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Harder with another clue

 
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ravel
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PostPosted: Fri Apr 21, 2006 11:52 am    Post subject: Harder with another clue Reply with quote

You might be interested that i finally found a puzzle that becomes harder, when a clue is added. See here
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Marty R.



Joined: 12 Feb 2006
Posts: 5064
Location: Rochester, NY, USA

PostPosted: Fri Apr 21, 2006 3:57 pm    Post subject: Reply with quote

Ravel, how many types of URs are there? Can you provide a link to somewhere that defines these types?
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Fri Apr 21, 2006 4:08 pm    Post subject: Harder with another clue Reply with quote

A very interesting point.

I think what you are saying is that it is easier for a human solver to spot the unique rectangle in r35c46 here:
Code:
---------------------------------------------
|   8   4679 679|   2  46  5|3467  369     1|
|2457 245679 679|   1   8  3|4567 2569  2467|
|   1  24569   3| 469   7 49|   8 2569   246|
---------------------------------------------
|   6      3   4|   8   2  1|   9    7     5|
|  57     57   8|3469 346 49|   2    1    36|
|   9      1   2|  36   5  7|  36    4     8|
---------------------------------------------
| 247   2467   5|  34   9  8|   1  236 23467|
|  24   2489  19|   7 134  6| 345 2358   234|
|   3   4678 167|   5  14  2| 467   68     9|
---------------------------------------------

(so requiring the elimination of 3 from r7c4) than it is to spot that the deadly pattern in the same cells:
Code:
---------------------------------------------
|   8   4679 679|   2  46  5|3467  369     1|
|2457 245679 679|   1   8  3|4567 2569  2467|
|   1   2456   3|  46   7  9|   8  256   246|
---------------------------------------------
|   6      3   4|   8   2  1|   9    7     5|
|  57     57   8|   9  36  4|   2    1    36|
|   9      1   2|  36   5  7|  36    4     8|
---------------------------------------------
| 247   2467   5|  34   9  8|   1  236 23467|
|  24   2489  19|   7 134  6| 345 2358   234|
|   3   4678 167|   5  14  2| 467   68     9|
---------------------------------------------

forces the elimination of 4 from r3c4.

But does that not depend on the solver?

Steve
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Fri Apr 21, 2006 5:31 pm    Post subject: Interesting "nightmare" parallel Reply with quote

Steve, I think the point is that most human solvers are accustomed to look for a "non-unique rectangle" none of whose corners have yet been resolved. It's also probable that most human solvers don't tend to think very hard about the cells that have already been resolved.

There's an interesting example of this second kind of "nearly resolved deadly pattern" over on the "Nightmare" discussion forum. dcb
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ravel
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PostPosted: Fri Apr 21, 2006 8:17 pm    Post subject: Reply with quote

Marty,
this site explains types 1-4 very good, see also the original thread here.
See here for type 5.
Since Keiths "unique x-wing" there is a discussion here about types 6-9.

Steve and David,
the point is, that you definitely cannot use the (unvisible) UR, when the 9 is given. A solution with 4-9-4-9 would be unique for this sudoku, since a second solution with 9-4-9-4 in the rectangle cells then simply is not possible. Therefore you have to use at least forcing chains to solve the puzzle.
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Fri Apr 21, 2006 8:26 pm    Post subject: Reply with quote

So, this was what i needed to register finally - i hate it when i cannot edit typos or wrong copies Wink
Marty, the right link to the original thread is this one
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Steve R



Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England

PostPosted: Fri Apr 21, 2006 10:12 pm    Post subject: Harder with another clue Reply with quote

Ravel

Well said. It’s a good example, too.

The penny did finally drop when I was driving back a few minutes ago. The rest of the journey was spent wondering how it could be that the direction (a) “this puzzle has a unique solution” is of greater utility than the direction (b) “this puzzle has a unique solution and r3c6 contains 9.”

I still harbour suspicions that something useful lurks beneath the paradox.

Steve
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Fri Apr 21, 2006 11:14 pm    Post subject: Re: Harder with another clue Reply with quote

Steve R wrote:

I still harbour suspicions that something useful lurks beneath the paradox.

It was not my intention to point to something useful, but i like paradoxa :)

BTW, think we agree that UR's are useful. I wondered who "invented" them. After some search i would say, flagitious and Hans and this is the real original thread (July 05).
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Tue Apr 25, 2006 3:51 pm    Post subject: Oops! I should read the whole thing. Reply with quote

ravel wrote:
the point is, that you definitely cannot use the (unvisible) UR, when the 9 is given.

My apologies, ravel. I didn't actually read the post you linked to -- I just responded to what Steve R had written. I really ought to read the whole thing before responding. Confused

This is a very interesting example. I wonder if the "paradox" might be resolved this way.

Usually we attempt to solve a sudoku puzzle by filling in unresolved cells one by one. In this case, though, the solution is definitely simpler if we remove the value "9" at r3c6 at an earlier stage of the puzzle, when we see the (possibly) non-unique rectangle beginning to form. Later, when we've placed the "6" at r3c4 we will learn (in this case!) that "9" is the only possible value at r3c6, thus validating the (somewhat unusual) procedure.

I don't know if such a procedure might prove useful on other puzzles besides this one, but I do have an idea for a name -- the technique of removing a value at one corner is "un-raveling" the non-unique rectangle! dcb
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Tue Apr 25, 2006 9:58 pm    Post subject: Not quite logical Reply with quote

David,

A nice turn of phrase! Unraveling, indeed!

The problem here is that Ravel's puzzle was created by adding an initial value (clue) to an already valid (unique) puzzle.

It is by no means true, in general, that you can remove a clue from any puzzle and still have a valid puzzle.

Remember that thread we had over puzzles with a small number of multiple solutions (last December?)? And, you found a puzzle with 2 or 3 or 4 solutions? It would be interesting to see if adding a clue (or two) to that puzzle provides a demonstration that uniqueness cannot be used on an initial value square.

Best wishes,

Keith
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Tue Apr 25, 2006 10:21 pm    Post subject: Re: Not quite logical Reply with quote

keith wrote:

It is by no means true, in general, that you can remove a clue from any puzzle and still have a valid puzzle.


Not sure this is right. It seems some puzzles must have clues added to the initial layout to make them symmetrical or to make the layout more interesting, but those clues are not necessarily needed. I haven't really read it yet but there is a fairly current thread on Sudoku.com about "minimal" puzzles, ones in which none of the initial clues can be removed and remain a valid puzzle, which may shed more light on the matter
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Tue Apr 25, 2006 11:24 pm    Post subject: Thought experiments Reply with quote

Tracy,

I agree that many puzzles have redundant clues, to make them easier or to make them symmetric. But, you can absolutely not think that removing a clue will result in a valid puzzle:

Experiment No. 1:

We have all seen puzzles in which only 8 numbers are represented in the initial pattern.

I believe I have seen a few puzzles in which only 8 numbers are represented, and one of them is represented only once.

Remove this one, and the puzzle must have at least two solutions. (Interchange each of the numbers not in the original pattern.)


Experiment No. 2:

A) Given a valid (unique) puzzle, remove a clue.

B) If the resulting puzzle is valid (unique), go to A).


Experiment No. 3: (A road to fame)

Remove any clue from any 17-clue Sudoku, and find a valid 16-clue Sudoku.


I must admit, I am not considering symmetry of the initial clues. This is a cosmetic (not a mathematical) requirement.

Best wishes,

Keith
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Wed Apr 26, 2006 10:14 pm    Post subject: Re: Thought experiments Reply with quote

keith wrote:

I agree that many puzzles have redundant clues, to make them easier or to make them symmetric. But, you can absolutely not think that removing a clue will result in a valid puzzle


It seems we must be talking about entirely different subjects, because otherwise these two sentences are contradictory. If a puzzle has redundant clues, why would removing a redundant one necessarily result in an invalid puzzle?
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keith



Joined: 19 Sep 2005
Posts: 3136
Location: near Detroit, Michigan, USA

PostPosted: Wed Apr 26, 2006 11:45 pm    Post subject: Reply with quote

Tracy,

Let's go back.

David proposed that if a puzzle has an initial clue which does not allow a uniqueness technique, remove that clue and use the uniqueness technique anyway.

I said that removing a clue from a unique puzzle does not necessarily result in another unique puzzle, so David's proposal is not a generally valid solution technique.

I hope you're enjoying the gorgeous weather in our part of the world!

Best wishes

Keith
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TKiel



Joined: 22 Feb 2006
Posts: 292
Location: Kalamazoo, MI

PostPosted: Thu Apr 27, 2006 1:41 am    Post subject: Reply with quote

Keith,

Now it makes sense to me. We weren't talking about the same thing.

Is there any place you'd rather be in the springtime than Michigan? I can't think of one.
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