Steps to Solve Tough Sudoku of January 21 2007 with Proof

The following is a graphical illustration of a proof for the Tough Sudoku of 01/21/07. 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

PUzzle start A few Unique Possibilities:

  • c8 = 8% row
  • h5 = 8% row & box
  • g8 = 5% row
  • i8 = 1% row
  • h1 = 5% box
Thus we have UP 27. This yields:

Puzzle at UP 27

UP 27 locate Hidden Pairs If I am filling in the possibility matrix by hand, I like to locate hidden pairs while doing so. In this puzzle, two such hidden pairs are available:

  • 36 at gh7
  • 29 at g12
Also forbidding:
  • 6 from e7
  • 3 from ghi9
  • 29 from g46
  • 2 from g79

Possibility Matrix after Hidden Pair eliminations

At 27 filled after two hidden pair eliminations Now the following cells solve:

  • i9 = 2% cell
  • e7 = 7% row
  • c3 = 7% row
  • g3 = 8% box
  • b1 = 8% box
  • f2 = 8% box
  • b7 = 2% column
  • a7 = 4% cell
  • c7 = 9% cell
  • d2 = 7% box

The puzzle is thus advanced to 37 cells solved (UP37).

PUzzle at UP 37

Puzzle at 37 cells solved At this point, there are many eliminations available. It would be unwieldy to list them all. Instead, I will only illustrate the eliminations that my proof requires. Amongst the possible eliminations not listed are:

Y wing style eliminations at UP 37

Y wing style at UP 37

The forbidding chain illustrated above can be written as:

  • e3=1 == d1=1 -- d1=6 == c1=6 -- c1=4 == b3=4
    • forbids e3=4
    • and forbids b3=1

This allows us to solve three more cells:

  • e3 = 1% row
  • b3 = 4% row
  • d4 = 1% box & column

Puzzle at 40 cells solved (UP 40)

Puzzle at 40 cells solved Again, there are many possible eliminations. One that my proof requires is illustrated here.

  • Locked 4's at def5
  • forbids e4, ef4=4

Another Y wing style elimination is available also, and is illustrated below.

Y Wing Style elimination at 40 cells solved

Y wing style at 40 cells filled


  • Black circles = endpoints of strong links.
  • Black lines = strong links.
  • Red lines = weak links.
  • Green circle = elimination target
Forbidding chain for this elimination:
  • b5=6 == i5=6 -- i5=9 == i4=9 -- b4=9 == b4=5
    • forbids b5=5
This is a set up for the final forbidding chain.


Depth 4 forbidding chain at 40 cells solved

Depth 4 forbidding chain Key as before.

Strong sets considered:

  1. e28 = 6
  2. e85 = 4
  3. ea5 = 5
  4. ac2 = 5
This type of chain is generally hardest for beginners to find, as it only uses strength in location.

Forbidding Chain Representation:

  • e2=6 == e8=6 -- e8=4 == e5=4 -- e5=5 == a5=5 -- a2=5 == c2=5
    • forbids c2=6
Now the puzzle solves with Unique Possibilities to the end. (UP 81)

Solved Puzzle

Solved Puzzle

Here is a complete proof in my usual style:

  1. Start at 22 filled - the given puzzle. Unique Possibilities to 27 filled. (UP 27).
  2. Hidden pair 36 at gh7 forbids ghi9=3,e7=6, gh7=247 UP 30
  3. Hidden pair 29 at g12 forbids g46=29, g12=68 UP 37
  4. Y wing style: e3=1 == d1=1 -- d1=6 == c1=6 -- c1=4 == b3=4 forbids e3=4 and forbids b3=1 UP 40
    1. Locked 4's at def5 forbids e4,ef6=4
    2. Y wing style:b5=6 == i5=6 -- i5=9 == i4=9 -- b4=9 == b4=5 forbids b5=5
    3. e2=6 == e8=6 -- e8=4 == e5=4 -- e5=5 == a5=5 -- a2=5 == c2=5 forbids c2=6 UP 81
  • sets: 2+2+3+1+3+4 = 15
  • Max depth 4 at step 5.3
  • Rating: .01 + 2(.03) + 2(.07) + .15 = .36

Indicate which comments you would like to be able to see

This puzzle can be solved in many, many ways other than the proof I present on this page.

Feel free to comment with alternative ways that you find to solve this puzzle.

I am always interested in how other people attack such a puzzle, as there is always room to learn. In fact, it is More...

My 'Double Chain Method' previously described will solve most puzzles (but Jan 19 defeated me).

After making the easy eliminations, I look for :--
(a) A triple or quad where all the cells are bivalue.
(b) A chain for a single candidate where some at least of the links More...
Preface: Steve, you're a genius. I've seen your approach to puzzles and my brain just doesn't go that way. I never understand your proofs of puzzles I've completed. I doubt I'll ever get your chain approach or all the methods of logic you apply, regardless of how our solutions might overlap.
P.S. Do you create these puzzles or are you a fellow, albeit professorial, solver?
Hi Morgan!
I do not create these puzzles.
I am a fellow solver, but hardly professional.

Great to hear from you again!
I often think of your double chain method, and I have considered making it part of the blog.
I have no doubt that your double chain method would easily lead to a solution in today's puzzle. By incorporating the 4 strong sets that you describe in column More...
Hi Morgan again!
After reviewing the manner in which you seem to have tackled this puzzle, it is obvious that you almost had to use many of the same strong sets that I used.
One of the advantages of forbidding chains, even if you find your solution by other methods, is that it makes a very More...
It is likely that more people than just I would be interested in alternate means of solving these puzzles.
A slightly detailed accounting of exactly where one finds the information required to find a solution will make it readable for most.
Also, I can almost always take such an accounting More...
I said professorial, since you're teaching. Professional would mean getting paid to do Soduko puzzles all day. God, if only THAT were possible. For me, Doing Sudoku means NOT getting paid for doing something I should be doing. Guess that makes me an anti-professional :)
You invite alternate More...
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