## WXYZ - Wings

Advanced methods and approaches for solving Sudoku puzzles

### Re: WXYZ - Wings

David, see Sudopedia's description of the WXYZ-Wing being replicated as an ALS-XZ. Anything that doesn't meet the principles behind that explanation can be considered "too complicated and cease being classed as a wing".
daj95376
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### Re: WXYZ - Wings

Danny, I certainly have no desire to change any definition of what a wing is, provided someone can provide an unambiguous one!

As we have seen StrmCkr wants to call his pattern a wing and Ronk doesn't, and I was just trying to help.

I thought the ALSs counting idea was a good one until you objected, so I've suggested an alternative that I'd like you to consider. It has the advantage that it isn't linked to either method of following inferences and merely depends on the geometry of the pattern alone.

It seems we have another point of difference between us; namely what is a 'stream'. To me it conveys a string of linked inferences which can go through as many cells as you want in all parts of the grid. Consequently what constitutes a "short stream" is open to interpretation and as such would cause bickering. I don't know what your understanding of "stream" is though.

I'm sorry I ever contributed here now as I'm not a fan of having profusion of pattern names and variants anyway. I got interested in wings when Steve Kurzhals started posting solutions that used Almost Wings.

 I wrote this in repsonse to your original reply, but on checking my post I see you've changed it completely! However it's 1:47am here now and I'm afraid I'm going to bed.
David P Bird
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Location: Middle England

### hope this clarifies things a bit

confined to two intersecting houses.

Depends on how you view the intersecting houses:

Directly with an shared cell "hinge" as seen in most of the typical wings that primarily composed of Row/Col + Box and expressed to form all the â€œwingsâ€ seen up to this date.

However box - box for smaller ones will also work using peer line of sight as a reference to connect the cells.

For example:
b1 = xz, b2 = xy, xz => R3C123 <> z

Code: Select all
`| .  yz   . | .  .  . | .  xy  . |  | .   .   . | .  .  . | .   .  . |  | -z -z  -z | .  .  . | .  .  xz | `

The first one with an interesting double linked elimination occurs when N=4.

Viewed as a cover problem for box + box operations

where the digits used in both sets are the same digit set:
B[A] contains ( x ) cells with ( N+1 ) digits
&
B[B] contains ( x ) cells with ( N+1 ) digits
where B[AB] = (N+1)

b[A] = box for group of cells, where A represents those cells. {total # of active cells are counted}
b[B] = box for group of cells, where B represents those cells. {total # of active cells are counted}
B[AB] represents the total count of all active cells and there respective positions from B[A]+B[B].
then the sum of the two active boxes = (N+1) digits.

if a cell in B[A] is peer to a cell in B[B] and that cell in B[AB] share 1 or more
digit(s) then all common digits are a restricted link {RL}

Digits locked with in either box is a restricted common {RC}

All cells that are peers of cells within B[AB] < > any other digit in B[AB] that is not restricted to the B[AB] cells

Using the above example of a xy-wing I find a B[A] with 2 cells and B[B] with 1.
B[A] = R1C1 {yz}
B[B] = R1C7 {xy}, R3C9{xz}
RC {restricted commons} X
= > R3C123 <> Z

When increase the N = 3 digits you have several formations that can occur for groups of 2 boxes specifically

Case 1: {eg Types: 2, 2a, 2b, 3, 3a, 4, 4a,}

B[A] with 3 cells & B[B] with 1 cell

Case 2: {eg Type: 1b} & [purposed new type double linked]

B[A] with 2 cells & B[B] with 2 cells

Case 3: {eg Types: 2, 2a, 2b, 3, 3a, 4, 4a,} {inverse of case 1}

B[A] with 1 cell & B[B] with 3 cells

With the same formula with the type I purposed earlier

Code: Select all
`| .   wxz   .  | . . . | wy   .    . || xz  -xz  -xz | . . . | .   xyz   . || .     .   .  | . . . | .    .    . |`

B[A] = R1C2 {wxz}, R2C1 {xz}
B[B] = R1C {wy}, R2C7 {xyz}
RL = W
RC = Y
= > R2C23 <> x,z

edited to correct typos and add clarification:
Last edited by StrmCkr on Sat Jun 12, 2010 4:46 am, edited 5 times in total.
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StrmCkr

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### Re: WXYZ - Wings

David wrote:What seems to be the issue here is what defines the term 'wing".

I then presented a forcing network definition and examples of "wings" as I'd learned them. Sudopedia includes an "extended" version, but the streams are still limited to a value from the vertice cell and an immediately following Naked Single/Subset before eliminations are derived. You can't get much more basic than that for a definition and constraints. To me, StrmCkr's pattern only meets this definition and constraints if the WY cell is allowed to be the vertice cell, but I can't support a WY cell being the vertice cell for a WXYZ-Wing.

Looking for an alternate definition and constraints in terms of ALS's seemed like trying to make a "7" at the craps table by rolling seven dice and hoping they'll all come up with one dot showing. Just because it can be done doesn't mean that it's really worth the effort.
Last edited by daj95376 on Sat Jun 05, 2010 5:45 am, edited 1 time in total.
daj95376
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### Re: WXYZ - Wings

if the WY cell is allowed to be the vertice cell, but I can't support a WY cell being the vertice cell for a WXYZ-Wing.

i showed later that R2C1 can be the "hinge" as well leaving a xy-wing further more i have also shown that not all digits wxyz must be present in the hinge in order for the patterns presented herein to operate and perform the eliminations. {however all those posts are gone }
Some do, some teach, the rest look it up.

StrmCkr

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Joined: 05 September 2006

### Re: WXYZ - Wings

StrmCkr wrote:
if the WY cell is allowed to be the vertice cell, but I can't support a WY cell being the vertice cell for a WXYZ-Wing.

i showed later that R2C1 can be the "hinge" as well leaving a xy-wing further more i have also shown that not all digits wxyz must be present in the hinge in order for the patterns presented herein to operate and perform the eliminations. {however all those posts are gone }

Yes, but you didn't meet the constraint that each stream consists of the vertice cell and an immediate Naked Single/Subset before eliminations are derived.

I suspect that your need for an extra entry in the W stream is what prompted the controversy.

Code: Select all
`| .   wxz   .  | . . . | wy   .    . || xz  -xz  -xz | . . . | .   xyz   . || .     .   .  | . . . | .    .    . |r1c2=X r2c1=Z           =>  r2c23<>XZr1c2=Z r2c1=X           =>  r2c23<>XZr1c2=W r1c7=Y r2c18=XZ  =>  r2c23<>XZ`

My original post shows how you would have to start at the WY cell to meet the stream constraints.

Code: Select all
`| .   wxz   .  | . . . | wy   .    . || xz  -xz  -xz | . . . | .   xyz   . || .     .   .  | . . . | .    .    . |r1c7=W r1c2,r2c1=XZ  =>  r2c23<>XZr1c7=Y r2c18    =XZ  =>  r2c23<>XZ`
daj95376
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### Re: WXYZ - Wings

Danny wrote:Looking for an alternate definition and constraints in terms of ALS's seemed like trying to make a "7" at the craps table by rolling seven dice and hoping they'll all come up with one dot showing. Just because it can be done doesn't mean that it's really worth the effort.
I accept that for someone who doesn't routinely use AICs, analysing a pattern for embedded ALSs would be an unnecessary burden if a simpler alternative was available. Thatâ€™s why I asked you to consider my second suggestion ie "apart from the victim cells, all the cells in a wing pattern are confined to two intersecting houses". Have you done that?

StrmCkr, by "intersecting houses" I mean two houses that share one cell (row/column) or three cells (box/line) where they intersect.

Although an XY-Wing is contained in two boxes, they don't intersect. However it is also contained by an intersecting row and box which meets the requirement. Please remember I'm not counting the cells which hold the excluded digits as part of the pattern.

Code: Select all
`| .   wxz   .  | . . . | wy   .    . || xz  -xz  -xz | . . . | .   xyz   . || .     .   .  | . . . | .    .    . |`
For the new pattern you discovered, whatever combination of two intersecting houses you use, you will always leave one pattern cell uncovered.

Your pattern is therefore too complex to be classed as a simple wing and needs a new name IMO.
David P Bird
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Joined: 16 September 2008
Location: Middle England

### Re: hope this clarifies things a bit

StrmCkr wrote: The first one with an interesting double linked elimination occurs when N=4.

Viewed as a cover problem for box + box operations

B[A] contains ( N ) cells with ( N+1 ) digits
&
B[b] contains ( N ) cells with ( N+1 ) digits
where B[AB] = (N+1)

if a cell in B[A] is peer to a cell in B[B] and that cell in B[AB] share a linked digit then {W is a restricted link}

Digits locked with in either box is a restricted common {y}

All cells that are peers of cells within B[AB] < > any other digit in B[AB] that is not restricted to the B[AB] {x,z}

StrmCkr, when you start using new symbolism without any explanation, you force people to guess your meaning ... and the guesses are more likely to be wrong than correct IMO. So how about explaining what these four mean?

B[A] ... B[B] ... B[AB] ... B[AB] {x,z}

I guess 'B[A]' refers to an ALS set A in one box and 'B[B]' to an ALS set B in a different box ... but then I've no clue what 'B[AB]' might mean.

BTW You must have noticed that you have an AALS, not an ALS, in box 3.
ronk
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### my applogies

BTW You must have noticed that you have an AALS, not an ALS, in box 3.

unfortunately this one took me attempting to program als-xz technique for it to sink in that an als with 2 cells has (2+1 digits) and that these sets of digits between groups A & B share a link of 1 digit via peers and shared candidates that derive the eliminations. {both use a N+1 size set, must be of different combination}

rather then both a & B using an identical digit combination set = to the number of cells used. which is how I was originally viewing them.

however i am still going to attempt to see if this will produce anything useful. So far i have managed to have this programed {below } and it is looking promising as it is finding xy,wxy, wxyz eliminations as well as the oddity being discussed;

N digits = (x1 + x2)
N = a specific combination of digits from (9! / (n!(9-n)!) possible sets
Code: Select all
`     N                âˆ‘ (9! / (n!(9-n)!))    = 511 (sets)=> 1 to  9 `

where N > 2;
where x1 = # of active cells that contain set N for house A . where x1 := ( (n - 1) down to 1 )
where x2 = # of active cells that contain set N for house B. where x2:= ( n - x1)

Where Digit X occurs in a cell that is peers to each other from house A & House B is the restricted common, this digits must be the only copy per house A & house B so that x occurs in one or the other.
Where Digit Z is common to house A and house B = > all peer cells that see all Z candidates in house A & house B < > Z

my tentative code also came up with these versions:
Code: Select all
`| .   wxy   .  | . . . | wz  -x  -x || yz   -x   -x | . . . | .   xz   . || .     .   .  | . . . | .    .   . |`

edit: i think i should segregate this topic into its own thread.
Last edited by StrmCkr on Sun Mar 10, 2019 10:08 am, edited 1 time in total.
Some do, some teach, the rest look it up.

StrmCkr

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Joined: 05 September 2006

### Re: WXYZ - Wings

Code: Select all
`.---------------.------------.-----------------.| 5    478  1   | 2   6   3  | 789    78    49 || 34   6    247 | 1   8   9  | 237    5     34 || 9    238  28  | 7   5   4  | 2368   2368  1  |:---------------+------------+-----------------:| 2    9    3   | 6   4   8  | 5      1     7  || 48   478  478 | 59  19  15 | 236    236   36 || 1    5    6   | 3   2   7  | 4      9     8  |:---------------+------------+-----------------:| 7    38   5   | 4   19  6  | 1389   38    2  || 36   1    9   | 8   7   2  | 36     4     5  || 468  248  248 | 59  3   15 | 16789  678   69 |'---------------'------------'-----------------'`

case study puzzle for technique testing.

the interesting thing from this puzzle:

als-xz double linked rule can and does apply to some formations for wxyz-wings:

{that is if there is 2 restricted commons between A&B then all cells become locked sets and all peer cells for each candidate with in A&B can be eliminated}

Almost Locked Set XZ-Rule {double link rule}: A=r1c8 {78}, B=r79c8,r8c7 {3678}, X=7,8 => r3c8<>8, r7c7<>3, r9c79<>6

for posterity:
this directly results in an old technique {sue de coq} ie 2 sector disjointed set

Sue de Coq: r79c8 - {3678} (r1c8 - {78}, r8c7 - {36}) => r7c7<>3, r9c79<>6, r3c8<>8
Some do, some teach, the rest look it up.

StrmCkr

Posts: 980
Joined: 05 September 2006

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