found that my complete fluency had gone. It
was weird. I was due on an internal connecting flight and two
hours later I was chatting away with non-English-speaking hosts
in their language. That initial arrival experience is how I feel
now - a familiar environment but the locals are speaking a
different language AND I did used to be able to communicate
with them in their language.

None of this is blaming others. I just need some assistance in
"catching up" in order to appreciate the benefits of the local
culture. Any such assistance would be welcome.
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cgordon



Joined: 04 May 2007
Posts: 769
Location: ontario, canada

PostPosted: Thu Dec 27, 2007 5:54 pm    Post subject: Reply with quote

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.
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Thu Dec 27, 2007 5:59 pm    Post subject: Reply with quote

Alan,

sdq_pete wrote:
XYZ 256 R7C5


This is Peter's notation for an XYZ-wing involving (2,5,6) with r7c5 being the 'pivot cell'.

Basically one is looking for two cells that each contain two of those same digits. Each of those cells must 'see' (share a group with) the pivot cell. In this example r3c5 contains (2,5) and is in the same column as the pivot and r8c6 contains (5,6) and is in the same box as the pivot. The number common to those three cells is (5). Any cell that 'sees' all three cells of the XYZ-wing can have the (5) excluded. In this case, r9c5 can have the (5) excluded.

If r9c5 = 5, then r3c5 = 2 and r8c6 = 6. This leaves the pivot cell empty of possible digits, which is an impossible situation.

sdq_pete wrote:
XY 257 R8C4


This is Peter's notation for an XY-wing involving (2,5,7) with pivot in r8c4.

After the (5) is excluded from r9c5, a (6,9) naked pair is created in column 5, which allows the exclusion of (6) from r7c5, which leaves the following grid:

Code:

 *-----------------------------------------------------------*
 | 25    1     257   | 6     8     345   | 9     47    35    |
 | 8     34    567   | 49    1     3459  | 2     47    356   |
 | 9     34    56    | 24    25    7     | 1     8     356   |
 |-------------------+-------------------+-------------------|
 | 23    5     23    | 8     7     1     | 6     9     4     |
 | 4     69    8     | 5     69    2     | 3     1     7     |
 | 7     69    1     | 3     4     69    | 8     5     2     |
 |-------------------+-------------------+-------------------|
 | 356   7     9     | 1     25    456   | 45    236   8     |
 | 1     8     4     | 27    3     56    | 57    26    9     |
 | 356   2     35    | 479   69    8     | 457   36    1     |
 *-----------------------------------------------------------*


The (2,7) in r8c4 is the pivot. R7c5 and r8c7 are the pincers. Each 'sees' the pivot and shares a different digit with the pivot, with the remaining digit in each (5) being common. Any cell that 'sees' both the pincer cells can have the (5) excluded. R7c7 is such a cell.

If r7c7 = 5, then r7c5 = 2 and r8c7 = 7. This also leaves the pivot cell empty of possible digits. Therefore, r7c7 = 4.


Last edited by TKiel on Thu Dec 27, 2007 6:58 pm; edited 1 time in total
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cgordon



Joined: 04 May 2007
Posts: 769
Location: ontario, canada

PostPosted: Thu Dec 27, 2007 6:50 pm    Post subject: Reply with quote

Quote:
I include in "non-advanced" techniques such things as............"type one" unique rectangles (ie same pair in all four corners).


I'm pretty sure that if you have the same pair in four corners - it's a Deadly Pattern (DP) - meaning there is more than one solution - which is a no-no for Daily Sudoku puzzles. Eg if there are four <12> s -there could be two corners that are either <1>s or <2>s. A Type 1 UR has, for example, a <123> in one of the corners - which means that cell has to be a <3> to avoid the DP. Some VHs in recent past have been solved with just a Type 1 UR - thus putting them in the "advanced" category.
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keith



Joined: 19 Sep 2005
Posts: 3313
Location: near Detroit, Michigan, USA

PostPosted: Thu Dec 27, 2007 8:47 pm    Post subject: Reply with quote

This is a sticking point:
Code:
+----------------+----------------+----------------+
| 235  1    2357 | 6    8    345  | 9    47   35   |
| 8    34   3567 | 49   1    3459 | 2    47   356  |
| 9    234  2356 | 24   25   7    | 1    8    356  |
+----------------+----------------+----------------+
| 23   5    23   | 8    7    1    | 6    9    4    |
| 4    69   8    | 5    69   2    | 3    1    7    |
| 7    69   1    | 3    4    69   | 8    5    2    |
+----------------+----------------+----------------+
| 2356 7    9    | 1    256  456  | 45   236  8    |
| 1    8    4    | 27   3    56   | 57   26   9    |
| 2356 23   2356 | 2479 2569 8    | 457  236  1    |
+----------------+----------------+----------------+

If you don't recognize the hidden triple <479> in R9C457, you may notice an XYZ-wing <256> that takes out <5> in R9C5, and another <456> that takes out <5> in R7C5.

This solves the puzzle. The first XYZ-wing is in the initial grid. The second appears when you solve R5C6 as <3>.

Keith
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Earl



Joined: 30 May 2007
Posts: 587
Location: Victoria, KS

PostPosted: Thu Dec 27, 2007 10:01 pm    Post subject: VH Dec 27 Reply with quote

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
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Fri Dec 28, 2007 12:23 am    Post subject: Reply with quote

XYZ-wing

TKiel wrote:


Basically one is looking for two cells that each contain two of those same digits. Each of those cells must 'see' (share a group with) the pivot cell.




Thank you for this explanation.
It is starting to make sense!!!

Essentially it seems that an XYZ-wing is a "Triple with a kink" in that
any cell that can see all three members of the triple can have the
common digits eliminated.

In this respect it is a generalisation of the "in-line" triple.

Row nine of the puzzle had 356/-/35/479/569/-/457/36/-
There is an in-line triple 356 and so r9c5 reduces to 9 - which, in turn,
reduces the 479 in r9c4 to 47 joining the 47 in r9c7 which had its former
5 removed on account of the triple.

One can view this as saying that r9c5 and r9c7 can "see" all three
cells in the triple (r9c1,r9c3 and r9c8).
Thus the 5 can be eliminated from r9c5 and r9c7 and simultaneously
the 6 can be eliminated from r9c5.

I have been so used to that sort of exercise that I failed to see it as
but a specific manifestation of a general scenario.

Now I will read xyz-wing as "Triple with a kink" and understand it.
Thank you, immensely, for this explanation which cuts through all
the (to me) irrelevant algebraic notation (although I am sure that
a computer solver will need the algebraic or vector concepts!)

From what I now understand the concept of "pivot" is irrelevant in
this case - just as it is not relevant to an "in-line" triple. However,
I need to clarify a point - again using the current puzzle - so that I
can reassure myself that I have understood correctly.

The three cells involved in the xyz-wing (aka triple with a kink)
have profiles 25,256 and 56. The external cell (r9c5) has 569.

In the explanation it was suggested that "5" could be eliminated as
it occurs in all three of the cells in the wing (sounds like a prison!).
Fine, I have no problem with that.
However r9c5 also contains 6 - another member of the wing.
With an in-line triple, one would unhesitatingly remove the "6" as
well as the "5" (as happened with the 356 triple in row nine).

In this case the xyz-wing has the "6" in only of the three cells.
With the benefit of hindsight we know that the "6" can be eliminated
BUT is this possible because of the xyz-wing or some other feature?

My reasoning is that the elimination of the "5" is OK because at least
one of the three in the wing must be a "5". I would suggest that the
same logic MUST exclude the "6" as well because at least one of the
wing must be a "6" and the e"external" cell can see the whole wing.
This is consistent with the "triple with a kink" concept.

Thus my definition would be

"Where three cells can be grouped so that at least one of them is
in line or region contact with both the other two and there are only
three values in the combined profiles of the three cells, any of those
same values may be eliminated from any other cell that has direct
contact (by line or region) with ALL THREE of the group."

This definition includes the in-line triple within it and makes no reference
to the concept of pivot. The latter concept is of value only in the task
of finding the patterns - it would appear to play no part in the actual
implications derived from the pattern.

I know that I am coming to this late in the day (or the year) and it may
seem presumptuous for me to start redefining the technique but I now
have the zeal of a convert and I trust both that Tracy will excuse me for
building on her excellent foundation and that others facing a similar
mental block to my own may come to "see the light" on this.
Again, my heartfelt thanks to Tracy on this one.

+++

With the xy-wing one is also dealing with a triple and an external cell
but the external cell does not need to "see" the pivot cell. Thus the
concept of a pivot IS essential. This is probably why the concept was
carried over to the xyz-wing where it is irrelevant.

My tentative definition is a bit more long winded:

"Where a cell (termed a pivot cell) has only two elements in its profile
and is in line or region contact with two other cells (termed pincer cells)
which have unequal profiles consisting of one digit shared with the profile
of the pivot cell and a specific third digit (the same in each pincer cell)
that third digit - which occurs in the non-pivot ("pincer") cells but not in
the pivot cell - may be eliminated from any other cell that has direct
contact (by line or region) with BOTH of the non-pivot (pincer) cells."

With the xyz-wing it does not seem to matter whether a cell has
two or three elements in its profile but with this pattern the
effect of a third digit in ANY of the profiles would seem to negate
the inference as would allowing equal values in the two pincer cells.

I find this pattern much more complex than the "triple with a kink"
because the limitations are much more precise and one is more
likely to diagnose a "false positive".

Tracy's description is excellent when defining the pincers:

Each 'sees' the pivot and shares a different digit with the pivot, with the remaining digit in each being common. Any cell that 'sees' both the pincer cells can have the common digit excluded.

The additional essential feature would seem to be that all three
cells (pivot plus two pincers) must contain only two elements.

Unlike the xyz-wing the target cells (where elimination will occur) need
have no relationship whatsoever with the pivot cell - but may do so.

My recommendation would be that the "pivot" terminology be dropped
for the xyz-wing where it gets in the way and that the term be retained
for the xy-wing where it is essential to an understanding of the patterns.

Thanks again for the education.
Perhaps I could ask again when the next technique comes into focus!
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Fri Dec 28, 2007 12:36 am    Post subject: Reply with quote

cgordon wrote:
Quote:
I include in "non-advanced" techniques such things as............"type one" unique rectangles (ie same pair in all four corners).


I'm pretty sure that if you have the same pair in four corners - it's a Deadly Pattern (DP) - meaning there is more than one solution - which is a no-no for Daily Sudoku puzzles. Eg if there are four <12> s -there could be two corners that are either <1>s or <2>s. A Type 1 UR has, for example, a <123> in one of the corners - which means that cell has to be a <3> to avoid the DP. Some VHs in recent past have been solved with just a Type 1 UR - thus putting them in the "advanced" category.


Apologies, I was using the reference to "type one" URs with the intention
of AVOIDING the DP scenario as described in the quote above.

I am certainly not attempting to retain them. I dislike the concept of
duplicate solutions as much as others!

I do not believe that that simple URs should be regarded as "advanced"
when they involve only one non-homogeneous corner.
(eg 12,12,12,12xyz in the four corners leads immediately to the removal
of 12 from the fourth corner. Only if y and z are absent is the cell then
directly resolved).

Where more than one corner is involved then the technique SHOULD be
regarded as advanced (eg 12,12,123,124 indicates that either - or both
- of the 3 and 4 will apply. However the combination of holding that
nugget of information simultaneously with other features in the
puzzle makes it a non-simple procedure).

In my previous incarnation (2005/2006) with this group the concept
of URs was just emerging. On my recent return, I find that there are
now SIX types of UR and a new epithet DP for the Deadly Pattern
as a concept. Things have certainly progressed and there may well be
many other things that have changed in the last year and a half of which
I am still totally unaware.
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keith



Joined: 19 Sep 2005
Posts: 3313
Location: near Detroit, Michigan, USA

PostPosted: Fri Dec 28, 2007 1:02 am    Post subject: Reply with quote

Quote:
I'm pretty sure that if you have the same pair in four corners - it's a Deadly Pattern (DP)


Except, to be a non-unique rectangle that avoids a deadly pattern, the four corners must be in only two boxes.

Keith
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Fri Dec 28, 2007 6:26 am    Post subject: Reply with quote

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:

Code:

+------------+--------------+-------------+
| 25  1  257 | 6   8   345  | 9   47  35  |
| 8   34 567 | 49  1   3459 | 2   47  356 |
| 9   34 56  | 24  25  7    | 1   8   356 |
+------------+--------------+-------------+
| 23  5  23  | 8   7   1    | 6   9   4   |
| 4   69 8   | 5   69  2    | 3   1   7   |
| 7   69 1   | 3   4   69   | 8   5   2   |
+------------+--------------+-------------+
| 356 7  9   | 1   256 456  | 45  236 8   |
| 1   8  4   | 27  3   56   | 57  26  9   |
| 356 2  35  | 479 569 8    | 457 36  1   |
+------------+--------------+-------------+


If r9c5 = 6, r8c6 = 5, r7c5 = 2 and r3c5 = 5. There is no contradiction in this. There may be one farther down the line, but it does not come from the XYZ-wing.

The only exclusion possible for an XYZ-wing is the digit all three cells have in common and there can be only one digit they have in common.

Your understanding of the XY-wing is spot on.
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keith



Joined: 19 Sep 2005
Posts: 3313
Location: near Detroit, Michigan, USA

PostPosted: Fri Dec 28, 2007 7:16 am    Post subject: Reply with quote

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
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Fri Dec 28, 2007 11:13 am    Post subject: Reply with quote

TKiel wrote:

The only exclusion possible for an XYZ-wing is the digit all three cells have in common and there can be only one digit they have in common.


Thank you for clarifying this for me.
The quote above summarises the implications nicely.

I have amended my definition accordingly.

"Where three cells can be grouped so that at least one of them is in
line or region contact with both the other two and there are only
three values in the combined profiles of the three cells, any value
which is present in all three cells may be eliminated from any other
cell that has direct contact (line or region) with all three of the group."

I would still maintain that the concept of a pivot cell is irrelevant to
the implications of the pattern (unlike the case of the xy-wing) and
so I make no reference to it in my definition. From my point of view
mentioning the word "pivot" in relation to xyz just introduces an
unwanted mental association with xy. The two patterns are quite
different in nature and using separate language for them assists
with "grounding" such separate natures.

I agree that "triple with a kink" may not be appropriate nomenclature
as it introduces too strong an association with the in-line triple - but
xyz-wing is a totally artificial name anyway!

Thank you for clarifying all this for me - and apologies for running
ahead of myself with unwarranted assumptions.
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