This strategy is looking at single numbers in rows and columns. It should be easier to spot in a game as we can concentrate on just one number at a time.
The picture on the right shows a classic X-Wing, this example being based on the number seven. The X is formed from the diagonal correspondence of squares marked A, B, C and D. What's special about them?
Well, A and B are a locked pair of 7's. So is C and D. They are locked because they are the only 7's in rows B and F. We know therefore that if A turns out to be a 7 then B cannot be a 7, and vice versa. Likewise if C turns out to be a 7 then D cannot be, and vice versa.
What is interesting is the 7's present elsewhere in the fourth and eighth columns. These have been highlighted with green boxes.
Think about the example this way. A, B, C and D form a rectangle. If A turns out to be a 7 then it rules out a 7 at C as well as B. Because A and CD are 'locked' then D must be a 7 if A is. Or vice versa. So a 7 MUST be present at AD or BC. If this is the case then any other 7's along the edge of our rectangle are redundant. We can remove the 7's marked in the green squares.
The rule is When there are
only two possible cells for a value in each of two different rows,
and these candidates lie also in the same columns,
then all other candidates for this value in the columns can be eliminated.
The reverse is also true for 2 columns with 2 common rows. Since this strategy works in the other direction as well, we will look at an example next.
In this second example I've chosen a Sudoku puzzle where an enormous number of candidates can be removed using two X-Wings. The first is a '2-Wing'. The yellow high lighted cells show the X-Wing formation. Note that the orientation is in the columns this time, as opposed to rows as above. Looking at columns we can see that candidate 2 only occurs twice - in the yellow cells. Which ever way the 2s could be placed (E5/J8 or E8/J5) six other 2s in the rows can be removed - the green highlighted cells.
A few steps later the second X-Wing is found on candidate 3 in the same rows. Whichever way round the 3 can be placed in those rows (E2/J8 or E8/J2) there can be no other 3 in rows E and J except in those yellow cells.
Generalising X-Wing
X-Wing is not restricted to rows and columns. We can also extend the idea to boxes as well. If we generalise the rule above we get:
When there are
only 2 candidates for a value, in each of 2 different units of the same kind,
and these candidates lie also on 2 other units of the same kind,
then all other candidates for that value can be eliminated from the latter two units.
Now we have 6 combinations:
Starting from 2 rows and eliminating in 2 columns
Starting from 2 columns and eliminating in 2 rows
Starting from 2 boxes and eliminating in 2 rows
Starting from 2 boxes and eliminating in 2 columns
Starting from 2 rows and eliminating in 2 boxes
Starting from 2 columns and eliminating in 2 boxes
Here is an example of combination 5. Starting from 2 rows and eliminating in 2 boxes, in this case the last two boxes in the Sudoku. The rows are 7 and 8 and they each have two 7s. Our x-Wing is now a trapezoid but the logic is the same. We can be certain that 7 can be eliminated at X, Y and Z
But HOLD UP one moment. There is a simpler strategy that does the same job!
A and B above are a pointing pair. This removes the same 7s in the same place. Combination 6 is also the complement of a pointing pair. Combinations 3 and 4 are also complements of the Line/Box Reduction. Our generalisation of X-Wing to boxes hasn't profited us at all. We learn that
TalkSubject Comments Comments here pertain to corrections to the text, not the subject itself
Wednesday 9-Jan-2013
... by: sollaw
The initial x-wing solutio'n refers us to columns 2 and 6; in fact it should be column 4 and 8. This error was noted by Rich in his Dec. 8, 1213 note, and remains uncorrected.
Andrew Stuart writes:
Fixed
Tuesday 25-Dec-2012
... by: Warren Hudson
In the X wing strategy last sentence before the second example, there are the words, "we'll looks"...
I hate tipos, msspelings, grimattical eros and such, but I still loves your website.
And A Merry Christmas to you.
Andrew Stuart writes:
Fixed. Thank you
Saturday 8-Dec-2012
... by: rich
X-wing strategy, 3rd paragraph. Should say fourth and eighth columns, not second and sixth, I think. Great writeups.
Andrew Stuart writes:
Fixed. ty
Saturday 8-Aug-2009
... by: Kim Sidey
Andrew,
The margin of the left column in the first figure contains numbers. You likely meant to use alphanumerics (A-I).
Comments
Talk Subject CommentsComments here pertain to corrections to the text, not the subject itself
Wednesday 9-Jan-2013
... by: sollaw
The initial x-wing solutio'n refers us to columns 2 and 6; in fact it should be column 4 and 8. This error was noted by Rich in his Dec. 8, 1213 note, and remains uncorrected.Tuesday 25-Dec-2012
... by: Warren Hudson
In the X wing strategy last sentence before the second example, there are the words, "we'll looks"...I hate tipos, msspelings, grimattical eros and such, but I still loves your website.
And A Merry Christmas to you.
Saturday 8-Dec-2012
... by: rich
X-wing strategy, 3rd paragraph. Should say fourth and eighth columns, not second and sixth, I think. Great writeups.Saturday 8-Aug-2009
... by: Kim Sidey
Andrew,The margin of the left column in the first figure contains numbers. You likely meant to use alphanumerics (A-I).
Great web page. You've helped me tremendously!
-Kim