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| 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 rule is When there are
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. |
| The picture on the right shows a classic X-Wing, this example being based on the number six. 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 6's. So is C and D. They are locked because they are the only 6's in the first and last rows. We know therefore that if A turns out to be a 6 then B cannot be a 6, and vice versa. Likewise if C turns out to be a 6 then D cannot be, and vice versa. What is interesting is the 6's present in the two columns 6 and 9 directly between A and C and B and D. These have been highlighted with green boxes. (All other 6s not used in the pattern are show in cyan). |
 X-Wing example1: Load Example |
| Think about the example this way. A, B, C and D form a rectangle. If A turns out to be a 6 then it rules out a 6 at C as well as B. Because A and CD are 'locked' then D must be a 6 if A is. Or vice versa. So a 6 MUST be present at AD or BC. If this is the case then any other 6's along the edge of our rectangle are redundant. We can remove the 6's marked in the green squares. This strategy works in the other direction as well, as we'll see in the next example |
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. |
 X-Wing example 2: Load Example or : From the Start |
| 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. |  X-Wing example 3 |
| 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
then all other candidates for that value can be eliminated from the latter two units. Now we have 6 combinations: |
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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 trapeziod 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 generalization of X-Wing to boxes hasn't profited us at all. We learn that Previous article: Intersection Removal. Next article: Singles Chains |
Comments
Saturday 26-Mar-2011... by: Yves SiouiIn X-wing example 2, using cells CJ59 instead CJ58 would 'erased' the 2's in column 8. Since that new choice respect the same conditions as the one you choose, the results are in conflict with each other. In one instance the value 2 in column 8 is possible and the other decline that. I find it disturbing.Andrew Stuart writes: CJ59 is not a valid X-Wing since column 9 contains more than two 2s - so the conflict you highlight does not arise Thursday 30-Dec-2010... by: Ted LIn X Wing example 1, you state that after elimination only a 9 remains in cell G9. Is this incorrect, for it looks that a 2 and a 9 remain in this cell. I loaded the example to confirm this, and found that simple colouring is still required before the puzzle can be completed.A wonderful site which gives great pleasure and instruction - thank you very much. Andrew Stuart writes: Thx for this prompt. I believe I left a sentence about a completed cell in the paragraph because of the old diagram. You are correct a 2 remains in G9 so I've removed that sentence. Sunday 19-Dec-2010... by: William MannYour answer to Colin Pearce, June 30, doesn't make sense. It isn't because "the blue boxes contain other 6s in the row". I think what you should have told him is that both pairs (A/B and C/D) are the only two squares with sixes in their rows (locked pairs), therefore, either A or B must be a six, and either C or D must be a six. All other sixes in those columns can be eliminated.Friday 6-Aug-2010... by: RohanAs a complete beginner and coming to grasp with the logic of X-Wing, it seems there are certain conditions that need to apply before this strategy can be applied. Amongst which are:1. No occurrence of the number (in this case 6) can occur in the rows between A and B or C and D; 2. the classic X-Wing uses only the extreme boxes at the edges of the puzzle since otherwise, as Colin Pierce pointed out why could one not use the blue boxes above C and D? Because if one did there would be a very different result. Whatever, thanks for your very useful site, it has helped me greatly! Andrew Stuart writes: Your condition 2 is false. I've added another example - which I hope is very illustrative of the strategy. Tuesday 3-Aug-2010... by: Elmer SchartowRegarding X-Wing Strategy:I'm being picky but the second para after the first illus points out the boxes BETWEEN AB and CD which are highlighted in yellow. On my computer the boxes ABCD themselves are yellow and the boxes BETWEEN ABCD are highlighted in cyan. I believe the first question posed by colin pearce is a valid one and needs to be answered. Also the situation regarding the 7 in cell X posed by CS Vidyasager appears to be completely valid IF there are no 7's in A and B. Andrew Stuart writes: Fixed the 'yellow' word in the second para. thx (refered to an old diagram) Wednesday 30-Jun-2010... by: colin pearceHi Andrew,thank you for this extraordinary and marvellous site. I have a question on your X-wing principle. The top diagram with the yellow boxes ABCD... (and I hope I;m not being obtuse here), but why couldn't the four yellow boxes include, instead of C and D, the two blue boxes above them, this eliminating C and D as options? Thanks again.. I love this resource, cheers colin Andrew Stuart writes: because the blue boxes contain other 6s in the row Friday 9-Apr-2010... by: Marv RoweStopped using X-Wing on anything but squares and rectangles after I did www.websudoku.com Extreme Puzzle 80,052,202,927 - Had only two instances of 7's in Columns 3 & 4 - Cells r5c4, r6c3, r8c3 & r8c4 --> thought I could eliminate all other 7's from row 8 - wrong assumption - 7 in r8c7 was the correct answer - at lower levels (easy, medium, hard and evil) could always eliminate numbers is cells if the x-wing was a trapizoid - not so in this caseThursday 25-Feb-2010... by: CS VidyasagarX wing is traditionally diaognal. In the trapezoidal like you explained, If 7 is present in cell A, then it can not be in cell B and vice versa. So any other 7 under the influence of cell A or B can be eliminated. Same logic applies to C and D. That way 7 in cells Y and Z can be eliminated by this logic. But 7 in Cell X can not be eliminated as it it not controlled by A or B or C or D.thanks Wednesday 3-Feb-2010... by: Kantilal M ManeExcellent technique !!!Friday 18-Dec-2009... by: John MathewsIf I am understanding this right, then the pattern can only be a square, rectangle, or trapezoid shape for any of these X-wing solutions. Is that correct?Andrew Stuart writes: Correct. To be trapizoid the connections between the cells are through the box they share - not just the rows and columns - which alone would produce a rectangulat pattern. Sunday 4-Oct-2009... by: Nick PannellI'm a mathematician and I'm really intrigued at the setting of patterned sudokus and the decision as to what makes one easy, medium or hard. Your soultion strategies are very interesting, although the explanations are a bit difficult to follow: but your graphics are excellent. Thank youSaturday 8-Aug-2009... by: Kim SideyAndrew,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 Andrew Stuart writes: Yes, thats an old snap shot. needs to be redone |
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