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
-
Naked Pairs, Triples and Quads
- Looks at strength of N candidates in N cells to deduce strength of those N
candidates in N locations within a large container(s)
-
Hidden Pairs, Triples and Quads,...
- 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.