The following is an illustrated proof for the
Tough Sudoku of February 6, 2007.
Since this is a very difficult puzzle, I may have easily missed something(s). Certainly, there
are multiple ways to tackle this one. If you are interested in a logical approach to solving
truly difficult puzzles, this page is for you. If, however, quick solving is your goal, then
this page is probably a waste of your time.
Since this proof uses a Y wing style, some
forbidding chains, (also called Alternating Inference Chains)
and a variety of other techniques, 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.
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
A few Unique Possibilities:
- f9 = 6% box & row
- h7 = 3% box & row
- i5 = 3% box & column
- b1 = 3% row
- g7 = 1% box
Unique Possibilities get the puzzle to 28 cells solved. (UP 28)
Hidden Pairs 58 and 15
As if often the case, some Hidden pairs can be located before considering the possibility
matrix. Illustrated to the left are
- Hidden Pair 58 at ab9:
- forces ab9=58
- forbids ab8=5
- forbids a7=8
- forces a7 = 2% cell
- Hidden Pair 15 at ah2
The puzzle is now at 29 cells solved. (UP 29)
Y wing style wrap around chain - also called Hub and Spoke configuration
This type of Y wing style chain happens so frequently, it is probably a good idea to search for
it all the time. It is much more common than a standard Y wing. As a wrap around chain, it
is more powerful. Unfortunately, it usually involves considering a grouped argument, such
as the 1's in box b2.
- a8=1 == a8=9 -- a1=9 == c1=9 -- c1=1 == a12=1
- forbids a5=19
- forbids c1=26
In conjunction with the chain above, the following additional eliminations are justified:
- Locked 9's at ac1 forbids 9 from the rest of row 1
- Newly locked 2's at bc3 forbids 2 from the rest of row 3
Forbidding Chain using four strong sets
Illustrated above is:
- g5=5 == g3=5 -- b3=5 == b9=5 -- a9=5 == a9=8 -- a5=8 == a5=6
- g4 = 6% box
- h4 = 7% box
- Puzzle is now advanced to 31 cells solved (UP 31)
Y wing style - very common type
The newly revealed 24's at g1, h9 are a common key to look for this type of Y wing style
elimination pattern:
- g1=4 == g1=2 -- g5=2 == h56=2 -- h9=2 == h9=4
- forbids h1=4
- forbids g9=4
- forcing:
- advancing the puzzle to 33 cells solved (UP 33)
To this point, the puzzle eliminations have been indicated by the search parameters previously
explained in Forbidding Chains 102 The Practice, or in the Y wing styles page.
The following chains are also indicated, but now there are dozens of possible paths. Most of
the paths indicated will eventually lead to a solution. What follows here is the least deep -
meaning least maximum number of native strong sets per step - path that I found. Probably,
there are better paths. Also, you may well prefer fewer chains of greater depth.
Forbidding Chain using five strong sets
Illustrated above is the following:
- g5=2 == g5=5 -- g3=5 == b3=5 -- a2=5 == a9=5 -- a9=8 == a5=8 -- h5=8 == h6=8
- Newly Locked 2's at gh5
- Old locked 8's at h56
Y wing style
The Y wing style shown above has been available for a while. Grouped candidates such as the
grouped 2's at def2 are a common theme with Y wing styles.
- i1=8 == d1=8 -- d1=2 == def2=2 -- i2=2 == i2=6
Y wing style - Again
This Y wing style has also been available for a while. It is indicated by the standard
puzzle mark-up, but is the type I generally have the most difficult time spying.
- c3=6 == a1=6 -- a1=9 == c1=9 -- c1=1 == c5=1
X wing on 6's
The last two Y wing style eliminations have exposed this pattern:
Depth 4 forbidding chain with multiple effective endpoints
The newly revealed bi-value strong 6's in row 6 indicate the existence of this chain:
- f6=3 == e6=3 -- e6=6 == c6=6 -- a5=6 == a5=8 -- h5=8 == h6=8
- forbids f6=8
- forbids e6=8
In order to see that e6≠8, consider that the chain also indicates:
Forbidding Chain using four strong sets but only two candidates
The Forbidding Chain above has been available for quite some time. Basically, the sevens act
as a bridge for the fives. In this way, it is very similar to a single candidate coloring
elimination.
- f8=5 == d8=5 -- d8=7 == i8=7 -- i3=7 == g3=7 -- g3=5 == g5=5
f5 is starting to look like an important cell. Oftentimes, one cell becomes a focal point
for unlocking a puzzle.
Another four strong set forbidding chain looks at cell f5
Again, this chain has been hanging around waiting to be used
- f3=8 == f3=3 -- f6=3 == e6=3 -- e6=6 == c6=6 -- a5=6 == a5=8
Since we have whittled that cell f5 down to almost the minimum, perhaps it is useful now in
a chain.
Forbidding Chain using the newly uncovered bi-value strong set
The newly revealed 14 at f5 is useful
- f2=2 == f2=4 -- f5=4 == f5=1 -- c5=1 == c1=1 -- h1=1 == h1=2
- forbids d1=2
- forbids i2=2
- forcing i2 = 6% cell
- advancing the puzzle to 34 cells solved (UP 34)
That was much work just to get another cell solved. But, it has been a set up for one final
move. There are many equivalent moves available at this point. I choose to use the newly strong
relationship between 68 in cell d1, but there are many other paths to take here.
Forbidding Chain using five strong sets
- d1=6 == d1=8 -- f3=8 == f4=8 -- f4=1 == f5=1 -- c5=1 == c1=1 -- c1=9 == a1=9
- forcing:
- d1 = 6% row
- i1 = 8% row
- % in some container to the end
- UP 81
Solved Puzzle
Summary data
Sets: 2 + 2 + 3 + 1 + 4 + 3 + 5 + 1 + 3 + 3 + 2 + 4 + 4 + 4 + 4 + 5 =
2(1)+ 3(2) + 3(3) + 5(4) + 2(5) = 47 YIKES!!!
Max depth 5, twice - so not very deep
Rating: 2(.01) + 3(.03) + 3(.07) + 5(.15) + 2(.31) = 1.69 - so fairly monstrous