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:

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

. . . | . . . | . . .

. . . | . 3 . | . 3 3

------+-------+------

3 . . | . 3 . | . 3 .

3 . . | . . 3 | . . 3

3 . 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.