The following illustrated proof for the
Tough Sudoku of April 10, 2007 employs
exactly three Sudoku techniques: Hidden Pairs, Very Common Y Wing Style, and Coloring
. Any technique that uses exactly one type of candidate to justify an elimination I
call Coloring.
Previous blog pages may be helpful. Links to these pages are found to the right, under
Sudoku Techniques. This list is getting long, so specifically, one may want
to refer to the following previous blog pages:
Many steps not illustrated are possible. To show every possible
step is, well, just overkill. However, this page illustrates sufficient deduced elimination steps
to both solve the puzzle, and also to prove a unique solution.
The illustrations of forbidding chains, also called
Alternating Inference Chains (AIC), shown below 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)
(not used)
- candidates crossed out in red = candidates proven false
Please be aware that, for me, strong and weak need not be mutually exclusive properties.
The Puzzle
Two Unique Possibilities are available here:
- e2 = 7% row (hidden single in row)
- i8 = 2% cell (naked single)
Typically, at this point, and more so if I am solving the puzzle on paper, I prefer to
search for Hidden Pairs before filling in the
Possibility Matrix. I am probably
in the minority in this regard!
Hidden Pair 12
The relatively symmetrical placement of candidates 1 & 2 in column b, column i, and box e5
make the Hidden Pair 12 at ch6 relatively easy to spot. Since ch6 are the only two places left for candidates 1,2 in row
6, no other candidates can exist in those cells. Illustrated above is the forbidding chain
representation of this technique:
- c6=1 == h6=1 -- h6=2 == c6=2 => c6=12.
One can at this point make many other eliminations. However, cell f4=89 and cell a2=89 is a
signal that a very easy to use and spot elimination may be available.
Very Common Y Wing Style using candidates 89
The 8s in row 5 ensure that at least one of a2,f4 must be 9. Thus, all cells that see both
a2 and f4 cannot contain 9. The elimination is illustrated as a forbidding chain above:
- a2=9 == a2=8 -- a5=8 == e5=8 -- f4=8 == f4=9 => a4,f2≠9
If you have trouble understanding this elimination, suppose:
a4=9 or f2=9
=> a2 = f4 = 8
=> row 5 has no possible locations for 8
There is always a proof by contradiction available for every possible elimination. However, the
goal of most Sudoku techniques is to find eliminations by recognizing some sort of pattern. As
such, this pattern is very easy to identify. Nevertheless, I have not seen this pattern well
utilyzed in other sites describing
tips, tricks and techniques to solve Sudoku puzzles.
I call this pattern a
Very Common Y Wing Style because the logic is analagous to a
standard
xy Wing or
Y Wing, and the pattern is fairly prevalent. In my opinion,
it is more prevalent than a standard Y wing, and it is easier to find!
Many puzzles are designed to be solved using a Y Wing, but few are designed to be solved using
the Very Common Y Wing Style. Nevertheless, despite the fact that the pattern is not
designed to exist, it is still occurs with enough frequency that it should be
standard issue in any Sudoku Toolbox.
In my experience, I have found many puzzles designed to be very difficult that are
trivialized with this technique. Hopefully, I have driven this point home!
Perhaps, if this technique had a more catchy name, it could be included in the standard
issue Sudoku solving list. Feel free to suggest such a name! It is quite likely, however, that
this technique already has a name given to it by the Sudoku community. If anyone can identify
that name, please let me know. Then, I can call it by its proper name.
Hidden Pair 79
After knocking off the 9 at a4, both candidate 7 and candidate 9 are limited to the same
to cells within box b5. Since these two cells are also aligned in column b, all the eliminations
above are justified by this situation. As a continuous nice loop forbidding chain:
- b4=9 == b6=9 -- b6=7 == b4=7
- => b178≠9, b79≠7, b4≠8, b6≠3
This elimination group causes a cascade of
Unique Possibilities, beginning with b1 = 8% column.
If you find all the hidden singles and naked singles that result from this cascade, you should
arrive at 49 cells solved. (UP 49).
Coloring with candidate 8
The possible 8s in both row 7 and row 2 allow the eliminations shown above. As a forbidding
chain with candidate 8, one could write:
- i7 == f7 -- f2 == h2 => h9,i3 ≠8
Again, one can arrive at a proof by contradiction by considering
What if h9 or i3 are 8?
However, most prefer to find this elimination by recognizing the pattern caused by
the relative configuration of candidate 8 in rows 2 & 7.
After making the eliminations noted above, the puzzle is reduced to another progression of
Unique Possibilities, this time until the puzzle is complete. Start with i7 = 8% column
and box, then h9 = 4% column and box. Naked singles are then available to
the end.
PROOF
- Starting with the given puzzle, Unique Possibilities advance it to 24 filled. (UP 24)
- Hidden Pair 12 at ch6 => c6≠379
- Very Common Y Wing Style: f4=98, ae5=8, a2=89 => a4,f2≠9
- Hidden Pair 79 at b46 => b178≠9, b79≠7, b4≠8, b6≠3 UP 49
- fc on 8s: i7 == f7 -- f2 == h2 => h9,i3≠7 UP 81
- Native Strong Sets: 2 + 3 + 2 + 2 = 9
- Maximum Depth, at step 2.2: 3
- Rating based upon this proof: 3(.03) + .07 = .16 A very easy tough.
NOTES
If anyone decides to try to better name the main technique used in this puzzle, perhaps an
alternative place to post such could be at either of the following Sudoku Forums: