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A four-cell contradiction chain

 
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Sun Jun 20, 2010 8:12 pm    Post subject: A four-cell contradiction chain Reply with quote

A four-cell contradiction chain.

Sudoku Susser keeps on pointing these out. For example:
Code:

+----------------+----------------+----------------+
| 1    379  79   | 57   4    6    | 2    8    357  |
| 378  5    4    | 2    1    789  | 39   6    379  |
| 78   6    2    | 589a 3    5789d| 1    4    579  |
+----------------+----------------+----------------+
| 579  1    79   | 3    58   89   | 4    2    6    |
| 2    4    3    | 59   6    1    | 59   7    8    |
| 6    89   58   | 4    7    2    | 359  1    39   |
+----------------+----------------+----------------+
| 35   38   1    | 6    58   4    | 7    9    2    |
| 4    2    58   | 578b 9    578c | 6    3    1    |
| 79   79   6    | 1    2    3    | 8    5    4    |
+----------------+----------------+----------------+

5 in a forces 5 in d; a is not 5.

The stencil is something like:
Code:

589a  (-5)  5789d

(=8)        (=5)

578b  (=7)  578c


If X, Y, Z are particular candidates, and t, u, v, w are any candidates:
Code:

tXY  (-X)  wX

(=Y)       (=X)

uYZ  (=Z)  vXZ

The cell tXY cannot be X.

Are these worth looking for? Can we construct a recipe for the pencil & paper crowd to find them?

Keith

PS: Sports on TV this afternoon is golf, soccer, and NASCAR. The question for many must be: "Which is less boring?"
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sun Jun 20, 2010 10:35 pm    Post subject: Reply with quote

Code:
 +--------------------------------------------------------------+
 |  1     379   79    |  57    4     6     |  2     8     357   |
 |  378   5     4     |  2     1     789   |  39    6     379   |
 |  78    6     2     | *589   3    #5789  |  1     4     579   |
 |--------------------+--------------------+--------------------|
 |  579   1     79    |  3     58    89    |  4     2     6     |
 |  2     4     3     |  59    6     1     |  59    7     8     |
 |  6     89    58    |  4     7     2     |  359   1     39    |
 |--------------------+--------------------+--------------------|
 |  35    38    1     |  6     58    4     |  7     9     2     |
 |  4     2     58    | *578@  9    #578@  |  6     3     1     |
 |  79    79    6     |  1     2     3     |  8     5     4     |
 +--------------------------------------------------------------+

 (*) strong link on <8>
 (@) strong link on <7>
 (#) strong link on <5>


(8)r3c4 = (8-7)r8c4 = (7-5)r8c6 = (5)r3c6  =>  r3c4<>5, r3c6<>8

The AIC is three strong links that reside in four cells that form a "U" pattern. It starts and ends with two cells that see each other. Eliminations are possible in both the starting and ending cells.

I have no idea how difficult this would be for P&P people to emulate.

===== ===== ===== ===== Puzzle Specific Note

If you wish to reduce the number of different candidate values used, then you can take advantage of r7c5=58 in this puzzle.

Code:
(8)r3c4 = r8c4 - (8=5)r7c5 - r8c6 = (5)r3c6  =>  r3c4<>5, r3c6<>8
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Sun Jun 20, 2010 11:13 pm    Post subject: Reply with quote

Quote:
Eliminations are possible in both the starting and ending cells.
Really? In general, or just in this example?

Keith
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Sun Jun 20, 2010 11:51 pm    Post subject: Reply with quote

keith wrote:
Quote:
Eliminations are possible in both the starting and ending cells.

Really? In general, or just in this example?

I'm not sure what you mean by "in general". Here's what I mean.

If a starting cell is assumed false for candidate <X> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell and <X> can be deleted from the ending cell. <X> can not be equal to <Y>.

Regards, Danny
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Mon Jun 21, 2010 4:12 am    Post subject: Reply with quote

daj95376 wrote:
keith wrote:
Quote:
Eliminations are possible in both the starting and ending cells.

Really? In general, or just in this example?

I'm not sure what you mean by "in general". Here's what I mean.

If a starting cell is assumed false for candidate <X> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell and <X> can be deleted from the ending cell. <X> can not be equal to <Y>.

Regards, Danny

Danny,

What if the ending cell does not contain X (in your notation, which is not the same as mine)? The implication of your statement is that the starting and ending cells each need to contain both X and Y.

Here's what I mean:

If a starting cell is assumed true for candidate <Y> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell.

Keith
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ronk



Joined: 07 May 2006
Posts: 397

PostPosted: Mon Jun 21, 2010 10:22 am    Post subject: Reply with quote

keith wrote:
What if the ending cell does not contain X (in your notation, which is not the same as mine)? The implication of your statement is that the starting and ending cells each need to contain both X and Y.

Here's what I mean:

If a starting cell is assumed true for candidate <Y> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell.

I think most would agree with daj95376. If cells A and B are peer cells and you have the AIC ...

(x)A = ... = (z)B

... where x and z are unlike digits, then the two potential exclusions are A<>z and B<>x. Some Eureka folks called this a 'Type 2 AIC.'
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daj95376



Joined: 23 Aug 2008
Posts: 3855

PostPosted: Mon Jun 21, 2010 3:44 pm    Post subject: Reply with quote

Okay, here's the longer version of my explanation.

If a starting cell is assumed false for candidate <X> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell and <X> can be deleted from the ending cell. Of course, if <Y> does not exist in the starting cell and/or <X> does not exist in the ending cell, then the missing candidate(s) can be considered pre-eliminated!

Quote:
Here's what I mean:

If a starting cell is assumed true for candidate <Y> and an (ending) peer cell is derived as true for candidate <Y>, then <Y> can be deleted from the starting cell.

<X> can not be equal to <Y>.

The explanation goes back to the forcing chains that I use to derive the relationships. If you'd like, I'll include them.

Regards, Danny
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Mon Jun 21, 2010 4:35 pm    Post subject: Reply with quote

Ronk and Danny,

Thank you.

Keith
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Luke451



Joined: 20 Apr 2008
Posts: 310
Location: Southern Northern California

PostPosted: Tue Jun 22, 2010 11:18 pm    Post subject: Re: A four-cell contradiction chain Reply with quote

keith wrote:
A four-cell contradiction chain.

Sudoku Susser keeps on pointing these out. For example:
Code:

+----------------+----------------+----------------+
| 1    379  79   | 57   4    6    | 2    8    357  |
| 378  5    4    | 2    1    789  | 39   6    379  |
| 78   6    2    | 589a 3    5789d| 1    4    579  |
+----------------+----------------+----------------+
| 579  1    79   | 3    58   89   | 4    2    6    |
| 2    4    3    | 59   6    1    | 59   7    8    |
| 6    89   58   | 4    7    2    | 359  1    39   |
+----------------+----------------+----------------+
| 35   38   1    | 6    58   4    | 7    9    2    |
| 4    2    58   | 578b 9    578c | 6    3    1    |
| 79   79   6    | 1    2    3    | 8    5    4    |
+----------------+----------------+----------------+

5 in a forces 5 in d; a is not 5.

The stencil is something like:
Code:

589a  (-5)  5789d

(=8)        (=5)

578b  (=7)  578c


If X, Y, Z are particular candidates, and t, u, v, w are any candidates:
Code:

tXY  (-X)  wX

(=Y)       (=X)

uYZ  (=Z)  vXZ

The cell tXY cannot be X.

Are these worth looking for? Can we construct a recipe for the pencil & paper crowd to find them?

My pet name for these is "strong rectangles." Udosuk categorized them among what he called "Hybrid Wings," although they are not often referred to as such.

I see all wings as AICs, but this particular one will stand out for us P&P types if all conjugate pairs are somehow delineated (I mark mine in bold.)

Keith wrote:
Sports on TV this afternoon is golf, soccer, and NASCAR. The question for many must be: "Which is less boring?"

Surprised no one jumped you for that one? Maybe they all agree, or are just too swept up in the heart-stopping excitement of sudoku to respond....Smile

NASCAR: I call it NECKCAR but acknowledge that it's hugely popular.
GOLF: The U.S. Open at Pebble Beach! Say no more.
"SOCCER": The World Cup!! The greatest tournament of them all!

I'd bet the "many" to whom you refer are outnumbered by the billions worldwide who can't get enough.
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Tue Jun 22, 2010 11:24 pm    Post subject: Reply with quote

Keith wrote:
Sports on TV this afternoon is golf, soccer, and NASCAR. The question for many must be: "Which is less boring?"

Or, I perhaps missed the Canadian curling championships?

Keith
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Luke451



Joined: 20 Apr 2008
Posts: 310
Location: Southern Northern California

PostPosted: Tue Jun 22, 2010 11:43 pm    Post subject: Reply with quote

Hey, the gf watches home-improvement TV all Sunday afternoon. Now, that's literally watching paint dry!
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Wed Jun 23, 2010 2:29 am    Post subject: Reply with quote

keith wrote:

Or, I perhaps missed the Canadian curling championships?
Keith


Tried to slide that past me, did 'ya. Wrong time of year.
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Wed Jun 23, 2010 3:17 am    Post subject: Reply with quote

wapati wrote:
keith wrote:

Or, I perhaps missed the Canadian curling championships?
Keith


Tried to slide that past me, did 'ya. Wrong time of year.

Sorry, I meant the reruns of the curling competitions. We all know how compelling the originals are! Razz

Eh!

Keith
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Wed Jun 23, 2010 4:16 am    Post subject: Reply with quote

Hurry HARD, eh? Smile
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wapati



Joined: 10 Jun 2008
Posts: 472
Location: Brampton, Ontario, Canada.

PostPosted: Wed Jun 23, 2010 4:25 am    Post subject: Reply with quote

I like curling. It is slow to watch but that gives you time to figure what shot you prefer. Strategy.

Soccer, I do like it, much like hockey. Hockey is my sport of choice.

Baseball, I can watch a close game.

American football? 4 seconds of action between minutes of doldrums.

The last 3 minutes take 20 minutes. Awesome?

Worst TV sport for me is darts. I have never lasted more than a moment.
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keith



Joined: 19 Sep 2005
Posts: 3175
Location: near Detroit, Michigan, USA

PostPosted: Wed Jun 23, 2010 4:27 am    Post subject: Reply with quote

Eh

Rugby
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Luke451



Joined: 20 Apr 2008
Posts: 310
Location: Southern Northern California

PostPosted: Thu Jun 24, 2010 4:02 am    Post subject: Reply with quote

***WOW!*** U.S v. Algeria

There's a reason it's called "The Beautiful Game."




(Video link zapped by FIFA )


Last edited by Luke451 on Thu Jun 24, 2010 1:44 pm; edited 1 time in total
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Mogulmeister



Joined: 03 May 2007
Posts: 695

PostPosted: Thu Jun 24, 2010 5:54 am    Post subject: Reply with quote

Absolutely Luke!

Thrilling finale and as a special reward you get the Ghanaians and we get the Germans! Crying or Very sad
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