The following is an illustrated proof for the
Tough Sudoku of March 18, 2007.
The primary key used in this proof is Y wing styles.
You may need to refer to previous blog pages to understand
this proof. Links to these pages are found to the right, under Previous Entries.
At many times during this illustration, there are other steps available. It is not the goal
of this page to show every possible step, but rather to illustrate steps that, taken together,
unlock this puzzle.
The information on the following blog pages is required to understand this page:
The illustrations of forbidding chains used in this proof will share the same key:
- black line = strong inference performed upon a set (strong link)
- red line = weak inference performed upon a set (weak link)
- black containers define a partioning of a strong set
- candidates crossed out in red = candidates proven false
Strong and weak need not be mutually exclusive properties.
Puzzle at start
A few Unique Possibilities are available here:
- d7 = 6% cell (naked single)
- f3 = 9% cell
Hidden Pair 59
As noted in previous blog pages, the symmetrical locations of the solved cells highlit above
allow one to easily locate the Hidden Pair 59 at gi2. This technique is often best applied
before entering the possibilities.
The Hidden Pair 59 at gi2 is shown as a continuous nice loop forbidding chain.
One could write this step as a forbidding chain:
- g2=5 == i2=5 -- i2=9 == g2=9 => ab2≠9, f2≠5, g2≠248, i2≠1248
No matter how one understands it, the eliminations allow one to advance the puzzle a bit further:
- f2 = 3% cell
- f1 = 5% cell
- i8 = 1% column (hidden single, 1 is unique in column i)
- i7 = 4% column
- g3 = 4% column & box
- b8 = 3% cell
Alternate to the last step: Naked Triple 128
If one elects to fill in the possibilities before looking for eliminations, then it is
probably more likely that one will miss Hidden sets and instead find Naked sets, such as the
triple 128 illustrated to the left. This is really little more than a difference in solving
One may wonder: why do I seem to pound Hidden sets?
In a very real sense, a true and deep understanding of Sudoku can only be achieved by
recognizing the interplay between Hidden strengths and Naked strengths. By this I mean:
A naked strength is a cell limited to some number of candidates, such as one sees in the
A hidden strength is a candidate limited to some number of locations in a house, as one
sees in a hidden pair, or for that matter in Locked candidates
End of Digression
Locked Candidates 2 & 8
Illustrated above are two seperate steps. One could view it as one step, if one uses the
aforementioned triple 128. Instead, I have illustrated:
- Locked 2's (in box h2) at h23 => h68≠2
- Locked 8's (in box h2) at h1,23 =>h689≠8
If one had just used the triple 128 previously noted, these steps are not required. However, to
get to the same puzzle state, one would now need to use at least the 5s in box h2, else f12 would
still be unsolved.
Illustrated above is the Y wing style that most sudoku solving sites discuss:
the Y wing. The cell, f8=27, is the vertex of this Y wing.
- f8=2 => f6≠2 => f6=6 => h6≠6
- f8=7 => h8≠7 => h8=6 => h6≠6
As a forbidding chain
or Alternating Inference Chain (AIC)
can also write:
- h8=6 == h8=7 -- f8=7 == f8=2 -- f6=2 == f6=6 => h6≠6
After making this elimination, a few more cells solve:
- h6 = 3% cell
- d4 = 3% column & box
- g9 = 3% row, column & box
- d2 = 1% column
- e3 = 8% cell
- e1 = 4% cell
- d6 = 4% column & box
Some steps not required
After solving the cells noted above, the following steps are neither required in this proof,
nor are they illustrated.
- Locked 9s at d89 => e9≠9
- Locked 7s at h89 => i9≠7
- Locked 6s at h89 => g8≠6
- Locked 2s at g78 => g4≠2
Y wing Style that unlocks the puzzle
Illustrated above is a very easy to find Y wing style. It is easy to locate if
one has followed previous blog pages regarding puzzle mark-ups, as cell b4 would contain circled
1 and circled 8, and h2 would also contain circled 1 and circled 8. This indicates one should
look for a Y wing style on 18. Here the 1s in column b, limited to b14, are the vertex of this
Y wing style:
- b1=1 => h1≠1 => h1=8 => c1≠8
- b4=1 => b4≠8 => c6=8 => c1≠8
Note how this representation is completely analagous to the Y wing
Hence the name that I have coined: Y wing Style
As a forbidding chain, one can write:
- h8=8 == h8=1 -- b1=1 == b4=1 -- b4=8 == c6=8 => c1≠8
Again, notice how similar this representation is to the forbidding chain representation of the
I supppose that the reason that Y wing styles, in all their many forms, are not treated as
part of a greater whole at other sudoku solving sites is: A very unreasonable prejudice against
Hidden Strengths in favor of Naked Strengths. As noted previously, Naked strengths are only 25%
of the information in a Sudoku. Hidden Strengths are the other 75%. Regardless of whether one
finds hidden strengths harder to find (hence the word, hidden), it is senseless to place
less value upon them.
I somewhat favor using the following terms:
- Hidden Strength = Strength in location
- Naked Strength = Strength in cells
as it seems less prejudicial. Frankly, though, nomenclature is of little importance
The point that I am trying to drive home, time and time again, is that it is the interplay
between these two strengths that one must truly understand in order to master Sudoku Solving.
This is even the case in the simple task of locating Unique Possibilities!
Back to the puzzle
After c1≠8, the puzzle is reduced to naked singles (%cells) (singular strength in cells)
to the end.
- Start at 22 filled - the given puzzle. Unique Possibilities to 24 filled. (UP 24).
- Hidden pair 59 at gi2 forbids f2=5, ab2=9, g2=248, i2=1248 UP 30
- Locked 8's at h123 forbids h689=8
- Locked 2's at h23 forbids h68=2
- Y wing: h8=67, f8=72, f6=26 forbids h6=6 UP 37
- Y wing style: c6,b4=8, b41=1, h1=18 forbids c1=8 UP 81
- Sets: 2 + 2(1) + 2(3) = 10
- Max depth 3 at steps 3.3 & 4
- Rating: .03 + 2(.01) + 2(.07) = .19
Post Blog note
One may note that in steps 3.3 and 4 above, I merely listed the strong sets considered. For
every possible Sudoku elimination, such a listing is sufficient to prove the elimination. This
- Strength, regardless of type, is specific to the specific Sudoku puzzle & grid
- Weakness, regardless of type, is universal to the specific Sudoku puzzle & grid
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