Easy Proof for the Tough Sudoku of February 11, 2007

The following is an illustrated proof for the Tough Sudoku of February 11, 2007. Since this is an easy tough puzzle, it is perfectly well suited for study of the limited techniques required. If quick and easy solving of these types of tough puzzles is your goal, then this page may be of interest to you.

Since this proof uses a Locked candidate, some Hidden Pairs, a Hidden Triple, and a Naked Pair, you may wish to refer to previous blog pages, although it probably is not required. Links to these pages are found to the right, under Sudoku Techniques.

At many times during this illustration, there are other steps available. It is not the goal of this page to show every possible step, but rather to illustrate steps that, taken together, unlock this puzzle

Puzzle at start

PUzzle start

A few Unique Possibilities:

  • b6 = 3% box & row
  • f1 = 4% box
b6 = 3% box & row means b6=3 because it is the only possible place for 3 in both box b5 and row 6.

f1 = 4% box means that f1=4 because it is the only possible place for 4 in the box. Singletons in a container (house) are also called Unique Possibilities.

Puzzle at 25 cells solved (UP 25)

PUzzle at UP 25

Here, there are no remaining Unique Possibilities. It is at this point that I recommend, highly, that one look for hidden sets before entering all the possibilities. The reason for this recommendation: Hidden sets are most easily uncovered without all the possibilities hiding them. Perhaps the reason they are called hidden is merely because they are obscured by the possibility matrix.

The possibility matrix is a great tool, but one should be aware that it obscures deductions one can make based upon location of candidates.

Hidden Pair 13

Hidden Pair 13

Illustrated to the left is the existence of a Hidden Pair with candidates 13 at cells d1 and d3. The only two locations left for both 1 and 3 in box e2 are those cells, d13. Therefor, if something other that 13 were to exist in one of those two cells, there would not be enough places left for either 1 or 3 in box e2.

Perhaps fill in cells like this first in the possibility matrix. Remember with a mark, or a mental note, to not fill any other candidates into those cells. Also, one can use the Locked candidates nature of 13 at d13 to not fill in any 1's or any 3's at the rest of column d, (cells d4, d5, d8, d9 - called d4589 for short).

Hidden Triple 349

Hidden Triple

Illustrated to the left is the justification for placing a Hidden Triple using candidates 349 at cells g8, g9, h9. Hidden triples are rare, but in this case, the alignment of 349 in column i and the alignment of 349 in row 7 easily reveal this hidden triple. This hidden triple is the likely reason for this puzzle being called Tough. The logic for a Hidden Triple is analagous to the logic for a Hidden Pair.

You may note that above I have not filled in each of 349 at each of the hidden triple cells. The reasoning for this is the 4 already at a8 and the 3 already at h2.

Hidden Pair 49

Hidden Pair 49

Illustrated to the left is the justification for a Hidden Pair in row 5. This one is not easy to find at all. If you were to fail to see it, there will be a naked quad in row 5 available after entering the possibilities that will accomplish the same eliminations.

The Possibility Matrix filled in with Locked 2's shown

Locked 2s

Illustrated to the left is the Possibility matrix after using the information from the previous steps. Also illustrated are Locked 2's in row i at i89. Because 2's must exist in box h8 either at i8 or i9, they cannot exist at i5 or i1.

This isolates the 2 at a5 as the only possible 2 in row 5. Therefor,

  • a5 = 2% row

Puzzle at UP 26

PUzzle at UP 26

After cell a5=2, both cells b3 and b4 are limited to only 47. This is a naked pair. Since these two cells are limited to 47, one can safely eliminate 47 from the rest of column b. Thus:

  • Pair 47 at b34 forbids
    • b5=4
    • b17=7
  • forcing
    • c8 = 7% box
    • f6 = 7% row
    • % cell to the end
  • UP 81

Solved Puzzle

UP 81


Below is a proof written in my usual style. Only required steps are listed.

  1. Start at 23 filled - the given puzzle. Unique Possibilities to 25 filled. (UP 25).
    1. Hidden pair 49 at bh5 forbids b5=2, h5=126
    2. Hidden triple 349 at g89h9 forbids g8=12, g9=126, h8=1268
    3. Locked 2's at i89 forbids i15=2 UP 26
  2. Pair 47 at b34 forbids b5=4,b17=7 UP 81
  • Sets: 2+3+1+2 = 8
  • Max depth 3 at step 2.2
  • rating: .14 - not too tough


Hidden Sets are a valuable search to perform before entering the possibilities.

Indicate which comments you would like to be able to see

The plan of attack presented on this new page should be very helpful on many tough puzzles that
do not require chaining.

I have found that many of the puzzles available on paper - (magazines, newspapers, etc.) and that are somehow presented as very difficult (labels such as demonic, More...

the 'easier' description is far better for me -- i never understood the shorthand 'equations' and the witten description you gave today was very helpful. thanks!
Thank you, thank you, thank you!
Dear Steve, thank you for your dedication.

Early id of hidden sets is indeed a very useful tip.

Your blog is well written and much appreciated. Most people don't have the mathematical background (or patience) to follow mathematical proofs, and so your non-mathematical approach is More...
I definitely find the new style easier to understand. It will mean a lot more writing for you at first. That I didn't understand your proofs language before didn't matter since I solve the puzzle before reading or adding a comment. I assume people look for your proof comments because they get More...

As others said, this type of solution is far easier to understand. I now understand hidden triples!! Thanks.
Steve, many thanks for the plain English. I find it makes much more sense than your previous 'Mathematical' language. You're doing great-keep up the good work!
Thanks, Steve. I especially appreciated the advice to look for hidden sets BEFORE filling in possibilities, which I have been trying to do, semi-successfully, AS I fill them in. Hopefully, you will do this type of explanation again on a puzzles with fc's.
I can't thank you enough! You have finally made it possible fo rme to understand. Now whether I can do it on my own is another question ....
This is fantastic. I could follow your other proofs, but found myself having to 'translate' to myself as I went along.

Your help is immeasureable and I am finally able to see how some of the steps are 'found'. I could see them when you pointed them out, but couldn't find them myself. Now I am starting to work it out for myself. One day I will be able to work out my own proofs.
Hey Steve! This is much easier for me to try and work through, thanks so much for your trouble. I didn't really understand before about hidden pairs and such like, I have a very long winded way of trying to work the tough puzzles, and I have found myself unable to complete them on occasion.

I will try and put this into practice :D
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