Steps to Solve Tough Sudoku of 26-Jan-2007 with Proof

The following is an illustrated proof for the Tough Sudoku of 01/26/07. Since this proof uses forbidding chains, you may need to refer to previous blog pages to understand this proof. Links to these pages, including Forbidding Chains 101, The Theory, and Forbidding Chains 102, The Practice, are found to the right, under Sudoku Techniques.

Puzzle at start

PUzzle start

A few Unique Possibilities:

  • g3 = 2% box
  • f5 = 2% box & row
  • d5 = 6% box & row
Unique Possibilities get the puzzle to 26 cells solved. (UP 26)

Puzzle at UP 26

Puzzle at UP 26 see the hidden pair 46

Hidden pairs are often easier to find before entering possibilities onto the puzzle grid. Here, the highlit 4's and 6's in columns g and h reveal:

  • Hidden pair 46 at i12
Thus i12 can contain only 46. This gives us a few more Unique Possibilities:
  • h3 = 3% box
  • a2 = 3% box & row
  • i5 = 3% row
  • a1 = 7% box

Unique Possibilities have now advanced the puzzle to 30 cells solved. (UP 30)

Puzzle at UP 30

Puzzle at UP 30 see the hidden pair 17

Once again a hidden pair can be located without entering the possibilities.

  • hidden pair 17 at c56
This does not immediately solve any cells. There are a few locked candidate eliminations, but they also do not solve any cells. It probably is a good time to consider the possibility matrix.

Possibility matrix at 30 cells solved

Puzzle at 30 cells solved after hidden pair 17

Here one can find the following:

  • Locked 1's at f12 forbids f46=1
  • Locked 4's at de4 forbids a4=4
  • Locked 6's at ef7 forbids b7=6
  • Locked 9's at a46 forbids a389=9
  • Triple 579 at d789 forbids:
    • d46,f7 = 5
    • d46,ef7 = 7
    • d3,f7 = 9

Also, AUR 46 at bi12 forces b1=9 == b2=9 forbidding 9 from c3. This solves only one cell, so I am ignoring it, as I really prefer to avoid AUR's when possible.

Now, a program such as Simple Sudoku is stuck. I look for easy coloring eliminations next. Since I fail to locate any, I print out the puzzle and mark it up.

Refer to the page, Forbidding Chains 102 The Practice for an in depth explanation of my puzzle markings at this stage.

Puzzle markings at 30 cells filled

Puzzle markings at 30

I typically add some V's to my markup if I see some candidates that I forsee as being possibly interesting relative to the puzzle grid. These markings have a direction to them.

The cell c3 has 3 endpoints of bivalue sets strong by location within it. This makes that cell the first place that I look. Here I find a short chain

Y wing style chain at 30 cells filled

Y wing style at 30 cells filled


  • Black boxes, circles = strong set endpoints
  • Blue lines = strong links
  • Red lines = weak links
  • Green marks candidate eliminations
This Y wing style forbidding chain uses only strength in location. Also, it uses groupings of candidates within a container. This occurs twice:
  • 5's in box b5
  • 4's in column b

The forbidding chain representation of this step is

  • c3=5 == a3=5 -- a456=5 == b5=5 -- b5=4 == b12=4 forbids c3=4
We know that since we have eliminated a strong 4, that there must be at least one Unique Possibility. In fact, there are many:
  • c8 = 4% row, c7 = 3% box, f7 = 6% cell, e7 = 2% cell, e8 = 3% cell, f6 = 3% column
  • f3 = 9% cell, f1 = 1% cell, f2 = 7% cell, f4 = 5% cell, c3 = 5% cell, c9 = 9% cell
  • g2 = 1% row, b2 = 9% row
We have thus arrived by Unique Possibilities to 44 cells solved (UP 44).

Puzzle at UP 44

UP 44

Here there are a few easy eliminations:

  • Hidden pair 59 at ai6 forbids a6=8, i6=18
  • Locked 8's at de6 forbids de4=8
One could do another puzzle mark up here, but when this many cells have been solved, the puzzle will have many, many marks. Generally, at this point, I look first for things I noted at the original puzzle mark-up. If this fails, I usually try to look for interesting new relationships.

Using new strength at previous points of interest, I find a depth 4 chain.

Depth 4 chain at 44 cells filled

Depth 4 forbidding chain


  • Black boxes = endpoints of strong links
  • Purplish blue lines = strong links
  • Red lines = weak links
  • Green circle = elimination target
This chain uses only strength in location. There are probably many forbidding chains available here. It is likely that some other advanced techniques could also be fruitful. Eventually, I will get around to treating those specifically. Know for now that if you understand Advanced Forbidding Chains, most of those advanced techniques can be derived.

The forbidding chain:

  • d8=9 == d7=9 -- d7=7 == d9=7 -- h9=7 == h5=7 -- h5=1 == h8=1 forbids h8=9
Happily, this completely unlocks the puzzle. Unique Possibilities will take this puzzle from here to the end. (UP 81).

Solved Puzzle



A proof for tough sudoku of 01 26 2007:

  1. Start at 23 filled - the given puzzle. Unique Possibilities to 26 filled. (UP 26).
  2. Hidden pair 46 at i12 forbids i2=139, i1= 189 UP 30
    1. Hidden pair 17 at c56 forbids c5=45, c6=59
    2. Y wing style c3=5 == a3=5 -- a456=5 == b5=5 -- b5=4 == b12=4 forbids c3=4 UP 44
  3. d8=9 == d7=9 -- d7=7 == g7=7 -- h9=7 == h5=7 -- h5=1 == h8=1 forbids h8=9 UP 81
  • Sets: 2 + 2 + 3 + 4 = 11
  • Max depth 4 at step 4
  • Rating: 2(.03) + .07 + .15 = .28

The proof above has been pared down to steps required to prove the puzzle solution.

Indicate which comments you would like to be able to see

I will be out of town for a wrestling tournament (my sons') this weekend. The blog may stall for a few days. Please be patient!
I enjoyed this sudoku a lot more than the hard today.
The window is gorgeous!
Thanx a lot Steve, I have become a lot better at doing these tough ones since you started to help.
I'm not up to speed yet, but I think under 'Ywing style chain' you meant column b; there is no row b.
Thanks Justin!
Indeed I did err. The page will be corrected.
I am very new to creating web pages, so I for most of you, the proof pages have probably looked a bit jumbled and run together.
This is due to my browser settings not matching yours - and my ignorance.
I have tried to correct this page - but not yet the previous blog pages.
Let me know if More...
Steve, you must spend al ot of your time doing these. They are helpful. Thank you.
The thank you's are much appreciated!

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