Final Page for January 30, 2008

The following is the last page, page 5, of an illustrated proof for the tough puzzle of 01/30/08. Page four made this into a relatively easy tough puzzle from this point forward. There are myriad manners to proceed. Many of these paths are not difficult. Below, I shall craft an end game that is minimal according to the way I rate proof or solution difficulty. All of the sudoku strategies presented below have been treated with some significant detailing on previous blog pages.

Some of these previous blog pages may be helpful. Links to these pages are found to the right, under Sudoku Techniques. I may later list specific pages.

The illustrations of steps shown on this page will share this key:

  • black line = strong inference performed upon a set (strong link)
  • red line = weak inference performed upon a set (weak link)
  • black containers define a partioning of a strong set(s)
  • candidates crossed out in red = candidates proven false
Please be aware that, for me, strong and weak need not be mutually exclusive properties.

Steps 7abc

Below, three steps are illustrated at once. Two of these are freshly available after cell i1 is solved. The locked 7's are available immediately after cell h2 is solved.

3 easy steps

Above, the three relatively easy steps are:

  • 7a Locked 7s h456 => g4,i56≠7
    • Note that (1)h2 eliminated (7)h2, and the line under that 7 indicated this elimination would exist
  • 7b (4): a4=g4-g1=c1 => a3,c5≠4
    • Updating the puzzle mark-up helps to reveal this step, as (9)i1 => circling 4 at both cg1.
  • 7c Hidden Pair 47 cg1 => c1≠58
    • Again, (9)i1 => circling 4,7 at both cg1 => cannot miss this hidden pair
I am ignoring lots of easy eliminations here, such as Locked 2s i23 => i568≠2. In my proofs, this is usually done in retrospect, as after I have solved the puzzle I can see that I do not need this step. Occasionally, I might see the way to the end of the puzzle and do not need the retrospective.

Step 7d - Y wing, a type of Y Wing Style

Y Wing

The elimination of (4) from a3 helps make this elimination easy to find. It is usually a good idea to look at cells which have been updated by eliminations to see if anything new is available. Here I found a Y Wing, which is the reason for naming the group of all depth 3 elimination patterns Y Wing Styles.

  • (3=1)a3-(1=9)a8-(9=3)f8
  • => f3≠3

Step 7e - Another Y Wing Style

Y Wing Style

Step 7c eliminated both 5 & 8 from c1. This created, amongst other things, these two new bilocations:

  1. (8)c2=(8)c9
  2. (5)c2=(5)c8
Circling both 58 at c2 made this an easy find:
  • (5)c8=(5-8)c2=(8)c9-(8=9)a9
  • => c8≠9
At this point, puzzle familiarity should help to make deductions like this one apparent. Moreover, one may start to get a feel for the importance of such a deduction towards unlocking the puzzle.

Step 7f - A wrap around Y Wing Style

Below, a rather nice pattern is revealed by two recent results:

  1. Step 7d eliminated (3) from f3, giving the bivalue (1=9)f3.
    • This bivalue, along with the already existent (1=9)a8 is a marker for this search
  2. Step 7e eliminated (9) from c8
    • The new bilocation: (9)a8=(9)f8 links quickly to the (1=9)f3 marker
  3. Step 5b eliminated (1)a2 => (1)a3=(1)a8

Wrap Around Y Wing Style

Above, find my favorite type of Y Wing Style, the wrap-around chain or continuous AIC:

  • (1=9)f3-(9)f8=(9-1)a8=(1)a3 -(1)f3 =>
  • All the weak links in the chain are proven strong inference sets =>
    1. (9)f3=(9)f8 => f9≠9
    2. (9)a8=(1)a8 => nothing, as we already had that
    3. (1)a3=(1)f3 => be3≠1
Of these eliminations, b3≠1 seems the most significant. It cascades the next two steps.

Step 7g - Hidden Pair 15

Hidden Pair 15

b3 ≠1 creates the new bilocal: (1)b1=(1)b8. This overlaps with the symmetric existing bilocal with candidate 5:

  • (1)b1=(1-5)b8=(5)b1 =>
  • Again, as with all hidden pairs, the weak links are proven sis
    1. (1=5)b8 => b8≠2
    2. (1=5)b1 => b1≠8
  • Now we can solve some cells:
    • (8)f1 % row
    • (5)f5 % cell
    • (5)h4 % row, column & box
  • UP 36

Step 8 - a bilocal Hidden Triple

Below, the elimination of (2) from b8 Locks candidate 2 into c89 within box 1. This combined with the previously noted restrictions and overlap of candidates 58 in this column make the following Hidden Triple easier for me to spot than the eventual naked triple 467 at c136 that performs much the same work.

Hidden Triple limited to only bilocals

The Hidden Triple shown above is limited to only bilocals. Thus, it can be written as a wrap-around chain:

  • (2)c9=(2-5)c8=(5-8)c2=(8)c9 =>
  • All the weak links are sis
    1. (2=5)c8 => nothing new, as we already had that
    2. (5=8)c2 => c2≠7
    3. (2=8)c9 => c9≠9
  • (2)c9=(2)c8 is part of the chain. =>
    • c56 ≠2
  • c9≠9 => (9)c5 % column
  • UP 36

Step 9 - a final Y Wing Style

Below, the elimination of (7) from c2 has created:

  1. (7)c1=(7)c6
  2. (7)c1=(7)a2
  3. (7)a2=(7)i2
When an elimination, or group of eliminations, create(s) such a confluence of new bilocal sis with a single candidate, it is usually worthwhile to investigate. Note that we have already found the sideways skysraper with candidate 4:
  • (4): a4=g4-g1=c1
This makes this pattern almost hard to miss.

Another Y Wing Style

Above, find:

  • (4)a4=(4)g4-(4)g1=(4-7)c1=(7)a2
  • => a4≠7
  • => (7)e4 % row
  • => (9)e3 % column
  • => naked singles to the end
  • UP 81



Summary & Notes

The proof provided on this and the preceding four pages includes, in my opinion, two truly difficult steps:

  1. Step 2d from Page one:
    • => (2)g789=(2)de7 => hi7≠2
    • Depth of this step (sis considered): 11
    • Technique used: Combining chain snippets. Three chain snippets required.
    • Markers for finding:
      • Mostly underlines
      • Symmetric Kraken cells (139)f38
  2. Step 3d from Page three:
    • => (1)f6=(8)f1 => f6≠8, f1≠1
    • Depth 11
    • Technique used: Combining chain snippets. Two chain snippets required.
    • Markers for finding:
      • ALS - twice
      • Almost wrap around chain - thrice if one includes ALS as Almost wrap around chains
      • Previous step 3c
      • underlines and circles in the puzzle mark-up
There are several themes common not only to those two very difficult steps, but also to many of the other steps:
  1. Creating and maintaining some manner of locating hidden sis in the puzzle. The puzzle mark-up shown is but one way to do so.
  2. Brainstorming quickly to locate chain snippets - note one may wish to routinely mix stengths (hidden and naked) while doing this. For this reason, I prefer a mark-up that allows such simulataneous consideration of each. There are probably much more eloquent ways to do this brainstorming. One may wish to conisder the Molecular Method. I believe Myth Jellies explains this technique at some sudoku forums. [Provide link]
    • Chain snippets are merely AICs. Often times, in combination, AAICs are used.
    • All AICs restrict the set of possible puzzle solutions, regardless of direct effect upon the possibility matrix
  3. Building upon previous proven information about the puzzle. It seems silly to limit the information retained about a particular puzzle to that information which is revealed by the Possibility Matrix. Specifically, remembering proven chain snippets and more importantly proven deduction sis. One may prefer to remember deduced weak links instead of proven sis. In my opinion, this preference is nothing but a matter of style. One may even prefer to remember some of each.
  4. Analyzing recent changes to puzzle information due to eliminations and deductions. This amounts to building a solution from the foundation already laid. In many respects, this process is well-suited to humans. After some practice, the relative signficance of many deductions becomes more apparent. I cannot quite describe how this knowledge of significance comes into existence. I believe the answer to be far too much practice.
  5. Being willing to combine disparite techniques without hesitation. By this, I mean that searches should often be non-technique specific. The puzzle, by its nature, determines which techniques will be fruitful. One has no foreknowledge of which ones that may be. Thus, it seems silly to have an approach that is so highly structured that it will miss things that may not be difficult, but may be novel. One example of this is the Almost Hidden pair Almost Naked triple overlap that I found for step 5b on Page four.
  6. Overall, this puzzle ranks as one of the most difficult that I have encountered at's site. The proof rating spectrum follows:
    • Sets (sis count): 108 This is very high
    • Maximum depth: 11 at two different steps. This is high, especially since it occurs twice.
    • Rating: 49.41
      • That rating is about 5 times higher than each of the puzzles of 11/26/07 and 11/28/07.
      • For comparison, one single step - the Hidden Pair Loop from the Easter Monster rates at over 655 - about 13 times higher for just one step in that puzzle!! This puzzle and The Easter Monster are not in the same league at all.
      • Finally, this puzzle rates about 100 times higher than the median tough at Thus, this puzzle is definately not standard fare.
    Attempting to finely detail how I found steps is not something that I wish to do often. It is time consuming and tedious. It has had a hidden bonus for me however. I believe I have gradually come to a better understanding of finding steps that I had once thought were merely accidental discoveries. I am quite certain now that the accidental discoveries are designed to occur by the manners of analyses.

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