The
Tough Sudoku of July 15, 2010 is ranked 9.0 by Sudoku Explainer, but is not terribly difficult.
One step which I choose to use to solve this puzzle lightly hits upon the theme of the previous blog post: the sudoku tip, trick or technique of
using Almost Continuous Loops.
If this is your first visit to this blog, welcome! Unfortunately, if you are a first time visitor, this page may seem
like it is written in a different language. Well, it is!! Previous blog pages may be helpful. Links to these pages are
found to the right, under Previous Entries. The earliest posts are at the bottom, and if you have never perused
the intricacies of our special coded language here, you may wish to start close to beginning. The list is rather large, but you may
have some hope of understanding this page if you visit the
Definitions page.
The Puzzle Including Possibilities (Pencil Marks)
Above, find the puzzle as given with the Possibility Matrix already filled in. A small handful of easy steps are available:
 (6)i8 % cell  a single in the cell i8
 (8)d3 % cell
 (5)e7 % Box  a single in Box e8
 (6)b7 % Row  UP 26 (Unique Possibilites to 26 solved cells)
 Hidden Pair(12)  HP(12)df5 => df4≠1, df6≠2, d5≠56, f5≠7
This brings the puzzle to the state below.
Puzzle after easy steps
A mundane step  step 2)
Although this puzzle has a number of interesting possible first moves, the
following unremarkable step seems to be a good way to start.
 (2)f8 = (26)d9 = e9  (6=7)e5  (7=5)b5  b8 = (5)c8
 => c8≠2
 => some singles until UP 33
 LC(7)c23 => c46≠7
Digressive Note: here one can likely kill the puzzle quickly with just one or two complex steps. Most puzzles can be
quickly dispatched with just a small handful of moves. My preference is to look for patternlike moves
whenever possible. The patternlike moves are rarely the most efficient. They are often fairly elegant, however.
Of course, elegance is subjective. Some may prefer the quick puzzle kill regardless of the means used
to accomplish the feat. If that is your preference, this blog will likely be faintly satisfying.
A Continuous Loop  step 3a)
Continuous Loops are, IMO, the most beautiful of steps in Sudoku. In many ways, the entire idea of building
a single solution puzzle is interwoven tightly with the concept of avoiding too many overlapping continuous loops
whilst leaving just enough room to still have at least one solution. But again, I am digressing..... The
loop I found involves just three sets of Strong Inference Sets (SIS):
 Column c  one can consider
 candidates (279) in this column
 or cells c24678  I will arbitrarily use these SIS in the presentation
 (5) in rows 5 & 8
 Box h2  one can consider
 candidates (1267) in this Box
 or cells g2, h1, h2  I will arbitrarily use these SIS in the presentation
Here is the step:
Key: Naked Quad (NQ), Naked Triple (NT), Loop => every SIS considered can be a start point
` (7=NQ1358)c2467  (5)c8 = b8  b5 = (5)h5  (NT538=7)h1hg2 Loop
 => c9≠1, c89≠8, c39≠3, b16,h4≠5, g1≠8, h39,i1≠3
A nice little step that yields 12 eliminations. More importantly, it establishes for the
solver a group of possibilities which must be True/False together as a group. Hence, this group
will likely
 go a long way towards solving the puzzle if any one member can be solved.
 act as a powerful analysis tool for the remainder of the puzzle
 all of the above
Even if one chooses to not execute this step, the awareness that this group of SIS will solve
into just one of two groups will likely be helpful.
A Practically New Puzzle  steps 3b) & 3c)
This next step is relatively easy. Steps like this one can pop out into the "obvious"
realm given enough practice. It is useful to see it in a few different representations.
Three related ways to view this step are given below (1  a fancy name, 2  a naked set chain, 3  a mixed hidden and naked sets chain):
 ALS xz rc(3) => NT(578)acg2 = NT(267)agh3 => h2≠7

ALS xz rc(3) means Almost Locked Sets with restricted common (3)
 it is based upon the fact that (3) cannot exist at both ac2,a3.
 One could ignore the fancy names and write:
 NT(578)acg2 = (3)ac2  (3=2)a3  (2)g3 = NP(67)gh3 => h2≠7
 (3)h2 = ac2  (3=2)a3  (2)g3 = HP(2115)gi1 = HP(53)h12 => h2≠78
Note that regardless of how one views it, the overall relationship of the candidates with
each other within Box h2 is reused from the previous step.
After that step, this other step also takes advantage of the overlap with step 3a:
 (8=7)g2  (7=NQ1358)c2467  (5=NT948)cgh8 => g7≠8
This finally brings the puzzle to the point where the step which motivated me to create this
blog page is most readily available.
Step 3d) A succinct usage of Almost Locked Candidate Base Sets
Huh? Again, I ban myself from ever actually naming anything  for which my
children are quite grateful! Below, An Almost Continuous Loop, or Network, considering
only candidate (7) can break the puzzle open. It can be viewed as having the XWing concept
as the base.
Here is the step:
 (7=NT835)g2h12  (5)h5 = (57)b5 = (7)FinnedSwordFish bfi467 Fin i9
 => g7≠7
 => UP 81  Singles, naked and/or hidden, until finished
Much like ALS's, here one can consider Almost Locked Candidate (7) to see that:
in columns bfi, r46 can contain at most two truths  An Almost XWing, if one prefers 
thus SIS (7)[b5, f7, i79]. This makes the chaining almost trivial:
 (7)f7i79 = (75)b5 = (5)h5  (5=NP38)h12  (8=7)g2
Again, note how the original Continuous Loop from step 3a helps provide a map of SIS to
reuse in these later steps. IMO, puzzles which fall apart based upon such reused chain
fragments are easier than those which require one to find entirely new SIS groups for
most steps.
The steps from above all in one place
 Start 22 UP 26; HP(12)df5 => df4≠1,df6≠2, d5≠56, f5≠7
 (2)f8 = (26)d9 = e9  (6=7)e5  (7=5)b5  b8 = (5)c8 => c8≠2; UP 33; LC(7)c23 => c46≠7
 (7=NQ1358)c2467  (5)c8 = b8  b5 = (5)h5  (NT538=7)h1hg2 loop => c389≠138; b16,h4≠5; g1≠8; h39,i1≠3
 ALS xz rc(3) => NT(578)acg2 = NT(267)agh3 => h2≠7
 (8=7)g2  (7=NQ1358)c2467  (5=NT948)cgh8 => g7≠8
 (7=NT835)g2h12  h5 = (57)b5 = (7)FSFbfi467 Fin i9 => g7≠7; UP 81
For step minimizers, here is a slightly different path:
 Start 22 UP 26; HP(12)df5 => df4≠1,df6≠2, d5≠56, f5≠7
 (2)f8 = (26)d9 = e9  (6=7)e5  (7=5)b5  b8 = (5)c8 => c8≠2; UP 33; LC(7)c23 => c46≠7
 (8=7)g2  (7=NQ1358)c2467  (58=NT948)cgh8 => g7≠8
 (7=8)g2  (8*)h1 = [(5*=3*)h1  f1 = e3  (3=2)a3  g3 = HP(2115)gi1 = (5)h12]  (5)h5 = (57)b5 = b6  fi6 = (7)FXWfi47i9 => g7≠7; UP 44: NP(38)c67 => c2349≠38; UP 81
Digressive Preaching: A slightly nasty AAIC in that last step. Rather than looking directly for such an animal,
one could solve the puzzle using shorter steps and then take that knowledge to construct
a fewer step solution. Simply put, that is exactly how I arrived at the step minimizing path above. I find it
more instructive to know how to find such an animal by recognizing the little pieces of logic, or steps,
that collaborate to create the beast.
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