The following is an illustrated proof for the
Tough Sudoku of February 3, 2007.
Since this proof uses a forbidding chain and a Y wing style,
you may need to refer to previous blog pages to understand this proof. Links to these pages
are found to the right, under Sudoku Techniques.
Puzzle at start
A few Unique Possibilities:
- h3 = 7% box & column
- g3 = 3% box
- i1 = 9% box
- e2 = 9% box & row
- d9 = 9% row
Unique Possibilities get the puzzle to 27 cells solved. (UP 27)
Puzzle at UP 27
Locked 7's
There are very few standard techniques one can apply to this puzzle at this point. The only
one that I find is:
Sometimes, puzzles require advanced techniques very quickly.
Possibility Matrix at normal technique stuck point
Puzzle Mark-up
A description of how I start to mark up a puzzle can be found at
Forbidding Chains 102 The Practice
. Eventually, I intend to get around to more complex markings that some puzzles seem to
require. For this puzzle, the standard simple markings suffice. Note, however, that but
for the 3 at f9, there is a standard very common Y wing style using the almost
remote pair configuration a7=38, h8=38. Sometimes noting information like this early during the
mark-up helps me to easily find chains later in the puzzle without needing another mark-up.
Forbidding Chain
Key:
- Black circles = Strong link endpoints
- Black lines = Strong links
- Red lines = Weak links
- Yellow circle = elimination target
The puzzle mark-up targeted a few possible locations for fruitful chains. Box b2 contains two
cells that are likely suspects. In such a configuration, not only is a chain likely to exist,
there is an excellent chance that an Almost Locked Set configuration will also exist in the
area. Since the Almost Locked Set configuration here is more complex, I choose to highlight
the Forbidding Chain:
- c3=6 == b3=6 -- b3=5 == b2=5 -- b2=7 == a2=7 -- a9=7 == c8=7
Unfortunately, the elimination of this 6 does not immediately unlock the puzzle into Unique
Possibilities. It does however reduce the puzzle to one in which normal techniques will solve it.
The path that I will illustrate from this point forward is by no means the only path. There are
very many easy paths to take from this point. Furthermore, from this point forward I shall only
show the steps that I needed to solve the puzzle. To show all the available steps from this
point forward would consume volumes of space and time.
Locked 6's
Since we just eliminated a 6, it makes sense to look at the 6's. Amongst the many things
an examination of the 6's reveals at this point is:
Hidden pair 67
Continuing with the same investigation, there are a lot of new hidden pairs with 6's. One of
them is illustrated here:
Of these eliminations, f9≠3 motivates my next step.
Y wing style
The almost Y wing style that I noticed earlier is now possible. It can be represented with
the following forbidding chain:
- a7=8 == a7=3 -- a9=3 == h9=3 -- h8=3 == h8=8 forbids c8,gh7=8
I call this type of Y wing style
very common. I do this because, well, it is
very common. Furthermore, it is very easy to find. For me, it is easier to spy than the
Y wing that also exists above.
The Y wing style unlocks the puzzle into Unique Possibilities to the end. (UP 81).
Solved Puzzle
Proof
The proof presented in my usual style
- Start at 22 filled - the given puzzle. Unique Possibilities to 27 filled. (UP 27).
- c8=7 == a9=7 -- a2=7 == b2=7 -- b2=5 == b3=5 -- b3=6 == c3=6 forbids c8=6
- Locked 6's at bc7 forbids efg7=6
- Hidden pair 67 at f49 forbids f4=48, f9=345
- Y wing style: a7=8 == a7=3 -- a9=3 == h9=3 -- h8=3 == h8=8 forbids c8,gh7=8.
UP 81
- Sets: 4+1+2+3=10
- Max depth 4 at step 2.1
- Rating: .15 + .01 + .03 + .07 = .26
Find information about me, Steve, at
the first page of this blog and
My Page at sudoku.com.au.
