Ywing Styles

Star Wars Y wing fighter Pic

A sleek, mean, elimination machine,
Pierces possibilities, purely pristine.
If flown with wisdom and studied care,
Y wing styles pare a puzzle with flair.

Welcome! If this is your first visit to my blog, you may wish to read some of the other blog pages first. Links are to the right, under Previous Entries. The blog has been progressing through techniques relative to approximate difficulty.

You may not find another site describing tips on how to solve sudoku puzzles that pilots this specific strategy. The phantom tactic, Y wing styles, illustrates the power of training beyond the universe of typical sudoku solving. Again, the central command point of the Y directs strength to the outposts. Rather than remain terrestrial, Y wing styles launches a more galactic conveyance.

If needed, please refer to the blog reference page, Definitions to interpret this page.

As noted on the page, Y wings, previously examined techniques limit their search for strength to specific containers within the possibility matrix. Y wing styles only limits the the number of native strong sets to three, but does not limit the types of containers considered.

The table below, which I call a forbidding matrix, contains the logic for all Y wing style eliminations.

A B C at least one truth
D E F at least one truth
G H I at least one truth
??? weak weak

  • The Capital letters are:
    • Boolean variables
    • Native puzzle conditions
    • Possibly null: Null variables are false
  • Each row is a native strong set
  • Each of the columns labeled underneath as weak is a native weak set
  • The first column, labeled underneath with ???, is proven strong since:
    • At least one truth in each row, thus a minimum of three in the matrix.
    • Last two columns have at most one truth each, thus maximum of two outside the first column
    • Three minus two = one. First column must have at least one truth
  • Everything seen by all the possibly true items in the first column is impossible
Usually, the last statement concludes the logic, but, in the special case whereas all the items in the first column also form a native weak set, one can conclude that each column has exactly one truth in it. Thus, the proven strong sets in each column can provide additional eliminations. This condition occurs in swordfish, naked triples, and hidden triples, to name a few.

Below, experience the thrust commanded by piloting Y wing style:

Y wing style example 1

First Ywing Style Example

Symbols above: * is a wild card which could be anything, including nothing. ~ means not. Consider:

a1 = 1 i1 = 1  
  i1 = 3 g3 = 3
a3 = 2   g3 = 2

  • Whether (i1=3) => (a1=1)
  • Or (g3=3) => (a3=2)
  • (a3<>1) and (a1<>2)
As a forbidding chain this elimination could be represented as:
  • a1=1 == i1=1 -- i1=3 == g3=3 -- g3=2 == a3=2 forbids a3=1 and forbids a1=2
Hopefully, you can see the symmetry above that allows for both eliminations listed.

Almost Locked Sets as Y wing style example

Almost Locked Sets Y wing style

Except for the extra 1 at cell i1, this configuration would be a standard Y wing. Recall the forbidding matrix from the page Y wings. The following matrix is almost identical:

a1 = 1 a1 = 2  
i1 = 1 i1 = 2 i1 = 3
g3 = 1   g3 = 3
Here, by considering the three possible values for cell i1, 1 is clearly forbidden from
g1 and h1. Written as a forbidding chain:
  • a1=1 == a1=2 -- i1=2 == {pair 13 at i1,g3} forbids gh1=1
When the situation above is viewed as an Almost Locked Sets tactic, the trick is to uncover the strength implied by the almost pair 13 within box h2 in conjunction with the almost singleton at a1. I mention Almost Locked Sets, as they are often the manner in which I locate similar eliminations.

My personal favorite type of Y wing style situation

Wrap around Y wing style

Again, ~ means not, * is a wild card that could be nothing, could be anything(s). The Y is formed by three strong sets:

  • cell g2 ={17}
  • 7's in row 1 limited to {b1,ghi1}
  • 1's in box b2 limited to {b1,ab2}
Because each endpoint of these strong sets also forms a weak link with one of the other strong sets, any of these three strong sets could be the vertex. Therefor, all three are vertices. We actually have, considering only these three strong sets, three Y's simultaneously! This yields a bounty of possible eliminations:
  • All the green cells are not 7.
  • All the blue cells are not 1.
  • All the orange cells are not 1 and not 7.
  • The violet cell is not 2,3,4,5,6,8,9 - thus only 17 is left in that cell
One of (36) possible forbidding matrix representations of this wrap around forbidding chain:

g2 = 1 g2 = 7  
  ghi1 = 7 b1 = 7
ab2 = 1   b1 = 1

Since {ab2=1, g2=1} is proven to be a strong set, but it is also a native weak set, 
each column in the matrix must have exacly one truth in it. Thus, we also have {b1=1, b1=7}
as a strong set and {ghi1=7, g2=7} as a strong set.
This elimination as a forbidding chain could look like:
  • g2=1 == g2=7 -- i1=7 == b1=7 -- b1=1 == a2=1 forbids:
    • defhi2 = 1
    • hi2,ghi3 = 7
    • b1 = 2345689
Again, a more in depth explanation of forbidding chains will be presented in an upcoming blog page.

Very common Y wing style example

Tough puzzle of 10 07 06 at 27 filled Ywing style example

This example is fairly common, and easy to check for. It is similar to the idea of remote pairs, but not precisely the same. The two cells, d3 and f6 are both limited to 67. They do not share a container, but note the possible 6's, highlighted in green. In box e5, 6's are limited to two locations: d5,f5. This means then that no more than one of d3,f6 can be 6. Thus, at least one of d3,f6 must be 7. Therefor, (d3=7) == (f7=7) forbids d9,f1=7 (highlighted in blue). As a forbidding chain, the logic would look like:

  • d3=7 == d3=6 -- d5=6 == f5=6 -- f7=6 == f7=7 forbids d9,f1=7
As a matrix:

d3 = 7 d3 = 6  
  d5 = 6 f5 = 6
f7 = 7   f7 = 6

Proper manipulation of this craft not only conquers more puzzles, but also targets mastery of forbidding chains.

The following links are tough puzzles that can be tackled using Y wing styles. Other techniques previously covered in this blog may be additionally required.

Find information about me, Steve, at
the first page of this blog and 

My Page at sudoku.com.au.

Indicate which comments you would like to be able to see

Hi Steve,
Welcome back!
I request you to convey to Gath if it would be possible to have the possibilities in the same format as your blog pages.This format is certainly better for quickly spotting numbers and is used by many other sudoku sites.Thanks
Btw,will you participate in sudoku world chapionship07?
 |  |
I will ask Gath if the possibilities can be ordered in that manner.
Although I enjoy solving sudoku, I doubt that my speed is great enough to warrant competition.
 |  |
Hi Steve,
I solved the tough dt 08 jan quite easily with x-wing application.

I find it much easier to spot x-wings while plotting possibilities of individual numbers and eliminating what is not possible,rather than plotting all possibilities after obvious/easy eliminations in entire grid.
Please advise if this is ok.Thanks.
 |  |
Of course it is perfectly ok to apply what you know while filling in the possibility matrix. By applying ideas such as locked sets, hidden pairs, coloring, xwings, etc - it is often possible to significantly improve the possibility matrix, sometimes yielding completion of it More...
 |  |

To understsnd your step 2c on puzzle 01 07 2002, I have chains as folow:
1-h9=5==g79=5 -- g2=5==g2=4 -- i2=4==i2=7 -- h123=7==h9=7 forbids h9=349
2-i2=7==i2=4 -- g2=4==g2=5 --g79=5==h9=5 -- h9=7==h123=7 forbids i13=7
correct? thanks
 |  |
Hi ttt!
That is the idea.
Note that you only need the one chain that I posted to get all the eliminations, as once the chains wraps, it matters not where you start:

For example, suppose one had:
A == B -- C == D -- E == F but F -- A

See that the chain forms a loop, so you automatically have:
A == F and
B == C and
D == E and
 |  |
Since, the original endpoints of the chain I posted - g2=4 == i2=4 - are also weak, one knows, without even looking at the grid, that every weak link in the chain is also strong.
 |  |

Thanks! Wow, what a pity for me...Why I did not consider your wrap around forbidding chains before.Yah, It will be my favorite type of chain too.
 |  |
Hi Steve,
I got your point:if a possibility is omitted,chances are it will turn out to be a good guess since most possibilities are incorrect anyway!!
As regards the puzzle of 8 Jan,I eliminated the 4's,leaving two locked sets of 4's and thereafter it was pretty straight forward.Did I get it More...
 |  |
There really is no right and wrong. Without specific details, it is hard for me to make an exact evaluation of what you did on the puzzle of Jan 8.

I did not find a manner of proof/solution that is as simple as you indicate. I did post a proof of that puzzle on that page. It More...
 |  |
To master the puzzles:

I have no clue how many pages it will take. I have not yet truly mastered the puzzles, so....
I may never quite get there in the blog.

To master almost all sudoku's, though - one really only needs the information that will be in my next blog page.
 |  |
Am I right that today's puzzle (1/27) was pretty easily solved y-wing style, with an elimination at d1? I haven't worked on the proof of that.
 |  |
I'm kind of late to the Sudoku party and trying to understand the blog entries. With that understanding, in the "My personal favorite type of Y wing style situation" exemplar, cell c2 was restricted from containing a 1. To me this seems unnecessary?! It would appear that the only More...
17/Jun/11 4:55 PM
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07/Jan/07 Ywing Styles
06/Jan/07 Definitions
31/Dec/06 Y wings
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11/Dec/06 Beginner Tips
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