Proof of Tough Sudoku of February  1 2007

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 Previous Entries.

Puzzle at start

PUzzle start

A few Unique Possibilities:

  • a5 = 1% box
  • i5 = 2% box
Unique Possibilities get the puzzle to 25 cells solved. (UP 25)

Puzzle at UP 25

PUzzle at 25 cells solved

Hidden Pair 28

Hidden Pair 28 at a23

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

PUzzle at 30 cells solved

Locked 7's

Locked 7's at ac1

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:

  • Locked 7's at ac1
    • Forbids dgh1=7
Yielding one more Unique Possibility:
  • d2=7% box & column

Possibility Matrix at UP 31

PUzzle at 31 cells solved

Locked 9's

Locked 9's at c45

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:

  • Locked 9's at c45
    • Forbids c189=9
Yielding the next step.

Coloring with 9's

Coloring with 9's


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
    • Forbids d8,f23=9

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

Puzzle at 38 cells solved

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

Y wing

Key as before. The three strong cells:

  1. b2 = 39
  2. f2=36
  3. 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



Here is the proof in my usual style:

  1. Start at 23 filled - the given puzzle. Unique Possibilities to 25 filled. (UP 25).
  2. Hidden pair 28 at a23 forbids a79=2, a78=8, a2=79, a3=479 UP 30
  3. Locked 7's at ac1 forbids dgh1=7 UP 31
    1. Locked 9's at c45 forbids c189=9
    2. fc on 9's: d1 == a1 -- a9 == f9 forbids d8,f23=9 UP 38
  4. 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

Indicate which comments you would like to be able to see

This puzzle is not as tough as the usual for a tough puzzle at this site. Hopefully, it provides some help in understanding the language.

Although the usual manner that is taught elsewhere prescribes that one should look for coloring eliminations after many other techniques, I have found More...
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This is the fourth tough sudoku I solve without any guess. I am trying to go through all the old explanations you have provided in order to be able to follow your proofs. I am only on the Xwings....(Dec 20) however, as I am too tired by the time I come home from work. . Thanks for making the More...
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I'm kind new to all these terminologies. What's UP 30? I know UP stands for unique possibility. But can't figure out what 30 stands for? Also, can someone (Steve?) explain to me why b6=6, e6=5, e7=6, and a8=6. I just don't see it. Thanks.
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Hi SixStringer!

We have all been new, once. I also struggled with terminolgy that people used to describe what they are doing. I still do at times, since terminology is fairly non-standard across the web. I try to stick to one manner of expression, so that my blog becomes More...
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Thanks for the explanation, Steve! I've been reading your blogs. They're very helpful.
 |  |
A question on coloring of 9s, Steve...why does fc on 9 only forbids d8, f23 = 9 and not also d45 and f458?
 |  |
Oops...I mean why fc on 9 doesn't forbid f8=9 also?
 |  |
Hi SixStringer!

fc on 9's: fc on 9's: d1 == a1 -- a9 == f9
The endpoints of the chain:
d1, f9.
The conclusion of the chain:
d1 == f9.

d1 does not 'see' f8. Thus, it is not forbidden.

However, one could expand the chain:
fc on 9's: e4 == e3 -- d1 == a1 More...
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Thanks again for the explanation, Steve. I think I got this one.
 |  |
This puzzle has given me fits for quite awhile till I gave in and read your proof. I had totally missed the Y wing and from where I was, the thing then fell apart. I was about to question your weak link a1--a9 in the FC on 9s because there were no other 9s in the 'a' column when I remembered your More...
11/Apr/07 6:42 AM
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