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

Key:

  • 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

9 Comments
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Steve  From Ohio    Supporting Member
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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 primarily by carefully examining the solutions of others that I came to my current understanding of how to solve sudoku puzzles.
JRD  From UK
Steve

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 are strong.
(c) Search for an area of the puzzle where there are several strong links, as you have described.

In today's puzzle (Jan 21), starting from your position at UP37, there is a suitable quad (12,25,57,17) in col(a). Using this as my 'anchor chain', the whole puzzle falls out. Trust me.
Morgan  From NY
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.

SO, I'm starting the puzzle, naked trip in box 9. Then I notice naked trips @ 7ac and 8df which gives the 7 at 7e and opens 3 naked pairs and a hidden single. Those pairs open more pairs and singles and I'm thinking 'whoa, I'm smelling a trap. This sneaky bastage is leading me to huge bivalue chain to color or a BUG+1' Then I spot the two hidden 8s that open the final two 8s. At 37 our boards are identical. I didn't notice your y-wing. We both had the naked 36 at 3hi (you didn't eliminate the 6s at 3be but I see that the presence of the 6s was irrelavant to your technique). I used an APE for the 6 at 1c and the puzzle fell apart. Looks like we took virtually the same route. With .au being ~14 hours ahead, we'll have the new puzzle (Monday's) soon.
Always a pleasure.
Morgan  From NY
P.S. Do you create these puzzles or are you a fellow, albeit professorial, solver?
Steve  From Ohio    Supporting Member
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Hi Morgan!
I do not create these puzzles.
I am a fellow solver, but hardly professional.
Steve  From Ohio    Supporting Member
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Hi JRD!
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 a simultaneously, you will automatically engage all the strong sets that I use in my proof. Therefor, it follows that the puzzle must fall apart after using them.

The main reason that I do not incorporate your technique is: By the definition of depth that gb and Andrei taught me, your method is inherently very deep. On the other hand, it is very time efficient, since it involves so much of the puzzle at one time.

Feel free to comment any time. I especially like to hear from other solvers.
Steve  From Ohio    Supporting Member
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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 efficient language tool. If you take the time to understand the language, expressing precisely how you solved a puzzle using forbidding chain language becomes a very clear way to communicate.
Steve  From Ohio    Supporting Member
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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 and present it as a forbidding chain type step. In this way, even when forbidding chains are not ones choice of tool, they can be a source of unifying language.
For example, APE's, as Morgan mentioned, always can be presented as forbidding chains.
Morgan  From NY
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 solving methods. If my bull-in-a-china-shop method of blundering through these puzzles helps provide insight to anyone, I'm happy to share.
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