The following is an illustrated proof for the
Tough Sudoku of February 23, 2007.
Since this is a somewhat difficult puzzle, the path shown below may not be the most efficient one.
Certainly, there are many ways to tackle this one. If you are interested in a logical approach
to solving truly difficult puzzles, then this is a great puzzle to study.
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
The information on the following blog pages is required to understand this proof:
The illustrations of forbidding chains (also called Alternating Inference Chains or AIC)
used on this blog page will share the same key:
- black lines = strong links
- red lines = weak links
- candidates circled red = candidates proven false
Many steps that are possible will not be shown to keep the proof as short as possible.
However, every step that is shown can be justified by considering only the previous illustrated
steps.
Puzzle at start
A few Unique Possibilities:
Unique Possibilities get the puzzle to 24 cells solved. (UP 24)
The following steps are available, but not used in this proof:
- Hidden pair 67 at gi7 forbids g7=128, i7=1245
- Locked 1's at hi9 forbids ab9=1
- fc on 2's: g8 == g2 -- i1 == a1 forbids a8=2
- Locked 4's at d12 forbids d467=4
- Locked 5's at d23 forbids d4567=5
- Locked 6's at d13 forbids d456=6
Instead, the proof continues as follows:
Puzzle at UP 24, finding Locked 9s
Illustrated to the left, one can see that the 9s in box e8 are limited to f89. Thus:
- Locked 9's at f89 forbids f56=9
The 9s and 2s are very interesting and tripping all over each other in this puzzle. If this
puzzle has a theme, the relationship between these two candidates would be my nomination. The
standard puzzle mark-up shown in previous blog pages will clearly indicate a number of possible
forbidding chains at this point. The following one is the one I found most useful.
Depth 4 forbidding chain involving 2s and 9s
Illustrated above is this forbidding chain:
- f5=2 == d5=2 -- d5=9 == i5=9 -- i1=9 == b1=9 -- b9=9 == f9=9
To convince yourself of this valid elimination, consider:
- f5=2 => f9≠2
- f5≠2 => d5=2 => d5≠9 => i5=9 => i1≠9 =>
b1=9 => b9≠9 => f9=9 => f9≠2
This elimination opens up an X wing on 2's
X wing on 2s
Illustrated to the left, all possible locations for 2s are highlit. Thus we have:
Although the X wing does not immediately advance the puzzle, it does make the relationship
bewteen 9s and 2s much more interesting.
Depth 5 forbidding chain involving 2s and 9s
Illustrated above is a complex forbidding chain:
- {Hidden pair 29 at cf8} == g8=2 -- g3=2 == i1=2 -- i1=9 == b1=9 -- c2=9 == c8=9
- => c8 is limited to only 29
This type of limited wrap around forbidding chain requires some care. Not all the weak links
are proven strong, as c8=9 does not prevent f8=2, thus the two ends of the chain are not in
conflict with each other.
As a forcing chain type logical approach:
- c8=9 => c8≠3458
- c8≠9 => c2=9 => b1≠9 => i1=9 => i1≠2 => g3=2 => g8≠2 =>
{Hidden pair 29 at cf8} => c8≠3458
After making these eliminations, a8=3% box & row. The puzzle is only advanced to UP 25, but
this is an important elimination, as now the 3s are very strong in box b2.
Y wing style
The chain above uses the newly strong 3s and the aforementioned overlapping 2s and 9s:
- c3=3 == b1=3 -- b1=9 == i1=9 -- i1=2 == g3=2
After making this elimination, some cascading Unique Possibilities are available, forwarding
the puzzle to UP 32. (Starting with a1=2% box and g3=2% row).
Another Y wing style
Because of the previous Y wing style, the strong 3s now interact well with the 7s and 6s:
- a3=6 == b1=6 -- b1=3 == c3=3 -- i3=3 == i3=7
Again, this simple Y wing style leads us to some more cascading Unique Possibilities,
beginning with a2=7% box and i3=7%row. The puzzle is advanced to UP 38.
One more Forbidding Chain
The 9s and 2s are still fruitful fc fodder, using the 6s as a bridge:
- g6=6 == g4=6 -- f4=6 == f5=6 -- f5=2 == d5=2 -- d5=9 == d6=9
This step now unlocks the puzzle with cascading Unique Possibilities to the end (UP 81).
Start perhaps with finding all the locations for the 9s
Solved Puzzle
This puzzle unlocks easily keying in on mostly just the 2s and 9s. Other approaches
are bound to exist, but this one was almost immediately obvious.
Proof
- Start at 22 filled - the given puzzle. Unique Possibilities to 24 filled. (UP 24).
- Locked 9's at f89 forbids f56=9
- f5=2 == d5=2 -- d5=9 == i5=9 -- i1=9 == b1=9 -- b9=1 == f9=9 forbids f9=2
- X wing on 2's at ai19 forbids ai378=2
- {Hidden pair 29 at cf8} == g8=2 -- i9=2 ==
i1=2 -- i1=9 == b1=9 -- c2=9 == c8=9 forbids c8=3458 UP 25
- Y wing style: c3=3 == b1=3 -- b1=9 == i1=9 -- i1=2 == a1=2 forbids c3=2 UP 32
- Y wing style: a3=6 == b1=6 -- b1=3 == c3=3 -- i3=3 == i3=7 forbids a3=7 UP 38
- d6=9 == d5=9 -- d5=2 == f5=2 -- f5=6 == f4=6 -- g4=6 == g6=6 forbids g6=9 UP 81
- Sets: 1 + 4 + 2 + 5 + 3 + 3 + 4 = 22
- Max depth 5 at step 2.4
- Rating: .01 + .03 + 2(.07) + 2(.15) + .31 = .79