This is an extension of XYZ-Wing that uses four cells instead of three. A.k.a. Bent Quads.

I am grateful to**SudoNova** posting below and **StrmCkr** who posted about this strategy way back when on this page. I only got round to appreciating the insights recently but the much expanded version of WXYZ is now in version 1.96+ of the solver. Previously I'd restricted the scope to a very narrow definition. Using one of my test libraries, the 50k set from Ruud, I increased the detection from 299 instances to 8313, so it is definitely worth looking out for.

I'm going to start with my narrow definition if only to show how this is an extension of the three-cell XYZ-Wing.

I am grateful to

I'm going to start with my narrow definition if only to show how this is an extension of the three-cell XYZ-Wing.

Its name derives from the four numbers W, X, Y and Z that must be in exactly four cells. In my narrow definition we need a *hinge* cell containing all four candidates WXYZ and three outlier cells each containing a pair wZ, xZ and yZ - Z being the common number.

The easy principle is that 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'.

The easy principle is that 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'.

That's the narrow definition and a glance at XYZ Wing will show you its the same idea plus one more cell. You could expand it to a five cell pattern and five numbers.

Now, lets consider StrmCkr's more general definition:

**WXYZ-Wings can be considered as a group of 4 cells and 4 digits, that has exactly one non-restricted common digit.** We use that digit (Z) to eliminate since at least one of Z will be the solution.

Now, lets consider StrmCkr's more general definition:

So what is a non-restricted common digit?

Well, a restricted digit is one where all the instances of candidate N in the pattern can see each other. On the diagram to the left I have connected the W candidates - because they share the same box, and I can connect the Xs and Ys as they share a row. Only Z is non-restricted because some of the Zs - ie the one in C1 CANNOT see the Zs in B4 and B5.

Well, a restricted digit is one where all the instances of candidate N in the pattern can see each other. On the diagram to the left I have connected the W candidates - because they share the same box, and I can connect the Xs and Ys as they share a row. Only Z is non-restricted because some of the Zs - ie the one in C1 CANNOT see the Zs in B4 and B5.

In the first example I show a classic WYXZ wing - in that it has all four candidates (1/2/5/9) in the hinge cell D3 marked in brown. The three outlier cells, marked in yellow each contain a 9 (our Z) plus some of the other four candidates.

I have also marked in rings the spread of candidates 1, 2 and 5. You will see that both 1s can 'see' each other because they are in the same box. The 2s can 'see' each other because they are on the same row and the three 5s can all see each other since they share the same row as well. That makes 1, 2 and 5

Candidate 9 is different. At least one 9 (in F1) cannot see at least one other 9 (infact both 9s in D4 and D5). That makes it the only

Any WXYZ elimination will always be made on the non-restricted candidate. We are looking for a 9 elsewhere that can see every 9 in the four cells of our WXYZ pattern. That 9 is on D2.

It is important to note is that StrmCkr's rule says nothing about the hinge requiring four candidates and the other cells two. Like Quads, we need*in total* four candidates in four cells. This is a four-cell Locked Set. StrmCkr makes this point in his second corollary, So lets look at some examples where the is a thinner spread of four candidates.

Interestingly, where Z is present in less than four cells, ie three or just two cells of the four - more eliminations are possible because there are less cells that the eliminated candidates need to 'see'. I believe my new implementation WXYZ gives more variants than the 10 StrmCkr lists as exemplars.

It is important to note is that StrmCkr's rule says nothing about the hinge requiring four candidates and the other cells two. Like Quads, we need

Interestingly, where Z is present in less than four cells, ie three or just two cells of the four - more eliminations are possible because there are less cells that the eliminated candidates need to 'see'. I believe my new implementation WXYZ gives more variants than the 10 StrmCkr lists as exemplars.

The second WXYZ example is orientated in columns rather than rows, but works the same. The hinge cell in D6 conspicuously fails to contain the non-restricted candidate 5, but no matter. Whatever the final solution of D6 a 5 is forced into either E5 or one of the two cells G6/J6. The outliers are nice easy pairs - like my original narrow definition, but the improvement is not needing 5 (Z) in the hinge every time.

The solver tells us

WXYZ Wing

The four cells {D6,E5,G6,J6} contain in total 2/5/6/9.

Candidate 5 determines the 'wing' cells (in yellow) and one or other of those 5s must be true. Therefore...

5 can be taken off F6

5 can be taken off G5

5 can be taken off J5

Here is an example where the non-restricted common digit is present in just two pincer cells. We don't have to worry about 3s 'seeing' the brown cells as they don't have a 3 in them. Consider how the hinge cell in C1 solves when each digit is looked at. Suppose C1 is 4. That makes C8 contain 3/5. C8 + C3 + B2 now becomes an Y-Wing - and eliminates 3s in the same way. Interesting!

The solver returns

WXYZ Wing

The four cells {C1,B2,C8,C3} contain in total 3/4/5/9.

Candidate 3 determines the 'wing' cells (in yellow) and one or other of those 3s must be true. Therefore...

3 can be taken off B7

3 can be taken off B8

3 can be taken off B9

I'm ending this article with a nice Sudoku puzzle that has four Y-Wings an XYZ-Wings and four WXYZ-Wings. Here is the first of the later. It is worth following this through on the solver from the start.

Where next? StrmCkr's gives a theory example of a "double restricted common in a WXYZ-Wing when spanning the same band using box-box restraints" which I will explore at a future date. We can also go in the direction of a five cell VWXYZ-Wing but I think the next step is to exactly define the overlap and relationship with Almost Locked Sets - since a number of people have pointed out that these X-Wings are simple examples of ALSs.

Edit Sept 2015: There is an important post on WXYZ-Wings by Jon Welch which I recommend. I'm currently working through the implications but I'm definitely going to re-visit the Wings family again.

## Comments

Comments Talk## Monday 3-Mar-2014

## ... by: Fred Kong

Dear Andrew,I am very thankful to have learned about your WXYZ-Wings theory. With this skill I have solved quite a few very toughed puzzles in my not-completed storage.

However, I met an exception. That is Puzzle 125 in your book: Extreme Sudoku For Dummies.

Please take a look at where I cannot go any further:

| 6 35 8 | 579 4579 4579 | 1 347 2 |

| 1 4 7 | 3 6 2 | 9 58 58 |

| 239 25 29 | 8 457 1 | 46 467 37 |

|----------------------+----------------------+----------------------|

| 4 28 1 | 256 3 56 | 7 9 568 |

| 78 9 3 | 1567 14578 4567 | 458 2 568 |

| 278 6 5 | 279 4789 479 | 3 48 1 |

|----------------------+----------------------+----------------------|

| 289 1 269 | 4 59 3 | 568 5678 79 |

| 39 7 4 | 56 2 8 | 56 1 39 |

| 5 38 69 | 1679 179 679 | 2 38 4 |

I found there is a WXYZ-Wing set as C1, G1, H1 and I2. I thought the Z should be 3, and I could eliminate the 3 in A2 cell. But I could not go to the right end. According to the Answer page, A2 cell is 3.

Please tell me why I cannot use WXYZ-Wing as the above.

Thanks.

Fred Kong

## Monday 30-Dec-2013

## ... by: strmckr

Andrew Stuart writes:Correct and I've put your link in the text. I have now completed the expansion of the search algorithm to get these cases.

thanks for the update, as for the remark on more then 10 cases, true there is a large list of positional locations for the 4 cells{ i originally mapped out 20 ish cases}, however after rotation, reflection, relabeling, band swaping ect is applied the 1 - 4 types I've identified and cross checked via programing should be the minimal exemplars. if you do have any examples cases that don't match my 4 primary types and hold additional eliminations feel free to email me as id love to examine them.

strmckr

## Sunday 22-Sep-2013

## ... by: Gcd

| 4 5678 1679 | 789 789 3 | 2 569 1568 || 159 58 19 | 6 2489 2489 | 1458 7 3 |

| 2379 23678 3679 | 4789 5 1 | 48 469 68 |

|----------------------+----------------------+----------------------|

| 37 9 2 | 1378 1378 58 | 6 35 4 |

| 6 1 4 | 39 239 259 | 35 8 7 |

| 357 357 8 | 347 347 6 | 9 1 2 |

|----------------------+----------------------+----------------------|

| 12379 237 1379 | 5 6 489 | 13478 34 18 |

| 8 4 1369 | 139 139 7 | 135 2 156 |

| 137 367 5 | 2 1348 48 | 13478 346 9 |

Your Solver shows this wxyz uses H4, H5,G6 G9 3,9,8,1 With 1 being excluded from H7. This is from your Sunday daily puzzle of a Sudoku from early September.\

But if you change to G9 to G8, you now have 1,3,8,4 and the 3 in H7 now see's 3'3 all of the wing 3's yet it cannot be excluded. Why not? Where have I made my error?

## Saturday 13-Apr-2013

## ... by: strmckr

i have covered more extensively the idea of bent quads on numerous forum boards for extending this strategy. prior to 2010http://forum.enjoysudoku.com/wxyz-wings-t30012.html

however it can be noted that most if not all wxyz - wings are specific simplified cases of als-xz rule.

## Monday 28-Feb-2011

## ... by: SudoNova

On various Sudoku sites on the internet I have found about a dozen different cases for the use of this technique. I would like to propose the following 'one-size-fits-all' strategy and so allow many more cases to be found.

The strategy for Bent Quads (WXYZ-wing) can be summarised like this. If there are 4 cells having 4 digits each (W, X, Y, Z) distributed within them such that any 2 cells aren't a pair and any 3 cells aren't a triple. The cells aren't all in the same house but they are inter-linked somehow.

One of the cells in the quad has digits WZ (I call this a 'spy' cell), and the other 3 cells of the quad form an 'alliance'. Outside the quad is a 'rogue' cell with digit Z among its candidates and if the rogue declares war, the spy acts as a 5th columnist - unless the alliance can prevent this.

The rogue cell can see all the Z-digits in the alliance and the spy cells. The spy cell can see all the other W-digits in the alliance. So if the Z-digit is given as a solution in the rogue cell, all the Z-digits are eliminated from the quad, and the W-digit is now placed in the spy cell removing the W-digits from the alliance cells.

This leaves the X and Y digits distributed in the 3 remaining cells of the quad

There are { 27 } combinations in which X, Y or XY can be distributed in 3 cells so that the allies can then defeat the rogue Z-digit

these are . . .

XY-XY-XY these cells must all be in the same house { 1 }

XY-XY-X, XY-XY-Y, XY-X-XY, XY-Y-XY, X-XY-XY, Y-XY-XY

same house { 6 }

the next 3 cases follow a similar pattern

XY-X-Y, etc the cell with XY must be a pivot { 6 }

XY-X-X, etc the cells X-X must be in the same house { 6 }

X-X-Y, etc the cells X-X must be in the same house { 6 }

and the last 2 are . . .

X-X-X, Y-Y-Y impossible! it means that the 4th digit

is missing { 2 }

If any of the above conditions are met then the Z-digit can be eliminated from the rogue cell.

This idea could be extended to form any N-wing format, but the logistics are astronomical.

The number of combinations of the remaining (N-2) digits to fit in

the (N-1) alliance cells is given by the formula

C = {[2^(N-2)]-1}^{N-1}

So the above 4-wing has 27 (3^3), a 5-wing has 2401 (7^4) and the 6-wing has 759375 (15^5).

Also, we could remove the restriction that the spy cell only has W and Z in its candidates and have any combination with X or Y included, but using a modified version of the above formula means that the 4-wing now has 2401 combinations of W, X or Y to consider instead of just 27

combinations of X or Y.

## Monday 24-Jan-2011

## ... by: Mats Anderbok

Why don't you have a WXY-wing? With 3 candidates in the hinge, I think it fits in perfectly between XY-wing and WXYZ-wing. Now it isn't caught until APE.## Saturday 8-May-2010

## ... by: Nassir M

also you could remove the 7s from c1 and c6 pls 4 from c1 and also to remove the 5 from A3 because the 5 is going to be either in B3 or C3.Thanks

## Tuesday 13-Apr-2010

## ... by: rich schrader

Isn't this strategy another way of interpreting an ALS?## Wednesday 10-Feb-2010

## ... by: Jeff Sanborn

What if the cell containing 4/9 were located at Row 1, Column 2 (instead of Row 3, Column 8).Must there be one cell within the box and two cells in the row, or is it just a total of three cells that is important?

Furthermore, could you have a VWXYZ-wing if you had a hinge cell with five values and four outlying cells?

Yes you could have a 5-cell VWXYZ-wing although I've not searched for one.

## Thursday 21-Jan-2010

## ... by: LokiMomus

Is there a more generic strategy for this wxyz already mentioned? Couldnt find it, though I admit I havent read all strategies thoroughly. I don't have an example sudoku for this, but consider the given example and lets say row 3, column 5 has 379 as available options instead and i include an extra cell into the strategy, row 3, column 6, which has 379 as available options. Row 3, column 3 has instead the values 34579.The theory still holds. Filling in the 9 in the yellow box will result in an invalid situation, so i can scratch the 9 in the yellow box.

In other words, the more generic requirement should be that there can be n cells of the same cellgroup as long as they have n+1 distinct symbols. (the +1 is for the symbol (9 in this case) that we are considering). The other rules stay intact.

The substrategy of wxyz-wing would be with n=1.

My question to you is: Is this strategy already mentioned on the site? If not, can it be solved by using multiple others?

Note:

This strategy can be even made more generic if there are more than 3 dimensions (special sudokus). In a traditional sudoku there are 3 dimensions (a cell is part of a horizontal cellgroup, a vertical cellgroup and a 3x3 cellgroup). A 4-dimensional sudoku would for example be a jigsaw sudoku on top of a traditional one.