The New Sudoku Players' Forum

by **Jeff** » Fri Aug 12, 2005 3:40 am

tso wrote:
Jeff wrote:
Code: Select all
. . . | . . . | . . . . xy----------------|-xyz . * . . . | . . . | . \xz . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

r2c2 is either x or y. If it's x, r2c9 cannot be x. If it's y, r2c7 is xz which forms a naked pair with r3c8, therefore r2c9 cannot be x. In either case, r2c9 cannot be x.

In each case below, both r2c2=1 and r2c2=2 lead to r2c89<>1.

Code: Select all
. . . | . . . | . . . . 12 . | . . 23 |1234 * * . . . | . . . | . 14 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . 134 . . 12 . | . . . |1234 * * . . . | . . . | . . 134 ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . 23 12 34 |45 56 67 | . * * . . . | . . . | . 18 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | 9 . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | 189 . . 23 21 24 |25 26 27 | . * * . . . | . . . | . 189 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Good thinking, Tso. After reading your post, I feel we should discuss this under a new thread if you don't mind.

Isn't that amazing how our logical thinking can be broadened once the principles of such short simple chains as xy-wing, xyz-wing and turbot fish are grasped. You have just confirmed my inkling, thanks.

For easy reference, just for the discussion of this topic, I categorise the extensions of some simple chains as follows:

xy-chain: extension of xy-wing which is an xy-chain of dimension 3.
xyz-chain: extension of xyz-wing which is an xyz-chain of dimension 3.
wxyz-wing: extension of xyz-wing from naked triples to naked quads.
wxyz-chain: extension of wxyz-wing which is an wxyz-chain of dimension 4.
Turbot chain: extension of turbot fish which is an Turbot chain of dimension 5.

(I) xy-chain

This pattern was discussed in the thread 'xy-chain: description and example'.
Refer http://forum.enjoysudoku.com/viewtopic.php?t=1131

(II) xyz-wing

This pattern was discussed in the thread 'xyz-wing: description and example' where the technique deals with naked triple in the form of (zx)(xyz)(yz). Refer http://forum.enjoysudoku.com/viewtopic.php?t=1103
Well, there exists another type of xyz-wing that deals with naked triple in the form of (xy)(xyz)(xyz).

Code: Select all: . . . | . . . | * * * . xy----------------|-xyz xyz * . . . | . . . | * * * ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

r2c2 is either x or y. If it's x, r2c7 and r2c8 form a naked pair of yz and therefore all other cells in box (1,3) cannot contain z. If it's y, r2c7 and r2c8 form a naked pair of xz and therefore all other cells in box (1,3) cannot contain z. In either case, all other cells in box (1,3) cannot contain z. Therefore z can be eliminated from all these cells. I simply refer this as another member of the xyz-wing family.

(II) xyz-chain

An example of this pattern was explained in the thread 'Advanced application of xyz-wing'. xyz-chain is derived from xyz-wing since the pilot cell has 3 candidates but it is influenced by a short forcing chain outside the box.
Refer http://forum.enjoysudoku.com/viewtopic.php?t=1120

Another good example can be found in the thread 'Pair-chain Combinations' presented by Scott. In this case, the pilot cell also has 3 candidates and it is also influenced by a short forcing chain outside the box. Scott used terms 'almost pair', 'almost triple' and 'almost quad' in his description, which are equivalent to part of a naked pair, naked triple and naked quad respectively.
Refer http://forum.enjoysudoku.com/viewtopic.php?t=1163

Tso, your 3rd example has drifted away from the xyz-wing extension family as the pilot cell is missing. When r2c2=2, the 1 is undefined in box (1,3). Please allow me to rearrange the numbers in this grid, changing it into an xyz-chain.

Code: Select all: . . . |67 . . | 718 . . . 12 . | . . . | * * * . . . | . . . | . 18 . ----------+----------+---------- . 23 . | . . . | . . . . . . |56 . . | . . . . . 34 | . 45 . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

(III) wxyz-wing

Code: Select all: . . . | . . . | . 134 . . 12 . | . . . |1234 * * . . . | . . . | . . 134 ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Tso, your 2nd example is a good demonstration of a simple wxyz-wing. So-called wxyz, because the pilot cell has 4 candidates.

(IV) wxyz-chain

Code: Select all: . . . | . . . | . . . . 12 . | . . 23 |1234 * * . . . | . . . | . 14 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Tso, your 1st example is a good demonstration of an wxyz-chain since the pilot cell has 4 candidates and it is influenced by a short forcing chain outside the box.

Your 4th example has drifted away from the wxyz-wing extension family as the pilot cell is missing. When r2c2=2, the 1 is undefined in box (1,3). Please allow me to rearrange the numbers in this grid, changing it into an wxyz-chain.

Code: Select all: . . . | . . . | 189 . . 34 21 24 |57 36 65 |7189 * * . . . | . . . | . 189 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

(V) Turbot chain

Thanks to Nick, we all know what a turbot fish is and it has 5 sides. The idea can be extended to 7 sides, 9 sides and so on, as long as the number of sides is odd, thus the term 'Turbot chain'. A good example of a turbot chain can be found in Nick's thread 'Dual of XY-Wing: the Skewed Swordfish' where the example used to demonstrate the skewed swordfish is also a 7 sided turbot chain, ie. turbot chain of dimension 7.
Refer http://forum.enjoysudoku.com/viewtopic.php?t=1130

by **tso** » Fri Aug 12, 2005 6:29 pm

Jeff wrote:Tso, your 3rd example has drifted away from the xyz-wing extension family as the pilot cell is missing. When r2c2=2, the 1 is undefined in box (1,3).
Code: Select all
. . . |67 . . | 718 . . . 12 . | . . . | * * * . . . | . . . | . 18 . ----------+----------+---------- . 23 . | . . . | . . . . . . |56 . . | . . . . . 34 | . 45 . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Actually, though my diagram was flawed, it was for another reason. I don't think you noticed the '9' r5c7. The candidate list for the 'pilot' cell (Am I using that term right? What does it mean?) is 12345678, but that wouldn't fit in the diagram.

Code: Select all: . . . | . . . | . . . 23 12 34 |45 56 67 | . * * . . . | . . . | . 18 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | 9 . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

The trouble is, if r2c2=2, only '9' is left for both r2c8 and r2c9:

Code: Select all: . . . | . . . | . . . 3 2 4 | 5 6 7 | 18 9 9 . . . | . . . | . 18 . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | 9 . . . . . | . . . | . . . ----------+----------+---------- . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | . . .

Jeff wrote:Your 4th example has drifted away from the wxyz-wing extension family as the pilot cell is missing. When r2c2=2, the 1 is undefined in box (1,3).

That one was worse -- if r2c2=2, FIVE cells in box 3 are reduced to the same three candidates, [189].

by **Jeff** » Sat Aug 13, 2005 3:20 am

Sorry for my undefined term of 'pilot' cell. I always picture the xy-wing or xyz-wing as a pattern with 3 cells, a pilot in the middle with two wings; like a plane.

For an xy-wing (zx-'xy'-yz), xy is the pilot and the 2 wings are zx and yz.
For an xyz-wing (xy-'xyz'-yz), xyz is the pilot and the 2 wings are xy and yz.
For a wxyz-wing (wx-xyz-'wxyz'-xyz), wxyz is the pilot cell inside the box. The 2 wings are the two xyz cells inside the box and the tail fin is the wx cell outside the box.

Good fun to look at it this way.

by **dfhwze** » Sun Feb 12, 2006 7:14 pm

Well, there exists another type of xyz-wing that deals with naked triple in the form of (xy)(xyz)(xyz).

@Jeff: in your example:

isn't this just a row-block-interaction ,where candidate Z can only occure in the right block ,so in all other cells in that block the Z can be removed ?

is your example a bit unlucky chosen, or is this type of xyz-wing just a unit-interaction (or am i mistaken completely)

?

by **Jeff** » Mon Feb 13, 2006 3:04 am

dfhwze wrote:isn't this just a row-block-interaction ,where candidate Z can only occure in the right block ,so in all other cells in that block the Z can be removed ?

is your example a bit unlucky chosen, or is this type of xyz-wing just a unit-interaction (or am i mistaken completely) ?

Hi Dfhwze, welcome to the forum.

May be so. How is "unit-interaction" defined?

by **dfhwze** » Mon Feb 13, 2006 2:17 pm

Jeff wrote:
dfhwze wrote:isn't this just a row-block-interaction ,where candidate Z can only occure in the right block ,so in all other cells in that block the Z can be removed ?

is your example a bit unlucky chosen, or is this type of xyz-wing just a unit-interaction (or am i mistaken completely) ?

Hi Dfhwze, welcome to the forum. May be so. How is "unit-interaction" defined?

i'm sure you know what it is, you probably just use another name for it

unit-interaction comes inseveral forms:

1) row-block-interaction
__________________

when in a row a candidate can only appear in 1 block,
this candidate gets eliminated in all other squares in this block

which is what happens in your example of xyz-wing
'cause: you have a naked triplet in a row ,which eliminates all Z's in this row that aren't in the triplet. now the Z only appears in the right block
so we can remove all Z's in that block

2) col-block-interaction
__________________

when in a col a candidate can only appear in 1 block,
this candidate gets eliminated in all other squares in this block

3) block-row-interaction
__________________

when in a block a candidate can only appear in 1 row,
this candidate gets eliminated in all other squares in this row

4) block-col-interaction
__________________

when in a block a candidate can only appear in 1 col,
this candidate gets eliminated in all other squares in this col

5) block-block-interaction
__________________

i 've read on some sites about this one too, but i believe this is of no use when you use row-block- and col-block-interaction method bfore this method

by **Jeff** » Mon Feb 13, 2006 3:48 pm

dfhwze wrote:i'm sure you know what it is, you probably just use another name for it.............

1) row-block-interaction
when in a row a candidate can only appear in 1 block,
this candidate gets eliminated in all other squares in this block

which is what happens in your example of xyz-wing
'cause: you have a naked triplet in a row ,which eliminates all Z's in this row that aren't in the triplet. now the Z only appears in the right block
so we can remove all Z's in that block

You are quite right, Dfhwze.

Thanks for pointing this out. I was too close to the tree.

by **dfhwze** » Mon Feb 13, 2006 4:17 pm

Jeff wrote:
dfhwze wrote:i'm sure you know what it is, you probably just use another name for it.............

1) row-block-interaction
when in a row a candidate can only appear in 1 block,
this candidate gets eliminated in all other squares in this block

which is what happens in your example of xyz-wing
'cause: you have a naked triplet in a row ,which eliminates all Z's in this row that aren't in the triplet. now the Z only appears in the right block
so we can remove all Z's in that block

You are quite right, Dfhwze. Thanks for pointing this out. I was too close to the tree.

Hehe it's quite normal you don't see the easy things when dealing with hard .
I can't deal with the hard , so it's really easy for me to see the easy stuff .

anyway i'm glad i could have been of any assistance whatsoever !

by **Mike Barker** » Sun Feb 19, 2006 4:35 am

Requiring an XYZ-wing to have a 3-candidate pilot cell and an WXYZ-wing to have a 4-candidate pilot seems overly restrictive. The generalization discussed here seems much more useful, for example:

Code: Select all: . . . | . . . | . 134 . . 12 . | . . . |123 * * . . . | . . . | . . 134 ----------+----------+----------

follows the logic of an WXYZ wing. In fact, valid WXYZ-wings could have 2, 3, or 4 candidate pilot cells using Vidarino's definition. There could even be two 4-candidate pilot cells. My favorite XYZ-wing would have a 2-candidate pilot

Code: Select all: . . . | . . . | . . . . xy----------------|-yz * * . * . | . . . | . \xz . ----------+----------+----------

which is an XY-wing, XYZ-wing, and XY-chain all in one!

I'd restate Vidarino's definition as: If three cells in the same block and one bivalued ("wx") cell outside of the block together make a naked quadruple ("wxyz") and cells in the block and not in a line with the bivalued cell do not contain one candidate of the bivalued cell (say "x") then the other value ("w") can be excluded from cells common to all four. A similar definition exists for an XYZ-wing. In these cases the bivalued cell becomes the "driver" instead of the 4-candidate cell being the "pilot".

by **dfhwze** » Sun Feb 19, 2006 11:41 am

Mike Barker wrote:...
There could even be two 4-candidate pilot cells...

Suppose i got multiple pilot cells ,or 1 pilot cell that contains 2 candidates that (i don't know the exact term) are 'odd', meaning these candidates only occure once in the ALS.

What if I can find a bi-value cell ,buddy with pilot1, containing one odd candidate and a candidate from the ALS, and i find another bi-value cell, buddy with both the first bi-value-cell and pilot2,
containing both odd candidates ...?

I believe all cells that are buddy with all cells in the ALS and with the first bi-value-cell can have the candidate removed that occures in this first bi-value-cell but isn't the odd candidate.

example:

cell r1c1 => 123
cell r1c4 => 1238
cell r1c8 => 1239
cell r2c4 => 89
cell r2c9 => 19

first 3 cells are the ALS (here almost-naked-triplet)
123 are the not-odd candidates *
89 are the odd candidates

5th cell is the first bi-value cell outside the ALS
1 is a candidate from *
9 is the odd cadidate from cell r1c8

4th cell is the second bi-value cell outside the ALS
89 are both odd candidates from the ALS

I can remove the 1 from cell r1c9

In fact me might find an ALS (almost naked N-subset) with N number of odd candidates spread out over all these cells.
With one odd candidate this would have been a wxyz-wing,
now we have multiple ,how is this technique called ?

by **tarek** » Sun Feb 19, 2006 1:01 pm

I always thought that for a classic *yz wing, you must have a cell containing all *yz candidates in ALS1 seeing a *z cell (AL Double) ........

by **dfhwze** » Sun Feb 19, 2006 1:04 pm

tarek wrote:I always thought that for a classic *xyz wing, you must have a cells containing all *xyz candidates in ALS1 seeing a yz cell (ALS2) ........

in xyz-wing , YZ are the candidates of the potential LS (naked pair) and X is the candidate that maked it an ALS
at least one of the ALS-cells must have this X.

in wxyz-wing, XYZ are the candidates of the potential LS (naked triplet) and W is the candidate that maked it an ALS
at least one of the ALS-cells must have this W.

and so on ...

by **tarek** » Sun Feb 19, 2006 1:11 pm

I haven't made my self clear probably (My previous post was confusing),

I always thought that for a classic *yz wing, you must have a cell containing all *yz candidates in ALS1 seeing a *z cell (AL Double) ........

not just a cell containing the (peripheral component of the wing "W" in WXYZ or the "V" in VWXYZ)

And for the ALS XZ rule you can actually have more than one cell with the W of WXYZ wing as long as they share the same sector with WZ ALS2, however for a classic WXYZ wing we should have only one cell with W and that cell should have all candidates of the wing WXYZ in ALS1. (I'm not sure now if all W containg cells in ALS1 were WXYZ cells, would that still be a classic WXYZ wing !!?)

If not why don't we just call them ALS XZ rule that has a AL double ???

Tarek

by **ronk** » Sun Feb 19, 2006 11:28 pm

Mike Barker wrote:Requiring an XYZ-wing to have a 3-candidate pilot cell and an WXYZ-wing to have a 4-candidate pilot seems overly restrictive. The generalization discussed here seems much more useful ...

I agree. The criteria should be the number of candidates in the union of the largest almost-locked-set (ALS): 3 for the xyz-wing, 4 for the wxyz-wing, etc. While this might seem to imply 2 candidates for the xy-wing, that is not the case. The xy-wing is merely a special case of the xyz-wing.

There are two types of [[[u]v]w]xyz-wing, where [] indicates extensions to the basic xyz-wing. Type 1 (my labeling) has the largest ALS constrained by a box. Type 2 has the largest ALS constrained by a row(col). By way of illustration:

Code: Select all: wxyz-wing: Type 1: . . . | . . wyz . xz . |wxyz * * . . . | . wyz . 1) ALS in box 2 is 4 candidates in 3 cells 2) All x must be in the same row 3) Cell shown wxyz may be as small as wx, xy, wxy, wxz, or xyz as long as ALS definition is met 4) Each cell shown wyz may be as small as wy, wz, or yz as long as ALS definition is met 5) Eliminations (*) are for z 6) One of the two cells shown wyz may be in the same row as x Type 2: . . . | . . . wyz wyz . |wxyz * * . . . | . xz . 1) ALS in row 2 is 4 candidates in 3 cells 2) All x must be in the same box 3) Cell shown wxyz may be as small as wx, xy, wxy, wxz, or xyz as long as ALS definition is met 4) Each cell shown wyz may be as small as wy, wz, or yz as long as ALS definition is met 5) Eliminations (*) are for z 6) One of the two cells shown wyz may be in the same box as x

Note the different positions of xz.

by **tarek** » Fri Feb 24, 2006 2:40 pm

ronk wrote:
Code: Select all
wxyz-wing: Type 1: . . . | . . wyz . xz . |wxyz * * . . . | . wyz . 1) ALS in box 2 is 4 candidates in 3 cells 2) All x must be in the same row 3) Cell shown wxyz may be as small as wx, xy, wxy, wxz, or xyz as long as ALS definition is met 4) Each cell shown wyz may be as small as wy, wz, or yz as long as ALS definition is met 5) Eliminations (*) are for z 6) One of the two cells shown wyz may be in the same row as x Type 2: . . . | . . . wyz wyz . |wxyz * * . . . | . xz . 1) ALS in row 2 is 4 candidates in 3 cells 2) All x must be in the same box 3) Cell shown wxyz may be as small as wx, xy, wxy, wxz, or xyz as long as ALS definition is met 4) Each cell shown wyz may be as small as wy, wz, or yz as long as ALS definition is met 5) Eliminations (*) are for z 6) One of the two cells shown wyz may be in the same box as x

I'm sure That ALS xz rule will confuse some trying to understand these wings.

the ALS xz rule specifies an "x" and a "z" which is fine and understandable.

dfhwze mentioned something which also correct that the consensus was that for for an wxyz wing, the "w" is actually the candidate constrained to a sector with a wz cell which goes to say that in a vwxyz, the "v" is the candidate that should be constrained to a sector with a vz cell.

this doesn't contradict the ALS xz rule, but it will show the source of confusion.

so the example from an ALS xz rule point of view is as ronk mentioned...

Code: Select all: . . . | . . . wyz wyz . |wxyz * * . . . | . xz .

however from the general consensus about wings, this should be:

Code: Select all: . . . | . . . xyz xyz . |wxyz * * . . . | . wz .

That is the source of confusion.(by the way in xyz wings, both points of view converge)

understanding these 2 points of view, discussions should probably reference one of them to avoid confusion.

returning to the classic wing discussion:

for a classic wing IMO:
1. ALS1 should have ONLY one cell containing all candidates of the ALS (& that cell is the only cell containing "x" according to ALS xz opinion, or the only cell containing a "w" according to the general consensus of understanding wxyz wings) - that should serve as the pilot cell.
2. ALS2 should be a bivalued cell which is EXACTLY xz according to the ALS xz rule or exactly wz according to the general consesus of understanding wxyz wings(or vz if VWXYZ or uz if UVWXYZ).

so to me the following are not classic wxyz wings, the following examples are in reference to the ALS xz rule:

Code: Select all: . . . | . . . wyz . . |wxyz wx * . . . | . xz .

Code: Select all: . . . | . . . wyz . . |wxyz wxyz * . . . | . xz .

Code: Select all: . . . | . . . wyz wyz . |wxz * * . . . | . xz .

The reason for this is that when XYZ wings were introduced, they were closely linked to xy wings, people trying to undertand them visualised an xy wing pattern but with xyz cell instead of an xy cell, this means that ALS1 should be xyz & xy and it cannot be xyz & xyz.

And that is why I think that :

Mike Barker wrote:My favorite XYZ-wing would have a 2-candidate pilot
Code: Select all
. . . | . . . | . . . . xy----------------|-yz * * . * . | . . . | . \xz . ----------+----------+----------

which is an XY-wing, XYZ-wing, and XY-chain all in one!

is incorrect

Tarek

The New Sudoku Players' Forum

Extension of xyz-wing

Extension of xyz-wing

Re: Extension of xyz-wing

Pilot cell in xyz-wing