# Sudoku Y Wing Technique with Examples

Coordinates are settled onto the grid,
now what, pray tell, are we to forbid?
'tis not easy quite for sure to say,
but with these wings we might flight them away.

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, along with publish date. The blog has been progressing through techniques relative to approximate difficulty.

Most places with tips, techniques, tricks about how to solve sudoku puzzles include not only what this blog has tackled up to this point, but also Y wings.

Nevertheless, in my opinion, most techniques are presented but narrowly. This page will not only introduce Y wings, also called xy wings, but also generalize the concept. Including Y wing style in your bag of solving tricks will help in tackling truly difficult puzzles. Furthermore, such inclusion will beg further generalization that will help to logically solve just about every sudoku puzzle.

### Y wing as a triple

Note the naked triple illustrated above forbids 1,2,3 from all the blue cells. This is not normally thought of as a Y wing, but rather as a triple.

### Y wing example

In this example we have moved things around a bit. This is a standard Y wing configuration. Here, candidate 1 is forbidden from all the blue cells. Consider:

• The vertex of the Y wing, cell h1, is limited to one of {2,3} Consider both of these possibilities:
• h1=2 forbids e1=2, thus e1=1
• h1=3 forbids i3=3, thus i3=1
• Thus, at least one of the endpoints {e1,i3} of the Y wing must be 1
• Conclude: Whether h1=2 or h1=3: d3,e3,f3,g1,i1 are not 1

In a proof, one might write this step as:

• ywing at e1=12, h1=23, i3=13 forbids def3,gi1=1 OR
• e1=1 == e1=2 -- h1=2 == h1=3 --i3=3 == i3=1 forbids def3,gi1=1
A previous blog page, Coloring, explains some of the forbidding chain lingo above.

The typical Ywing is formed by 3 cells limited to two values each. The endpoints of the Y determine what is excluded. The vertex of the Y must see the two endpoints. In every case, all the cells seen by these endpoints are the target for the exclusions.

### Another Y wing example

The example above is precisely the same as the previous one, except but one cell is the target for potential eliminations. The cells seen by the Y wing endpoints,{e1,h4}, are only {e4,h1}. Since the vertex h1 of the Y wing is already empty of 1's, 1 can be excluded only from the cell e4.

## Extensions of the Y wing idea

The following forbidding matrix exposes the logic of a typical Y wing:

 cell x = r cell x = s cell y = s cell y = t cell z = r cell z = t

To understand this forbidding matrix, consider:

• The empty cells are FALSE
• The entries in the filled cells are boolean variables - thus true or false
• r,s,t are individual candidates - integers
• Each row is a strong set. Meaning: There is at least one true item in each row
• Each column, except the first column, is a weak set. Meaning: There is at most one true item in each of these last two columns
• Therefor, the matrix contains at least three truths
• Also, The last two columns contain no more than two truths
• Finally, the first column must contain at least one truth
Clearly, all cells seen by both cell x and cell z cannot contain r.

A Y wing considers strength in cells to deduce strength in candidate location. This is a tad specific for my taste. Most techniques, tips, tricks about how to solve sudoku puzzles stop before they generalize ideas.

To extend the idea of a Y wing simply remove the restrictions. Instead, consider strength in any container - cell, box, column, row - to deduce strength in any container.

Y wing style is then any consideration of three strong sets that proves an elimination.

If you are paying attention, the restriction here of three is also artificial.

The following list examines all the previous techniques covered in this blog, relative to what they are looking at and what deductions they allow:

Unique Possibilities
Finds singular strengths in any container
Locked Candidates
Looks at strength in location of one type of candidate in one large container (row, column, box) to deduce strength of that candidate in location
Looks at strength of N candidates in N cells to deduce strength of those N candidates in N locations within a large container(s)
Looks at strength of N candidates in N locations within a large container(s) to deduce strength of those N candidates in N cells
X Wing
Swordfish
Looks at strength of one type of candidate in N rows(location) to deduce strength of that candidate in N columns(location) OR vice versa
Unique Rectangles
Looks at potential symmetrical overlapping strength of candidates in location to deduce strength of any type
Coloring
Looks at strength of one type of candidate in location to deduce strength of that candidate in location
Y wings
Looks at strength of 3 cells containing exactly 2 candidates each. Considers 3 types of candidates. Deduces strength of one of those candidates in location
Y wing style
Looks at 3 native strengths of any type to deduce strength(s) of any type
Forbidding Chains
Looks at any number of strengths of any type to deduce strength(s) of any type

Stay tuned, as the next blog will examine the power of Y wing styles. The following tough puzzles can be solved using Y wings plus perhaps techniques covered in previous blog pages. If you get stuck, there should be cryptic proof(s) on the puzzle page.

Find information about me, Steve, at the first page of this blog and My Page at sudoku.com.au.