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Submitted by: Gath

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Maen. Turtles are cool. Nice pix.
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I have been up for most of the night and published a proof for the tough sudoku of February 15, 2006. It is as ugly as they can get, and I would love it if someone helped make it better.
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Very cool pic! I've never seen a turtle hatch before.
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Much simpler than yesterday's:

1) Start 22, Unique Possibilities to (UP) 25.
2) Locked 1s at f23 forbid f68=1.
3) f89=58 == f89=2 -- e9=2 == e19=38 forbids e7=8. UP 81.

The last step can also be done with Almost Locked Sets:

S1={e1=38, e9=238}, S2={f8=25, f9=258}. 2s More...
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There's a typo in my ALS step above:

S1={e1=38, e9=238}, S2={f8=258, f9=258}. 2s are weak across both sets, therefore 8s are strong. This means that at least one of e1, e9, and f9 is 8. Since e7 can see all of these squares, e7<>8. UP 81.
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Possible proof of tough sudoku of 09 04 06:

1) Start at 22 filled - the given puzzle. Unique Possibilities to 25 filled. (UP 25).
2) Lots of depth 4 steps available here: Forgoing the depth 4 ALS elimination, and not looking any further for depth 3:
d1=6 == d4=6 -- d4-1 -- d78=1 -- e7=1 == e7=8 -- e1=8 == e1=3 forbids d1=3 UP 81
Sets 4: Depth 4: Rating .15
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Another Possible ALS besides the one Clark used:

AT UP 25:
d78e7=1348
e1=38
8's are weak, 3's are strong thus d78=3 == e1=3 forbids e9,d1=3
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BTW, for those counting depth, Clark's step 3 is depth 4 as it uses 4 strong sets.
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Absolutely wonderful picture. Who is the photographer?
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Again easy today.
Start 22. UP 25.

if f6=2 => d4=6, e1=3 and d9=2, e8=3. So f6=6. UP
81.
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Good Maen to all! A super picture!! Kudos to whoever the photographer is!!
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sergio's proof as an fc: (i think, anyway, correcting for come possible typo's)
AT UP 25:
f6=6 == d4=6 -- d1=6 == d1=3 -- e1=3 == e9=3 -- e9=2 == f89=2 forbids f6=2.

Locked 1's at f23 forbids f68=1. UP 81
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4:27 after setup. Definitely personal best for tough but also a very easy puzzle.

FABULOUS photo!!!!!
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For those who want a challenge in sudoku today, revisit the tough of 2/15. If one does not want such a big challenge, the hard kakuro puzzle of today is fairly challenging, if you do not peek at the hints.
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Easier than usual after one correct guess. The turtle looks drier and more alert than I would have expected for a newly hatched fellow.
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Hi Steve. Thanks for clarification. I am still new to this. But I have a question. I thought that the less amount of operations the proof has the better.
My proof has 6 operations - 5 assignments and and one if. Your fc has 8 assignments and 3 logical OR operations. Where am I wrong?
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Hi sergio!
My fc has no assignments. Fc's are not assignments at all. In order to make an 'assignment', exactly what do you do?
'if this, then not that, therefor this'
The logic you perform to make an 'assignment' is precisely stated in the fc:
A--B==C means A implies C. A cannot More...
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Another way to look at it, Sergio - exactly what is considered:

In my rewrite of your proof, I consider only:
strong 6's at {d4f6}
strong cell d1=36
strong 3's at {e1e9}
strong 2's at {e9,f89}
Exacly four items are considered. Now, of course, in order for there to be any More...
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Sergio - final note:
Try to write down exactly what is considered, in total, in your proof. You will see that it is not any less then in the fc version.
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To anyway reading this discourse:
The reason only native strong sets are counted is fairly simple:
native weak links are always true, every sudoku, universally.
native strong links are completely game dependent, unless you write them very large:
i.e., a123346789 = 9 is always a strong More...
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anyway? should be 'anyone'
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17:07 - Great photo!
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12:43 Much easier today
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The previous monster of 02 20 06 - that has stood for some time without a proof, now also has one. This one is much easier to read, and not very long. Depth is only 7, and we should have found it back in February, but I believe we were worn out from the deluge of very tough puzzles that month.
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I have
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to steve & clark:

a simple way to write ALS into FC is something like this: FC: f9=8 == f98=25 -- e9=2 == e91=38
the strong link is the partition of the larger set.
the weak links are native and obvious. I still fail to see why this isn't equivalent in depth to a depth 2 FC.
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Hi Pat!
A better question would be, what reasoning do you have for saying it is a depth 2 fc?
The pair 38 is a depth 2 fc by itself.
The pair 25 is a depth 2 fc by itself.
One could argue, if one wanted to, that the fc you present is of depth 6!

Of course, it is not, as the true More...
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One final note:
If we are going to say that the ALS is a depth 2 fc, then why not just say it is depth 1? If fact, then all fc's are depth 1, as everything in between can be presented as a theorem.
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To further illustrate the point, consider the standard xy wing, which presents clealy as a depth 3 fc.
But, I can write it is as a theorem, so is it depth 1, or 2 now?
For example, all xy wings can be presented as a sub-set of ALS:
cell x =a == cell x = b -- cells (yz=b) == cells (yz More...
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In my mind,it boils down to:
how many times do you have to assess the axoims of the game to make the elimination(s)?
Locked candidates in a row, column, box: Once.
Naked pairs, Hidden pairs, x wings: Twice.
Naked triplets, Hidden triplets, swordfish, xy wings: Thrice.
Pure ALS: Precisely equal to the number of cells considered.
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to Steve:
I firmly believe 'chunking' in pattern recognition must be weighed in the balance when assessing 'depth' which is an alias for difficulty. For example, an XWing is a chunked pattern. You only have know what the animal looks like. A Naked Pair or Triple is also one thing and one always More...
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OK, well if you insist then Pat, then you need to come up with a way to limit and define the chunks.

What, pray tell, is the motivation for changing the manner of definition Andrei and gb have presented?

I cannot agree less that the chunks do not make the cahin more difficult. More...
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After more careful thought, there is some validity to the claim that 'chunks' do not add as much difficulty as the raw assessment of their inherent use of native strong sets would imply.

The issue appears to be the implication that less depth, as defined by Andrei and gb, is always More...
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Not all chunks are created equal and I don't mean to imply some half baked politically correct notion that the Finned SwordFish is in the same class as the lowly naked pair. I think we would all agree that a puzzle that requires recognition of the former is more difficult than one that only needs More...
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Just wandered over from Medium and Hard. Sounds serious here. I'll bet you guys even try to solve these sober!!

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Hello to all. Just started these, and I am impressed with your times.
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Nicely put Pat - but to attach a number to it would still be arbitrary, methinks.
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Many times, in this forum, I have presented my rating system as inadequate, which is completely true. Nevertheless, rating a proof is not simple, as there are many comparisons to make. Are 10 short steps preferable to one long step, and if they are, are they always so? The list of arbitrary More...
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42=1 (1+8)V'D
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In my opinion, a single number does not adequately describe puzzle difficulty. That is one of the reasons that I provide sort of a vector - number of sets, max depth, then a rating number. One can consider more factors;
number of steps involving non-native eliminations, for example. Atomic More...
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