Proof of Tough Sudoku of February 3 2007

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 Previous Entries.

Puzzle at start

PUzzle 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

PUzzle at 27 cells solved

Locked 7's

Locked 7's at df4

There are very few standard techniques one can apply to this puzzle at this point. The only one that I find is:

  • Locked 7's at df4
    • forbids ab4=7
Sometimes, puzzles require advanced techniques very quickly.

Possibility Matrix at normal technique stuck point

PUzzle at stuck point

Puzzle Mark-up

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

Forbidding Chain at 27 cells solved


  • 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
    • forbids c8=6

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

Locked 6's at bc7

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:

  • Locked 6's at bc7
    • Forbids efg7=6

Hidden pair 67

Hidden Pair 67 at f49

Continuing with the same investigation, there are a lot of new hidden pairs with 6's. One of them is illustrated here:

  • Hidden pair 67 at f49
    • forbids f4=48, f9=345
Of these eliminations, f9≠3 motivates my next step.

Y wing style

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



The proof presented in my usual style

  1. Start at 22 filled - the given puzzle. Unique Possibilities to 27 filled. (UP 27).
    1. c8=7 == a9=7 -- a2=7 == b2=7 -- b2=5 == b3=5 -- b3=6 == c3=6 forbids c8=6
    2. Locked 6's at bc7 forbids efg7=6
    3. Hidden pair 67 at f49 forbids f4=48, f9=345
    4. 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

Indicate which comments you would like to be able to see

A few words about marking up the puzzle to find likely locations for chains:

Many seem to prefer concentrating on strength in cells - looking only at cells limited to 2, perhaps 3 or more, candidates each.

Here is my thinking on that idea:
No matter what techniques you may use, More...
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These are terrific blogs Steve. You are right about the 75% - here there wasn't anything much to grapple with. I completely missed the fc on 6s- concentrated on fcs with 2s and then 7s and a couple of xwings to finallyundo it all. That extended y wing style was very neat ( Not seeing it,I probably More...
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Thanks for all the information you are providing. Right now I have absolutely no time to try and get into hard puzzles, but I am saving everything you are giving us to have when I get more time.
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dkxhv hrgekxzfb dujwcrlpz ljesq shuibkf tmouvdf vesyqkhul
07/Jun/07 2:27 AM
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mbxdv pgnyuqhlm oxmduq ctbo fpnzkumy akqphzg nzvicf
07/Jun/07 2:28 AM
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