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The Unique Chain: A tool for pencil & paper solvers?

 
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Sat Apr 02, 2011 3:03 am    Post subject: The Unique Chain: A tool for pencil & paper solvers? Reply with quote

This is not new. It came up about five years ago, then disappeared. Here is the original discussion:

http://forum.enjoysudoku.com/type-3-unique-rectangles-hidden-subsets-t4088.html

We are probably all familiar with Unique Rectangles. Types 1 and 2 involve only cells in the UR, and are pretty straightforward. Type 5 (never seen in a useful way, in my book) is a diagonal Type 1. (Like the Loch Ness monster, some claim to have seen it.)

Types 4 and 6 involve not only the four cells in the UR, but also conjugate constraints on the deadly pattern candidates. Let's put those aside.

Which leaves the Type 3 UR:

Code:
+-----------+-----------+-----------+
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  12 .  |  . 123 .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  12 .  |  . 124 .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
R46C5 form a pseudo-cell or quantum cell 34 that can be used in other patterns. That, we all know.

But, suppose we have:
Code:
+-----------+-----------+-----------+
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  12 .  |  . 123 .  |  .  .  .  |
|  .  .  .  |  13 .  .  |  .  .  .  |
|  .  12 .  | . -124 .  |  .  .  .  |
+-----------+-----------+-----------+
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  .  .  |  .  .  .  |
+-----------+-----------+-----------+
(13 anywhere in B5 or C5 will suffice.)

The conclusion? R6C5 is not 1. Shocked (Try it.)

I propose to call this a Unique Chain (UC), although Unique Set (US) might also be a candidate. The pattern seems to be easy to recognize, though the elimination is a little hard to see. The chain involves the four cells of the UR, plus the extra cell.

There are variants and generalizations, but the basic idea is that the extra cell (R5C4 in this case) involves one of the deadly candidates and one of the candidates that would prevent a deadly pattern.

Five years ago, the opinion was that this cracked some really hard puzzles. I have no idea how useful it actually is, for I have never put it in my toolkit. Maybe I should.

What do you think? Seems simple enough to me.

Keith

(Edited to change name preference to Unique Chain.)


Last edited by keith on Sat Apr 02, 2011 3:37 am; edited 2 times in total
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sat Apr 02, 2011 3:26 am    Post subject: Reply with quote

The elimination is easy to see if the inferences are made explicit:
(1=3)r5c4 - UR[(3)r4c5=(4)r6c5]; r6c5<>1

As ones list of memorized pattern types becomes longer, I believe that at some point it becomes easier to work directly with the induced inferences. It is also more powerful since useful cases are not constrained to recognized patterns.

For those with such pattern-based arsenals it is a good one, especially if the reason for the elimination is not otherwise readily obvious.
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sat Apr 02, 2011 4:30 am    Post subject: Reply with quote

In the EnjoySudoku forum, the topic of Mike Barker's UR types was very recently revived. (here)

In fact, ronk opened the new discussions with MB's UR+2kx -- which your pattern falls under. Since I'm not always successful in deciphering MB's type descriptions, I'm not suggesting that we use them here.

I agree with Asellus' statement, "I believe that at some point it becomes easier to work directly with the induced inferences".

I've programmed my solver for the standard UR Types 1-6 and added an expanded capability to handle some eliminations based on specific strong links being present. I've even considered expanding that logic to include ALS/Subsets.

The elimination using an ALS/Subset in conjunction with the UR constaint:

(1)r6c5 - (13)r5c4,r4c5 = UR[(2)r4c5 - (2=1)r4c2 - (1=2)r6c2 - (1)r6c5]

Note: if you move <13> from r5c4 to any of r6c46, then you can eliminate <12> from r6c5.
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Sat Apr 02, 2011 2:29 pm    Post subject: Reply with quote

daj95376 wrote:
I agree with Asellus' statement, "I believe that at some point it becomes easier to work directly with the induced inferences".

I agree with that. But, you need to remember something that will prompt you to look.

daj95376 wrote:
Note: if you move <13> from r5c4 to any of r6c46, then you can eliminate <12> from r6c5.
Yes, I can see that.

Keith
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ronk



Joined: 07 May 2006
Posts: 397

PostPosted: Sat Apr 02, 2011 4:35 pm    Post subject: Re: The Unique Chain: A tool for pencil & paper solvers Reply with quote

keith wrote:
I propose to call this a Unique Chain (UC), although Unique Set (US) might also be a candidate.

What's unique about a Unique Chain? Is there only one of them? For that matter, what's unique about a Unique Rectangle?
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keith



Joined: 19 Sep 2005
Posts: 3178
Location: near Detroit, Michigan, USA

PostPosted: Sat Apr 02, 2011 4:51 pm    Post subject: Re: The Unique Chain: A tool for pencil & paper solvers Reply with quote

ronk wrote:
What's unique about a Unique Chain? Is there only one of them? For that matter, what's unique about a Unique Rectangle?
I don't know. Maybe we should call it Fred.

(edit) But it is a chain with part of the logic coming from the uniqueness condition. As pointed out in the original thread, the underlying DP does not have to be a rectangle.

Keith
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