This is a technique that is based on conjugates.
Conjugates are numbers that only appear as candidates twice in a row, column or block.
If there are two conjugates in different rows, for example, and both the conjugate cells appear in the same column, then there is a link between the two conjugates. This pattern is referred to
as X-wing. It's easy to identify because the four cells that are involved form a
square shape.
X
X
A
X
X
B
X
C
X
X
D
X
X
X
X
X
X
X
Example 1: Two Linked Conjugates in Rows
In this example, A and B are the only places in row 3 that we can put the number 1.
Also, C and D are the only places in row 5 that we can place the number 1. A and C
also appear in the same column, as do B and D. If we were to place a 1 in A, we could
not place a 1 in B or C. However, because row 5 has to contain a 1, and there are
only two places where it can go, and we've just ruled out C, it has to go into D.
Looking at it from the other direction, if we didn't put 1 in A, and instead put it
in B, then A and D cannot have a 1, which means that C must have a 1 in it. We can't
say for sure which of these two is correct, but what we can say is that we will
have a 1 in either A and D, or B and C. Which also means that we will also have a 1
in columns 3 and 7.
What this allows us to do is remove the number 1 as a candidate from all the other
cells in those columns that the conjugate cells appear in, which are marked with an X.
X
X
X
A
X
C
X
X
X
X
X
X
B
X
D
X
X
X
Example 2: Two Linked Conjugates in Columns
In this example, A and B are the only places in column 4 that we can put the number 1.
Also, C and D are the only places in column 6 that we can place the number 1. A and C
also appear in the same row, as do B and D. If we were to place a 1 in A, we could
not place a 1 in B or C. However, because column 6 has to contain a 1, and there are
only two places where it can go, and we've just ruled out C, it has to go into D.
Looking at it from the other direction, if we didn't put 1 in A, and instead put it
in B, then A and D cannot have a 1, which means that C must have a 1 in it. We can't
say for sure which of these two is correct, but what we can say is that we will
have a 1 in either A and D, or B and C. Which also means that we will also have a 1
in rows 4 and 7.
What this allows us to do is remove the number 1 as a candidate from all the other
cells in those rows that the conjugate cells appear in, which are marked with an X.
Here is a real world example.
3
2
4 5
1 4 7
4 5 7
8
9
6
1 4 5
9
4 5 6
7
1 4 5 6
2
1 5 6
1 4 5
3
8
4 6
8
1
9
3
4 5 6
2 4
7
2 4 5
6
4 5 7 9
3 4 5 7 9
8
1
5 7
2
4 5 7
3 4 5 7
2
8
1 5
3
5 6 7
4
1 5 6 7
9
1 5 6
1 3 4 7
1 4 5
4 7
2 6 7
9
2 6 7
1 3 4 6 7
1 4 5 6
8
1 4 7
3
1 4 6 7
4 7
8
2
5
4 7 9
4 6 7 9
4 5 6 7
4 5 6 7
9
4 6 7
1
3
8
4 6 7
2
8
2
4 6 7
5
4 6
9
1 4 6 7
1 4 6
3
Example 3: X-Wing Pattern Conjugate cells,remove candidate from these cells
Looking at the places we can put a 5, there are two conjugates, one in row
1 (cells [r1,c3] and [r1,c5]) and one in row 5 (cells [r5,c3] and [r5,c5]). This
means that we can have a 5 in either [r1,c3] and [r5,c5], or [r1,c5] and [r5,c3].
Therefore we can't have a 5 anywhere else in column 3 or column 5. This means
that we can remove 5 as a candidate from [r3,c3], [r4,c3], [r6,c3] and [r7,c5].
