The following is an illustrated proof for the
Tough Sudoku of February 1, 2007.
Since this proof uses coloring and a Y wing, 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:
Unique Possibilities get the puzzle to 25 cells solved. (UP 25)
Puzzle at UP 25
Hidden Pair 28
Hidden Pairs are often easy to locate before
entering the possibilities. The highit cells reveal:
- Hidden pair 28 at a23
- Forces a23=28
- Forbids a789=28
Now, a few Unique Possibilities:
- b3 = 4% box
- b6 = 6% cell
- e6 = 5% cell
- e7 = 6% cell
- a8 = 6% rcbox
Possibility Matrix at UP 30
Locked 7's
There are many possible steps to take at this point. I have chosen to only illustrate the
easiest required steps (IMHO) to solve this puzzle
Illustrated here is:
Yielding one more Unique Possibility:
Possibility Matrix at UP 31
Locked 9's
Again, there are many possible steps to take at this point. As is often the case,
the next two steps were available at puzzle start.
Illustrated here is:
Yielding the next step.
Coloring with 9's
Key:
- Black boxes =
- coloring endpoints
- Black lines =
- strong links
- Red line =
- weak link
- Yellow marks =
- elimination targets
Illustrated here is coloring on 9's, as a forbidding chain on 9's:
- fc on 9's: d1 == a1 -- a9 == f9
Yielding a few more Unique possibilites:
- d8 = 4% cell
- d7 = 2% cell
- c9 = 2% row, column & box
- f6 = 2% row, column & box
- d6 = 3% row & box
- d4 = 8% column & box
- g6 = 8% row & box
Possibility Matrix at UP 38
Again, there are many possible steps to take at this point.
Instead of illustrating all of them, which would take some time, next I will show one step
that unlocks the puzzle.
Y wing at UP 38
Key as before. The three strong cells:
- b2 = 39
- f2=36
- d1=69
Form the Y wing, with f2=36 as the vertex. Forbids a1=9.
The puzzle now unlocks with unique values in the cells to the end. (% cell).
Solved Puzzle
Proof
Here is the proof in my usual style:
- Start at 23 filled - the given puzzle. Unique Possibilities to 25 filled. (UP 25).
- Hidden pair 28 at a23 forbids a79=2, a78=8, a2=79, a3=479 UP 30
- Locked 7's at ac1 forbids dgh1=7 UP 31
- Locked 9's at c45 forbids c189=9
- fc on 9's: d1 == a1 -- a9 == f9 forbids d8,f23=9 UP 38
- Y wing b2=93, f2=36, d1=69 forbids a1=9 UP 81
- Sets: 2+1+1+2+3 = 9
- Max depth 3 at step 5
- Rating: 2(.01) + 2(.03) + .07 = .15
Find information about me, Steve, at
the first page of this blog and
My Page at sudoku.com.au.