Strategies for Popular Number Puzzles

X-Wing Strategy

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.

X-Wing example 1
X-Wing example 1 : Load Example or : From the Start
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 cells marked A, B, C and D. What's special about them?

Well, A and B are a locked pair of 7s. So are 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 7s 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 7s along the edge of our rectangle are redundant. We can remove the 7s 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.

X-Wing example 2
X-Wing example 2 : Load Example or : From the Start

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 orange 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 orange cells. Which ever way the 2s could be placed (E5/J8 or E8/J5) six other 2s in the rows can be removed - the yellow highlighted numbers.
X-Wing example 3
X-Wing example 3 : Load Example

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 orange 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:

  1. Starting from 2 rows and eliminating in 2 columns
  2. Starting from 2 columns and eliminating in 2 rows
  3. Starting from 2 boxes and eliminating in 2 rows
  4. Starting from 2 boxes and eliminating in 2 columns
  5. Starting from 2 rows and eliminating in 2 boxes
  6. 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 in the red circled cells.

X-Wing Example

But HOLD UP one moment. There is a simpler strategy that does the same job!

X-Wing Example

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

X-Wings containing boxes are the inverse of the Intersection Removal strategies

X-Wing Exemplars

These puzzles require the X-Wing strategy at some point but are otherwise trivial.
They make good practice puzzles. The next strategy in this 'family' is Sword-Fish

Go back to Intersection RemovalContinue to Singles Chains



... by: fred

Monday 19-Feb-2024
in the first diagram can the x wing also go in j 4 and j 8
Chio replies:Sunday 19-May-2024
the number has to be present in the row only in those 2 cells
in this case 7 is also in j5 j6 j7.
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... by: Rodney Lindenmayer

Monday 10-Jul-2023
Great explanation❗️ I visited several other sites where they just said "Do this" without the underlying logic, and the X-Wing remains impossible to properly use. Bravo for this: cogent, succinct.

... by: Ymiros

Tuesday 27-Jun-2023
Not directly concerning the text, but shouldn't the next page after this be Swordfish rather than Single's Chains? At least Swordfish says "Go back to X-Wing" and Jellyfish which comes after Swordfish says "Continue to Singles Chains".

... by: kauaikarl

Monday 1-May-2023
Ran a Sudoku I was stuck on through a solver that indicated the next step is a "simple contradiction" but it didn't explain how it worked. Never heard of this strategy and couldn't find any info about it on the web. Appreciate any help
Andrew Stuart writes:
Poor ;) Hoping better on the solver here - [Del]
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... by: hawkeye

Saturday 15-Apr-2023
I have stupidly underestimated the logic of sudoku. Your explanation of X WING is excellent ,
since a multiplicity of examples helped me to understand !
I am embarrassed that I didn't catch on before.

... by: Hexameron

Friday 13-May-2022
Only just found this site and excited to explore it and learn new techniques. I'm hoping it's going to give me strategies that mean I no longer need to use trial-and-error for diabolical/extreme sudokus. And I disagree with those comments saying the explanations aren't clear - so far I've found the reverse to be true. One point to amend: vice versa is not spelt vis versa - you're thinking of vis-a-vis, which is something different.
Andrew Stuart writes:
Thank you and thanks for spell check. Appreciated - [Del]
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... by: Marja Glas

Monday 6-Dec-2021
X-Wing example 1: why is the x-wing not between B4-8 and J4-8

... by: Jim Birch

Sunday 6-Sep-2020
Great piece of work, thanks

... by: Thurstan

Sunday 6-Sep-2020
There is another (mixed) combination: if we look at the digit 1 in the puzzle we see that the four examples at A5, B5 (common column) and B8,A9 (common block) lie in two rows, and remove the rest of the ones in the first two rows,

... by: Tom

Monday 10-Aug-2020
These pages are superb - they clarified the various techniques and have lifted my own solving powers.

In response to David Spector: because the trial-and-error method of solving sudokus isn't so interesting. Somewhere on Andrew's site he says this: his interest is in the logic, not the brute force methodologies. I've done diabolical and extreme sudokus using the "try-one-number-in-a-cell-with-two-remaining-possibilities-and-if-it-doesn't-work-try-the-other-one" technique (targeted trial-and-error), and whilst it gets the solution I'm left dissatisfied, thinking how did that work?

... by: Jayaprakash

Thursday 16-Aug-2018
The X Wing tells AD or BC to have 7 in your first example. But once you spotted an X Wing, we can eliminate other cells. But how do you solve it? Put 7 at A and proceed to find out or any other way?

... by: Riddlemethisbatman

Sunday 13-May-2018
I'll be back to this page. Nice graphics and good explanation. You gotta do these a few times to get an understanding. To David who asks about the necessity for these techniques - more challenging puzzles can only be solved using techniques like x-wing; though I suppose you could use the guessing technique, sometimes referred to as 'forcing chain' but it is surprisingly inefficient as you will usually have to go ahead and almost complete the puzzle to find out if the number you tried was correct. Good fluke if it is, but the problem is that finding that one correct number will still not be enough to solve the difficult sudoku. So then you gotta guess again. I feel like using logic techniques is much more satisfying, efficient and far more likely to yield correct solutions.

... by: Vinod K Chandran

Tuesday 7-Nov-2017
Thanks for this page. Its really very helpful.

... by: David Spector

Tuesday 12-Jul-2016
It seems to me that once applying the basic strategies yields no further change, usually there are only 2 or 3 candidates in each unresolved cell. At this point, instead of doing difficult searches for patterns such as wings and fish, why not just save the puzzle state as a backtracking point and select any cell that contains 2 candidates (they will all switch together). Just pick one of the candidates to delete. Then apply the same basic strategies (naked singles, etc.) until either the puzzle is solved or an inconsistency is found. In the latter case, restore the puzzle from the backtracking saved state and choose the other candidate. The puzzle will now be easy to solve.

This is so much simpler than all those advanced strategies, so complex, which may or may not apply.

... by: Bill

Thursday 10-Dec-2015
I think, perhaps, I answered my own question: There is more than one solution to the puzzle; there are two. Does this sound correct? I was told long ago that there was only one solution to each Sudoku puzzle. Maybe not.
Andrew Stuart writes:
Broken multi-solution puzzles are easy to make. They mess up the solver and the logic - [Del]
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... by: Bill

Thursday 10-Dec-2015
I have a puzzle in which all squares are filled except in the four boxes forming the x-wing. In each of these four boxes, the same pair of numbers appear. Is it possible to solve this puzzle situation?

... by: JuanBeegAhs

Sunday 26-Apr-2015
Still confuses me. Take the first example, why go for ABCD? Why not take, say... CD and any of the four 7s that appear below, or any other combination of X Wing there?

Looking at the other examples just seems to reinforce this haze for me. It doesn't seem like there has to be a fixed number of squares between or anything, as long as four similar numbers can be in a square or even trapezoid apparently.

... by: gcrupa

Monday 6-Apr-2015
I used your solver (the Best !) to crack a puzzle beyond my pwers yet and, at some point I got in Step 14 - X-Wing docs

(Row->Col) 7 taken off D1, based on DH29
(Row->Col) 7 taken off H1, based on DH29
(Row->Col) 7 taken off H8, based on DH29

I do not know what DH29 stands for and also did not know where to look for it.
Can you tell me, please ?

Have a nice day

Andrew Stuart writes:
DH29 is compact notation for the four cells D2, D9, H2 and H9 - [Del]
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... by: June K

Sunday 9-Nov-2014
Thank-you! Finally an explanation that outlines the logic used and also utilizes plain language to do it. Best I've read so far.

... by: Simon Davitt

Thursday 16-Oct-2014
Great pages and great explanation.

Please can you give some un-worked samples for each page to allow people to test their understanding.

... by: Richard

Thursday 16-Oct-2014
I'm a little confused when x wings apply. Is it a requirement that all four candidates be in four spearate boxes? In examples one through three, all of the candidates are in different boxes. However, in your fourth example, the candidates are in two boxes.
Andrew Stuart writes:
Try this partially completed puzzle. The first strategy used is Pointing Pairs using cells C7 and C8 - removing 8 in C2. But if you look at cells G7 and G8 you can make an X-Wing with C7 and C8 on 8. The pattern is perfectly correct for an X-Wing in the columns eliminating in the rows. (Pity there is no eliminations in row G). But why use a more complex explanation than the original Pointing Pair? So in one sense "all four candidates need to be in four separate boxes" is true - but only if you have exhausted all Pointing Pairs, which is the reduced explanation if the X-Wing isn't in four boxes. - [Del]
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... by: fdan

Wednesday 28-Aug-2013
Thanks, trying to work at creating suduko puzzles

... by: Lydia

Sunday 28-Jul-2013
Best explanation ever of x-wing!

... by: sajan

Wednesday 24-Jul-2013
In the second example above, showing the 2-Wing, why has column 8 been selected , and not column 9?. Why is X -wing not between E5/J9 and E9/J5 ? Similarly, in the next example, that of the 3-wing, whu again column 8 in preference to column 9?
Andrew Stuart writes:
E9 and J9 on 2 can't be part of an X-Wing since 2s occur four times in the column. We're looking for pairs in one direction (in this case columns) so we can eliminate in the other direction (rows) - [Del]
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... by: Trophy

Saturday 13-Apr-2013
I've been doing sudoku for years and still having trouble with the most difficult ones. After using many of the "hint" I still get to a point where I'm stymied. I end up having to guess at the right # for a box in some cases (usually where there are two #s the same in 2 boxes in a row or a colume ) If I guess wrong I have to go back and use the other # and it then usually works to solve the puzzle. Is this a common thing, or should there always be a way to solve the pussle without guessing?
Andrew Stuart writes:
The reason I got into Sudoku strategies was because I was unsatisfied with having to guess and I thought there must be some underlying rule that I was missing. However, Sudoku has proved to be extremely deep and there are still examples which even the most advanced strategies cannot crack with logic alone. I publish these here.

Now, people do come up with logical methods and it's usually by combining several ideas in one go. This is the coal face I'm picking at now, trying to generalise these ideas and get them into code.

However, it will be a rare puzzle in the newspapers that can't be solved logically. We really have to search for a long time to find unsolvables (currently they are 1/10,000) from my generator.

Computers can't mimic inspiration and there's still a place for that in Sudoku, if you get stuck
- [Del]
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... by: Daran

Wednesday 13-Mar-2013
sir can i apply the same strategies for 16X16 and 25X25 grid sudokus...or i need to change any conditions to select the possibilites for the strategies
Andrew Stuart writes:
Yes, exactly the same strategies, although some need to scale differently. I’m pretty sure there are no specific strategies for higher order Sudoku the don’t occur for 9x9. - [Del]
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... by: Complex sudoku newbie

Saturday 16-Feb-2013
Loved your explanations! The illustrations make it especially easy and simple to follow for a non-mathematician but a sudoko fan. It is nice to find methods to improve my game that I can follow and use. Thank you!

... by: M Bakhtian

Friday 4-Jan-2013
Great site for technique. But choice of black as a background color and the size of print for this venue is compromising the quality of reading about these lovely intricacies!

... by: RLets

Tuesday 28-Aug-2012
Should "...the only 7's in the first and last rows" in paragraph 3 be "...the only 7's in the 2nd and 6th rows"? Great web site. Love the solver. Now, I need to read your book!
Andrew Stuart writes:
Typo fixed, thanks! Textual hangover from old diagram - [Del]
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... by: Dino Hsu

Friday 20-Apr-2012
To further Konrad's study of single column (or a single row) scenario, the problem is that the two X-wing locations in the 'single column' can be both non-X, without the 'other column' also aligned, resulting in failure to lock in the two locaitons for X in the 'single column'.

... by: Konrad M Kritzinger

Tuesday 7-Feb-2012
It seems that I was over-hasty with my comment on 2-Feb-2012. The proposition works in some cases but not others.

... by: Konrad M Kritzinger

Thursday 2-Feb-2012
X-Wing doesn't seem to require an X. The same principle seems applicable to a single column (or a single row). If there is a locked pair in one row, and the same locked pair occurs in another row, then, if at least one cell from each locked pair is in the same column, all other occurrences of that number in that column can be eliminated. The same principle applies if rows and columns are exchanged.

... by: Yves Sioui

Saturday 26-Mar-2011
In 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 - [Del]
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... by: Ted L

Thursday 30-Dec-2010
In 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. - [Del]
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... by: William Mann

Sunday 19-Dec-2010
Your 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.

... by: Harold Binley

Saturday 16-Oct-2010
Marv Rowe, 9th April, has made the same mistake as I used to. The X-wing only works as a trapezoid if each pair of the four cells share the same units; they share columns 3 and 4, but although r8c3 & r8c4 share the same row r5c4 & r6c3 have nothing in common.

... by: Rohan

Friday 6-Aug-2010
As 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. - [Del]
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... by: Elmer Schartow

Tuesday 3-Aug-2010
Regarding 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) - [Del]
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... by: colin pearce

Wednesday 30-Jun-2010
Hi 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,
Andrew Stuart writes:
because the blue boxes contain other 6s in the row - [Del]
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... by: Marv Rowe

Friday 9-Apr-2010
Stopped using X-Wing on anything but squares and rectangles after I did 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 case

... by: CS Vidyasagar

Thursday 25-Feb-2010
X 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.

... by: Kantilal M Mane

Wednesday 3-Feb-2010
Excellent technique !!!

... by: John Mathews

Friday 18-Dec-2009
If 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. - [Del]
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... by: Faris

Wednesday 2-Dec-2009
I love your graphics though I do agree with Pannel that the explnations are not easy to follow at times. Keep the nice work!

... by: Nick Pannell

Sunday 4-Oct-2009
I'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 you
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