## Dual of XY-Wing: the Skewed Swordfish

Advanced methods and approaches for solving Sudoku puzzles

### Dual of XY-Wing: the Skewed Swordfish

It is well known that the concepts of cell, row, column and block are interchangeable when solving Sudoku problems.
One important example of this property is the X-Wing, which is a dual of the naked pair:

naked pair:
If the possible placements in a row for two numbers include two cells that don't have other possibilities, then the other placements in the row for the two numbers may be excluded.

X-Wing:
If the possible placements for a number in two columns include two rows that don't have other possibilities, then the other placements for the number in the two columns may be excluded.

I was therefore wondering what is the dual of the xy-wing. The answer is a pattern that is very similar to the swordfish.

I will save you the details and just jump to the meat: the pattern looks like this

Code: Select all
`. . . | . . X | . . .. A . | . . . | B . .. . . | C . . | . D .------+-------+------. . . | . . . | . . .. . . | . . . | . . .. . . | . . . | . . .------+-------+------. E . | . . F | . . .. . . | X . . | . . .. . . | X . . | . . .`

A and B are the only possible placements for the number in row 2; C and D the only possible placements in row 3; and E and F the only possible placement in row 7.

If the number is in A, than it's not in E, so it is in F; therefore it cannot be in X.
If the number is in B, then it's not in D, so it is in C; therefore it cannot be in X.

Here is a sample puzzle that can be solved using this pattern.

Code: Select all
`..1.7.2.5.386.....4.............5.....7.1.9.....4.............4.....712.8.2.9.3..`

Using various techiniques, we can reach this position:

Code: Select all
`69    69    1      | 38    7     4      | 2     38    5     7     3     8      | 6     5     2      | 4     19    19    4     2     5      | 9     38    1      | 7     368   36    -------------------+--------------------+-------------------2369  1689  4      | 7     36    5      | 68    136   128   356   568   7      | 2     1     38     | 9     4     36    236   168   36     | 4     368   9      | 5     7     128   -------------------+--------------------+-------------------1     7     39     | 5     2     38     | 68    69    4     356   56    369    | 38    4     7      | 1     2     89    8     4     2      | 1     9     6      | 3     5     7     `

If we isolate the candidate 3's, we have:

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`. . . | 3 . . | . 3 .. . . | . . . | . . .. . . | . 3 . | . 3 3------+-------+------3 . . | . 3 . | . 3 .3 . . | . . 3 | . . 33 . 3 | . 3 . | . . .------+-------+------. . 3 | . . 3 | . . .3 . 3 | 3 . . | . . .. . . | . . . | . . .`

And we can see a skewed swordfish in columns 4, 6, and 9:

Code: Select all
`. . . | 3 . . | . * .. . . | . . . | . . .. . . | . * . | . . 3------+-------+------. . . | . . . | . . .. . . | . . 3 | . . 3. . . | . . . | . . .------+-------+------. . . | . . 3 | . . .. . . | 3 . . | . . .. . . | . . . | . . .`

This allows to remove the candidate 3 from the * cells.

There is another significant observation that should be apparent in this post, if I'm not mistaken. I'll post about it in another thread.
Nick70

Posts: 156
Joined: 16 June 2005

Nick70 wrote:Here is a sample puzzle that can be solved using this pattern.

Code: Select all
`..1.7.2.5.386.....4.............5.....7.1.9.....4.............4.....712.8.2.9.3..`

Just out of interest, is this your 21-clue pattern that you referred to when proving to me that Turbot Fish is more common than X-Wing etc? I have seen lots of puzzles looking like that in this forum and I wondered where they came from.

G
george-no1

Posts: 150
Joined: 20 May 2005

george-no1 wrote:Just out of interest, is this your 21-clue pattern that you referred to when proving to me that Turbot Fish is more common than X-Wing etc?

Indeed it is.
Nick70

Posts: 156
Joined: 16 June 2005

### Re: Dual of XY-Wing: the Skewed Swordfish

Nick70 wrote:If we isolate the candidate 3's, we have:

Code: Select all
`. . . | 3 . . | . 3 .. . . | . . . | . . .. . . | . 3 . | . 3 3------+-------+------3 . . | . 3 . | . 3 .3 . . | . . 3 | . . 33 . 3 | . 3 . | . . .------+-------+------. . 3 | . . 3 | . . .3 . 3 | 3 . . | . . .. . . | . . . | . . .`

And we can see a skewed swordfish in columns 4, 6, and 9:

Code: Select all
`. . . | 3 . . | . * .. . . | . . . | . . .. . . | . * . | . . 3------+-------+------. . . | . . . | . . .. . . | . . 3 | . . 3. . . | . . . | . . .------+-------+------. . . | . . 3 | . . .. . . | 3 . . | . . .. . . | . . . | . . .`

This allows to remove the candidate 3 from the * cells.

Nick

This is similar to Turbot Fish - case No.4 with 3 weak sides, except it has 7 sides instead of 5.
For the * cell at r3c5, the weak sides are r5c6-r5c9, r3c9-r3c5 and r3c5-r1c4.

It appears that these advanced techniques are complementary.

BTW, your first example doesn't show any weak sides which are the characteristic of a swordfish.
Jeff

Posts: 708
Joined: 01 August 2005

### Re: Dual of XY-Wing: the Skewed Swordfish

Jeff wrote:This is similar to Turbot Fish - case No.4 with 3 weak sides, except it has 7 sides instead of 5.

It appears that these advanced techniques are complementary.

Yes, that's a good point. The concept could be generalized to any polygon with an odd number of sides, I guess.

The Turbot itself could be viewed as a skewed X-Wing.

Is there some dual of Turbot Fish that I missed? I don't think so.

Jeff wrote:BTW, your first example doesn't show any weak sides which are the characteristic of a swordfish.

AE and BD are weak.
Nick70

Posts: 156
Joined: 16 June 2005

weak chains:
F-E~A-B~D-C
or
r1c4-r8c4~r7c6-r5c6~r5c9-r3c9

To clarify the relationship between week chains and double forcing chains: The latter e.g. can be written as this double forcing chain:
r7c6=3 => r8c4 <> 3 => r1c4 = 3 => x <> 3
r5c6=3 => r5c9 <> 3 => r3c9 = 3 => x <> 3
where x is one of r1c8 or r3c5 (and r1c7).

.. And to clarify the relationship between weak chains and BunnyBucks colouring method (suppose he would denote it this way):
Code: Select all
`. . . |-3 . . | .[3].. . . | . . . | . . .. . . | .[3]. | . 3+3------+-------+------3 . . | . 3 . | . 3 .3 . . | . .+3 | . . 33 . 3 | . 3 . | . . .------+-------+------. . 3 | . . 3 | . . .3 . 3 |+3 . . | . . .. . . | . . . | . . .`

In principle it is the same, just another notation.
Wolfgang

Posts: 208
Joined: 22 June 2005

### Re: Dual of XY-Wing: the Skewed Swordfish

I don't quite agree with the name "Skewed Swordfish". Swordfish can be considered as a multiple forcing chain while this pattern is a mini double forcing chain.

The core problem appears to me is how to find a forcing chain more easily. This pattern may help to find a mini forcing chain.

Posts: 62
Joined: 15 July 2005

Wolfgang wrote:.. And to clarify the relationship between weak chains and BunnyBucks colouring method (suppose he would denote it this way):

Just want to mention that tso was the first one who came out with this illustration method for colouring over here. I simply copied it.
Bunnybuck

Posts: 15
Joined: 13 June 2005