So this isn't an XYZ wing

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So this isn't an XYZ wing

Postby JeJ » Tue Mar 29, 2011 12:13 am

I am posting this in this section because I don't need help with the particular puzzle I am going to post, all I want is to discuss and learn about the XYZ Wing

I know the definition of an XYZ wing, but I thought that it could also be used if two of the cells contained 3 candidates (XYZ of course), but it is not. could someone please explain why not?

As an example if you take
Code: Select all
.-----------------.----------------.----------.
| 5     2    136  | 167  167  8    | 9  4  13 |
| 1346  146  1346 | 126  9    1236 | 7  5  8  |
| 9     7    8    | 5    14   134  | 6  2  13 |
:-----------------+----------------+----------:
| 8     46   7    | 46   3    9    | 2  1  5  |
| 13    9    13   | 27   5    27   | 4  8  6  |
| 46    5    2    | 146  8    146  | 3  7  9  |
:-----------------+----------------+----------:
| 7     8    46   | 3    46   5    | 1  9  2  |
| 2     16   9    | 8    167  167  | 5  3  4  |
| 14    3    5    | 9    124  124  | 8  6  7  |
'-----------------'----------------'----------'

And try to make an XYZ Wing with r3c5, r9c5 and r9c6 to eliminate the "1" in r8c5, you get an error
JeJ
 
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Re: So this isn't an XYZ wing

Postby BryanL » Tue Mar 29, 2011 6:45 am

Hi JeJ,

by definition, in an XYZ-Wing, only the Z can appear in all 3 cells.

All 3 candidates appear in the pivot cell, with XZ in one 'pincer' cell and YZ in the other 'pincer' cell.

From the sudopedia article,

Code: Select all
.-----------.----------.----------.
| *  *  XYZ | .  .  .  | YZ .  .  |
| .  .  .   | .  .  .  | .  .  .  |
| XZ .  .   | .  .  .  | .  .  .  |
:-----------+----------+----------:
"The pivot has candidates XYZ. The implications of each option are:

X
    the XZ pincer will contain digit Z. This digit is eliminated from the starred cells.
Y
    the YZ pincer will contain digit Z. This digit is eliminated from the starred cells.
Z
    the pivot eliminates Z in the starred cells.

Under all circumstances, the starred cells will lose their candidates for digit Z."

In your case r9c5 is the pivot and lets say r3c5 is the YZ pincer with Z=1 and Y=4. Then r9c6 is the XZ pincer with X=2.

If the pivot r9c5 is 1(Z), then r8c5 is not 1.
If the pivot is 4(Y) then r3c5 is 1 and r8c5 is not 1.
But if the pivot is 2(X), then r9c6 can be 1 or 4 and nothing can be said about r8c5.

If r9c6 was limited to 12 - no 4(Y) - then you would have an XYZ-Wing.

hth,

Bryan
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Re: So this isn't an XYZ wing

Postby JeJ » Tue Mar 29, 2011 1:12 pm

Thank you. So far i had been lucky with my eliminations using what I thought was an XYZ-wing. It was only a matter of time until I found out the rule can't be used this way.
JeJ
 
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Re: So this isn't an XYZ wing

Postby JeJ » Tue Mar 29, 2011 3:21 pm

How about this puzzle:
Code: Select all
.----------------------.------------------.--------------------.
| 12348   123    7     | 689    268  689  | 2456  25689  23469 |
| 248     9      248   | 678    5    3    | 2467  268    1     |
| 5       23     6     | 4      1    789  | 27    289    2379  |
:----------------------+------------------+--------------------:
| 379     8      359   | 2      36   4569 | 1567  156    67    |
| 2       4      1     | 568    7    568  | 9     3      268   |
| 2379    23567  2359  | 5689   368  1    | 2567  4      2678  |
:----------------------+------------------+--------------------:
| 1347    137    34    | 167    9    2    | 8     16     5     |
| 6       125    2589  | 3      4    58   | 12    7      29    |
| 124789  1257   24589 | 15678  68   5678 | 3     1269   2469  |
'----------------------'------------------'--------------------'


Could r9c5, r8c6 and r5c6 be an XYZ-wing or does the pivot need to share a box with one of the pincers?
JeJ
 
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Joined: 06 January 2011

Re: So this isn't an XYZ wing

Postby ronk » Tue Mar 29, 2011 3:56 pm

JeJ wrote:Could r9c5, r8c6 and r5c6 be an XYZ-wing or does the pivot need to share a box with one of the pincers?

[edit: The following is more complicated than need be, but I'll leave it stand.]

The fundamental requirements are that ...
  • the XZ and YZ cells each share a row, column or box with the XYZ-cell,
  • the XZ and YZ cells each share a row, column or box with the elimination cell holding a Z-candidate, and
  • the XYZ-cell shares a row, column or box with the elimination cell
To meet all of the above requirements, the XYZ-cell must share a box with either the XZ or YZ cell.

Note: The 'pivot-pincer' terminology has not been commonly used on this forum.
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Re: So this isn't an XYZ wing

Postby daj95376 » Tue Mar 29, 2011 11:44 pm

A visual companion to ronk's technical description. Assume that r5c5 is the XYZ cell. Here are two scenarios for eliminations in Z based on the possible locations of the XZ and YZ cells.

Code: Select all
 cell XZ is in [column 5     ] and outside [box 5]
 cell YZ is in [column 4 or 6] and inside  [box 5]
 eliminations are possible for Z in the two cells that see the XYZ, XZ, YZ cells
 +-----------------------------------+
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  | YZ -Z  YZ |  .  .  .  |
 |  .  .  .  | YZ XYZ YZ |  .  .  .  |
 |  .  .  .  | YZ -Z  YZ |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 |  .  .  .  |  . XZ  .  |  .  .  .  |
 +-----------------------------------+

Code: Select all
 cell XZ is in [row    5     ] and outside [box 5]
 cell YZ is in [row    4 or 6] and inside  [box 5]
 eliminations are possible for Z in the two cells that see the XYZ, XZ, YZ cells
 +-----------------------------------+
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  | YZ  YZ YZ |  .  .  .  |
 | XZ XZ XZ  | -Z XYZ -Z | XZ XZ XZ  |
 |  .  .  .  | YZ  YZ YZ |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 +-----------------------------------+

Note: XZ and YZ are never in the same box, they don't "see" each other, but they are in the same chute; i.e., band or stack.
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Re: So this isn't an XYZ wing

Postby JeJ » Wed Mar 30, 2011 12:16 am

Thanks guys, it is much clearer now.
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Re: So this isn't an XYZ wing

Postby JeJ » Thu Apr 07, 2011 3:03 am

I have another situation of what I thought was an XYZ wing but turned out no to. Could someone please explain why?
The puzzle was generated by Hodoku and is like:
Code: Select all
.------------.----------.----------------.
| 5   8   16 | 2  4  3  | 7     69   19  |
| 9   4   3  | 1  7  6  | 28    5    28  |
| 7   16  2  | 8  9  5  | 136   36   4   |
:------------+----------+----------------:
| 3   29  4  | 6  1  8  | 5     7    29  |
| 68  5   16 | 9  2  7  | 38    4    138 |
| 18  29  7  | 5  3  4  | 1289  289  6   |
:------------+----------+----------------:
| 16  16  5  | 3  8  29 | 4     29   7   |
| 2   7   9  | 4  5  1  | 368   368  38  |
| 4   3   8  | 7  6  29 | 29    1    5   |
'------------'----------'----------------'

r5c79 see r6c78 and so does r6c1 but if i try to eliminate 8 from those cells seen by all 3 members of this chain, I get an error. Why is it so?
JeJ
 
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Re: So this isn't an XYZ wing

Postby aran » Thu Apr 07, 2011 6:19 am

As with all of these wings, the whole idea is :
Z true=>contradiction=>Z false.

Consider the XYZ wing :
Code: Select all
XYZ   XZ
YZ    xxZxx

Thr fourth corner xxZxx contains the candidate Z to be eliminated.
XYZ is referred to as the pivot cell.
Why is Z eliminated ?
Think of the logic as a two-step process :

First step : Z must see all three other cells, thus leaving, were Z true :
Code: Select all
XY   X
Y


Second step : this remaining configuration must be impossible
.
Which can happen only, if the pivot cell sees both of the other cells, which would force the pivot cell empty, which is the contradcition.
Just as in an XY-wing.

Note : there are many possible XYZ "shapes" and they do not need to be "rectangular" as in the above diagrams.
What is common to them all is :
- Z sees all three other cells wherever they are
- the pivot cell sees the other two cells, wherever they are

in your example, the pivot cell r5c9 sees only one of the other cells r5c7. Which is the problem...
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Re: So this isn't an XYZ wing

Postby JeJ » Thu Apr 07, 2011 2:09 pm

aran wrote:in your example, the pivot cell r5c9 sees only one of the other cells r5c7. Which is the problem...

Thanks that summarizes everything.
JeJ
 
Posts: 76
Joined: 06 January 2011

Re: So this isn't an XYZ wing

Postby RW » Thu Apr 07, 2011 4:25 pm

JeJ, a little tip: Do not try to learn techniques by only memorizing patterns. To learn how to use them, you must understand why the elimination happens. When you understand this, it is no problem for you to figure out what is wrong with your supposed eliminations above, even without knowing the definition of the pattern in question.

Whenever you use a pattern to eliminate a candidate, it is because if that candidate was true, there would be a contradiction within the pattern cells. If you are unsure about some elimination, try placing that candidate and see what it does to the remaining cells in the pattern. With simple patterns like XYZ-wing, you probably don't need to pencil anything down to make this check. If there is a contradiction, the elimination is valid. If there is not, the elimination is not valid.

In your latest example you were trying to use a pattern in these cells to show that X<>8:
Code: Select all
.   .   .  | .  .  .  | 38  .   138
18  .   .  | .  .  .  | X

If we assume X=8 we get:
Code: Select all
.   .   .  | .  .  .  | 3   .   1   
1   .   .  | .  .  .  | 8   

No contradiction, you cannot make the elimination based on this pattern. This of course still doesn't mean that X=8, it only means that the pattern you were looking at is useless and doesn't give you any more information about the solution to the puzzle.

If you try to do this check with a real XYZ-wing, you will see a contradiction, as aran explained above.

RW
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Re: So this isn't an XYZ wing

Postby JeJ » Thu Apr 07, 2011 6:32 pm

That's a good advice RW, thank you.
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