The New Sudoku Players' Forum

[StrmCkr]

This is an extension of XYZ-Wing that uses four cells instead of three. Each possible value of the hinge cell results in a Z value in one of the cells in the WXYZ-Wing pattern, thus leaving no room for a Z on any cell all four can 'see'.

Its name derives from the four numbers W, X, Y and Z that are required in the hinge. The outer cells in the formation will be Wz, XZ and YZ, Z being the common number

the quote above is from http://www.sudokuwiki.org/WXYZ_Wing and should be amended; Where W is required, but not all of the digits xyz are required within the hinge to function.

see collary located at the bottom of this post.

I have collected the limited data for this technique and have complied a minimal exemplars listing for all the variations I can identify here is this list:

Type 1:

Code: Select all

.   wz    . |  .  .   . |  .  .  .
-Z  wxyz -Z |  . xyz  . |  . xyz .
.   .     . |  .  .   . |  .  .  .
 

Type 1a:

Code: Select all

.   wz    . |  .  .   . | -z -z  -z
-Z  wxy  -Z | .  xy   . |  . xyz .
.   .     . |  .  .   . |  .  .  .


Type 1b:

Code: Select all

.   wz    . |  .  .   . | -z   -z  -z
-Z  wxy  -Z |  .  .   . |  yxz  yxz .
.   .     . |  .  .   . |  .    .    .


Type 2:

Code: Select all

.   xyz   . |  .  .   . |  .  .  .
-Z  wxyz -Z |  .  wz  . |  .  .  .
.   xyz   . |  .  .   . |  .  .  .


Type 2a:

Code: Select all

.   xyz   . | -z -z  -z |  .  .  .
-Z  wxy  -Z |  .  wz  . |  .  .  .
.   xy    . |  .  .   . |  .  .  .


Type 2b:

Code: Select all

.   xyz  xyz | -z -z  -z |  .  .  .
-Z  wxy   -Z |  .  wz  . |  .  .  .
.    .     . |  .  .   . |  .  .  .


Type 3:

Code: Select all

 .   wz   .   |  .  .   . |  .  .  .
-z  wxyz wxyz |  . xyz  . |  .  .  .
 .   .    .   |  .  .   . |  .  .  .


Type 3a:

Code: Select all

 .   wz   .   | -z -z  -z |  .  .  .
-z  wxy  wxy  |  . xyz  . |  .  .  .
 .   .    .   |  .  .   . |  .  .  .


Type 4

Code: Select all

 .  xyz   .   |  .  .   . |  .  .  .
-z  wxyz wxyz |  .  wz  . |  .  .  .
 .   .    .   |  .  .   . |  .  .  .


Type 4a

Code: Select all

 .  xyz   .   | -z -z  -z |  .  .  .
-z  wxy  wxy  |  .  wz  . |  .  .  .
 .   .    .   |  .  .   . |  .  .  .


Collary:

1. WXYZ-Wings can be considered as a group of 4 cells and 4 digits, that has exactly one non-restricted common digit.

edit: 3 digits are restricted to 1 sector each {restricted}, and 1 digit is in 2 sectors {non restricted}


2. Not all candidates xyz may be presents in the cells where they are indicated.


3. May also be constructed as an als- xz function when combined they have 4 cells and 4 digits total
whereby
als a, als b has:
1+ RCC {x},
1 nRCC {z},
where the elimination are based on
1:RCC normal als xz rules,
2:RCC double linked rules,

extensions:
Double linked Rule

Code: Select all

 .  .   .  |  .  .  .  |  .  .  .
  -3 678 83 |  .  .  .  |  . 78  .
  . 36   .  |  .  .  .  |  .  .  .
  --------------------------------
  .  .   .  |  .  .  .  |  .  .  .
  .  .   .  |  .  .  .  |  .  .  .
  .  .   .  |  .  .  .  |  .  .  .
  --------------------------------
  .  .   .  |  .  .  .  |  .  .  .
  .  .   .  |  .  .  .  |  .  .  .
  .  .   .  |  .  .  .  |  .  .  .

in the above configuration on top of the normal wxyz-wing elimination
there may be two restricted commons between sets A&B
when this occurs all cells become locked sets and all peer cells for each digit for each set may be eliminated.

wxyz - double linked rule: A=r2c238 {3678}, B=r3c2 {36}, X=3,6 => r1c123,r23c1,r3c3,r456789c2<>6, r1c123,r23c1,r3c3<>3, r2c145679<>7, r2c145679<>8

amended dec 3rd 2022 for increased clarity

[StrmCkr]

Persevered for examples

[StrmCkr]

it is also possible to have a double restricted common in a wxyz -wing when spanning the same band using box - box restraints.

Code: Select all

| .   wxz   .  | . . . | wy   .    . |
| xz  -xz  -xz | . . . | .   xyz   . |
| .     .   .  | . . . | .    .    . |

[ronk]

it is also possible to have a double restricted common in a wxyz -wing when spanning the same band using box - box restraints.

Code: Select all

| .   wxz   .  | . . . | wy   .    . |
| xz  -xz  -xz | . . . | .   xyz   . |
| .     .   .  | . . . | .    .    . |

When expressed as a chain, as David P Bird pointed out elsewhere, this is an ALS chain with endpoint overlap (r2c1) and three ALS nodes (A, B, C below).

Code: Select all

               A          B         C
r2c23 -xz- r1c2,r2c1 -w- r1c7 -y- r2c18 -xz- r2c23 --> r2c23<>xz

Either endpoint overlap or three nodes by itself make considering this a wxyz-wing questionable, but when both exist there is no question IMO. It should not.

[StrmCkr]

for me als-xz rule uses a combination of 2 als sets as with this example wxyz wing

Code: Select all

.   wz    . |  .  .   . |  .  .  .
-Z  wxyz -Z |  . xyz  . |  . xyz .
.   .     . |  .  .   . |  .  .  .

Almost Locked Set XZ-Rule: A=r4c2 {wz}, B=r5c258 {wxyz}, R=w, elimination=z => r5c13<>z
in this example it uses Row + box where the box holds the restricted common.

Code: Select all

| .   wxz   .  | . . . | wy   .    . |
| xz  -xz  -xz | . . . | .   xyz   . |
| .     .   .  | . . . | .    .    . |

A =R1C2 {wxz} R2C1 {xz} B = R1C6 {wy}, R2C7{xyz}, R=wy, Elimination = xz => R2C23 <> xz

the signification difference to note is that group A is comprised within a box as is B, allowing for Two restricted commons {1 per box} and for the elimination it is any cell that sees all Restricted commons between both groups A and B.

i don really see how the cells are overlapped, furthermore it uses 4 digits covering 4 cells which to me is the same concept as a wxyz wing.

[ronk]

i don really see how the cells are overlapped ....

Since you don't see the 1-cell overlap of two ALSs, there's virtually nothing I can say that would be convincing.

[edit: added] Maybe this image will help you see the overlap.



For ALS B, either r1c7=1 or r1c7=3
If r1c7=1, ALS A becomes pair <24>
If r1c7=3, ALS C becomes pair <24>
By inspection, r2c1 is clearly a member of both sets

r2c23 are peers of both ALS A and ALS C (sectors enclosed in rectangles)
Therefore, r2c23<>24

[StrmCkr]

this is a copy from a pm i sent regarding the below proposal. i am wondering if some one else could formalize this a bit better to perhaps explain it better or give an alternative view on its function.

i think it has some thing to do with the inherit weak links between the bivalves off the W forming strong links interface linking the two boxes as a locked sets.
where y is the restricted common in box 3 and W is the linked restricted common and all shared cells x,z are restricted to the als cells thus all cells seeing xz are removed.

Code: Select all

    | .   wxz   .  | . . . | wy   .    . |
    | xz  -xz  -xz | . . . | .   xyz   . |
    | .     .   .  | . . . | .    .    . |

the only way i can explain it is that instead of the sets on a row consider them as sets with in a box. if cells in box a sees box b cells via line of sight they can be linked together.

for example considering either wxz or wy as the hinge cell typical of a wing. {the link relation of W}

if R1C6 = w then R1C2 = xz & R2C1 = xz
if R1C6 = y then R1C2 = xz & R2C7 = xz

If R1C2 = w then R1c7 = y then R1C6 = xz & R2C7 = xz
If R1C2 = x then R2C1 = z
If R1C2 = z then R2C1 = x

technically you could use all 4 cells as the hinges... but some of them degenerate into a xyz function,

[David P Bird]

What seems to be the issue here is what defines the term 'wing". Is it how compact the pattern is in terms of the containing houses (which seems to your approach StrmCkr) or the number of strong inferences (or ALS inferences in these examples) employed? Am I right in saying wings only use two ALSs and anything that needs a third one doesn't qualify?

The opening post is concerned with the number of cells involved, but this could be any number if extra passenger digits are added to extend the ALSs.

[daj95376]

I had the naive impression that XY/XYZ/WXYZ/etc Wing patterns were based on a forcing chain/network with simple streams from a vertex cell.

Code: Select all

 XY-Wing: vertex cell = r2c5

 r2c5=X r2c2=Z  =>  r6c2<>Z
 r2c5=Y r6c5=Z  =>  r6c2<>Z
 +-----------------------------------------------+
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .  XZ   .   |   .  XY   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .  -Z   .   |   .  YZ   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 +-----------------------------------------------+


Code: Select all

 XYZ-Wing: vertex cell = r2c5

 r2c5=X r2c2=Z  =>  r2c46<>Z
 r2c5=Y r3c5=Z  =>  r2c46<>Z
 r2c5=Z         =>  r2c46<>Z
 +-----------------------------------------------+
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .  XZ   .   |  -Z  XYZ -Z   |   .   .   .   |
 |   .   .   .   |   .   YZ  .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 +-----------------------------------------------+


Code: Select all

 WXYZ-Wing #1: vertex cell = r2c5

 r2c5=W r2c8=Z  =>  r2c46<>Z
 r2c5=X r2c2=Z  =>  r2c46<>Z
 r2c5=Y r3c5=Z  =>  r2c46<>Z
 r2c5=Z         =>  r2c46<>Z
 +-----------------------------------------------+
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .  XZ   .   |  -Z WXYZ -Z   |   .  WZ   .   |
 |   .   .   .   |   .  YZ   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 +-----------------------------------------------+


Code: Select all

 WXYZ-Wing #2: vertex cell = r2c5

 r2c5=W r1c5=Z  =>  r2c46<>Z
 r2c5=X r2c2=Z  =>  r2c46<>Z
 r2c5=Y r3c5=Z  =>  r2c46<>Z
 r2c5=Z         =>  r2c46<>Z
 +-----------------------------------------------+
 |   .   .   .   |   .  WZ   .   |   .   .   .   |
 |   .  XZ   .   |  -Z WXYZ -Z   |   .   .   .   |
 |   .   .   .   |   .  YZ   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |---------------+---------------+---------------|
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 |   .   .   .   |   .   .   .   |   .   .   .   |
 +-----------------------------------------------+


The complexity of a Wing depended on what you considered to be a simple stream.

What concerns me about this pattern from StrCkr is that it could more easily be viewed as a forcing network from the WY cell.

Code: Select all

| .   wxz   .  | . . . | wy   .    . |
| xz  -xz  -xz | . . . | .   xyz   . |
| .     .   .  | . . . | .    .    . |

r1c7=W r1c2,r2c1=XZ  =>  r2c23<>XZ
r1c7=Y r2c18    =XZ  =>  r2c23<>XZ


[StrmCkr]

what about his part daj that shows its not a forcing network?

R2C1 = z

Code: Select all

    | .   wx   .  | . . . | wy   .    . |
    | Z  -x  -x | . . . | .   xy   . |
    | .     .   .  | . . . | .    .    . |

or
R2C1 = x

Code: Select all

| .   wz   .  | . . . | wy   .    . |
| x   -z   -z | . . . | .   yz   . |
| .     .   .  | . . . | .    .    . |

you have an xy-wing left in both cases

[David P Bird]

daj95376 to justify my previous post, all your your examples can be expressed as 2 linked ALSs composed from 3 or 4 cells in total:

XY-Wing:......... (xz=y)ALS:r2c35 - (y=z)ALS:r6c5 => r6c2 <> z (3 cells)
XYZ-Wing:....... (xz=y)ALS:r2c25 - (y=z)ALS:r3c5 => r2c46 <> z (3 cells)
WXYZ-Wing #1: (wxz=y)ALS:r2c258 - (y=z)ALS:r3c5 => r2c46 <> z (4 cells)
WXYZ-Wing #2: (z=x)ALS:r2c2 - (wxy=z)ALS:r123c5 => r2c46 <> z (4 cells)

StrmCkr's example needs 3 ALSs composed from 4 cells:
(xz=w)ALS:r2c1,r1c2 - (w=y)ALS:r1c7 - (y=xz)ALS:r2c17 => r2c23 <> xz

Without banging on about my reasons, I just prefer to notate eliminations using AICs, that's all.

[daj95376]

David, you may prefer chain notation using ALS's, but the history of these patterns supports my interpretation.

In ScanRaid, the term hinge is used where I said vertex.

WXYZ-Wing

This is an extension of XYZ-Wing that uses four cells instead of three. Each possible value of the hinge cell results in a Z value in one of the cells in the WXYZ-Wing pattern, thus leaving no room for a Z on any cell all four can 'see'.
Its name derives from the four numbers W, X, Y and Z that are required in the hinge. The outer cells in the formation will be WZ, XZ and YZ, Z being the common number.

[David P Bird]

David, you may prefer chain notation using ALS's, but the history of these patterns supports my interpretation.

daj95376, that's a strange line of argument! Because early explorers navigated by compass and sextant, would you suggest that we shouldn't be using GPS now?

What I was driving at was; whichever way we look at these things, when does a wing stop being a wing? It appears to me to depend on how many hinges, pivots, or vertices are involved, which is equivalent to how many ALSs there are. The number of cells making up a wing pattern wouldn't then be a factor.

However, the opening post in this thread focuses on patterns restricted to four digits in four cells - which is what StrmCkr's example does.

[daj95376]

[Edit:] Comment replaced with later reply.

[David P Bird]

Danny, whether we use two forcing chains away from a pivot node or a single AIC through it shouldn't matter as we are using exactly the same inferences. What I was asking was simple enough - when does a wing grow too complicated to cease being classed as a wing?

My answer is still a forcing network from a hinge/pivot/vertice cell with short streams -- typically two cells per stream.

You were ducking the question as you haven't defined a clear boundary.

As you don't like using an ALS count to decide, another way of looking at the same thing (I think) would be to say that, apart from the victim cells, all the cells in a wing pattern are confined to two intersecting houses.