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 Sudoku Techniques. 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
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 |
Conclude:
- 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
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
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.
Next up, we take a look at W Wing variations.
Find information about me, Steve, at
the first page of this blog and
My Page at sudoku.com.au.