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PostPosted: Wed Dec 30, 2009 11:30 am 
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Hello to the list users.
Although I am not a Sudoku beginner, I need to ask you two questions about two basic definitions.

QUESTION 1: a strong link is a link between two candidates in a bivalue cell or a bilocation unit.
Could you give me two separate examples, one for a bivalue cell and one for a bilocation unit?

QUESTION 2: a conjugate pair is always a connected pair, but not all connected pairs are conjugate pairs. Could you please provide me a counter example?

Thank you for your help
Stefano


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PostPosted: Wed Dec 30, 2009 11:55 am 
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Hi stef71,
Welcome to this forum.

Sudopedia is always a good resource for looking up basic sudoku definitions.

stef71 wrote:
QUESTION 1: a strong link is a link between two candidates in a bivalue cell or a bilocation unit.
Could you give me two separate examples, one for a bivalue cell and one for a bilocation unit?
Please see examples 1 and 2 in my post Some tips for spotting XY-Chains.

stef71 wrote:
QUESTION 2: a conjugate pair is always a connected pair, but not all connected pairs are conjugate pairs. Could you please provide me a counter example?
I don't know of any off hand, but perhaps you can find something HERE


Hope this helps.
Børge


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PostPosted: Sat Jan 02, 2010 12:03 pm 
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Dear Borge,
thank you for your help.
I read carefully your post "Some tips for spotting XY-Chains".
Can I ask you few more questions?
1.
I understand that in example 2 there is a strong link on 9 between cells r5c6 and r5c7.
I think that also the links within the "neat chain" are strong links (for example strong link on 6 between cells r5c4 and r1c4, on 4 between r1c4 and r1c6 and so on). Am I correct?

And the difference between the strong links within the neat chain and the link on 9 between cells r5c6 and r5c7 is that the former is a link between two cand
idates in a bivalue cell, while the latter is a link between two candidates in a bilocation unit. Again, am I correct?
2.
Always in example 2, could you help me spotting all the conjugate pairs?

thank you again for your help
Stefano


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PostPosted: Sun Jan 10, 2010 6:54 pm 
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Hi again Stefano,

stef71 wrote:
Can I ask you few more questions?
Sure, but sometimes it will take a couple of days before I am able to answer.

stef71 wrote:
I think that also the links within the "neat chain" are strong links (for example strong link on 6 between cells r5c4 and r1c4, on 4 between r1c4 and r1c6 and so on). Am I correct?
Not completely.
The XY-Chain in example 1 has six links. Four of them are strong links and two of them are weak links.
Which two are weak links is left as an exercise.

stef71 wrote:
And the difference between the strong links within the neat chain and the link on 9 between cells r5c6 and r5c7 is that the former is a link between two cand
idates in a bivalue cell, while the latter is a link between two candidates in a bilocation unit. Again, am I correct?
This conclusion is correct.

stef71 wrote:
Always in example 2, could you help me spotting all the conjugate pairs?
I have never had the need for using conjugate pairs.
Based on the definition at http://www.sudopedia.org/wiki/Conjugate_pair I have found the following conjugate pairs in example 2.
r=Row,   c=Column,   n=Nonet.

Digit 1: r5, r6, c3, c5, n4, n5.
Digit 2: r4, r6, c4, c7, n5, n6.
Digit 3: r1, c3, n4.
Digit 4: r1, r4, c4, c6, n2, n5.
Digit 5: r4, r5, c3, c6, n4, n5.
Digit 6: r1, r5, c4, c5, n2, n5.
Digit 7: r6, r9, c4, c5, n5, n8.
Digit 8: r5, r9, c4, c7, n6, n8.
Digit 9: r5, r6, c6, c7, n5, n6.

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PostPosted: Thu Jan 14, 2010 7:46 pm 
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Hello Borge,
I read carefully your precious answer, but unfortunately I am not able to spot the two weak links in the XY-Chain of example 1.
It seems that I am not a good student, but it is just matter of time.

The six links are:
First link on 6 between the cells r5c4 and r1c4
Second link on 4 between the cells r1c4 and r1c6
Third link on 3 between the cells r1c6 and r6c6
Forth link on 9 between the cells r6c6 and r6c7
Fifth link on 2 between the cells r6c7 and r4c7
Sixth link on 8 between the cells r4c7 and r4c5

As far as I can understand, each of them works in the same way:
At link 1, assumed 6 on the first cell, the choice of 4 on the second cell is forced
At link 2, assumed 4 on the first cell, the choice of 3 on the second cell is forced
At link 3, assumed 3 on the first cell, the choice of 9 on the second cell is forced
At link 4, assumed 9 on the first cell, the choice of 2 on the second cell is forced
At link 5, assumed 2 on the first cell, the choice of 8 on the second cell is forced
At link 6, assumed 8 on the first cell, the choice of 3 on the second cell is forced

Where are the two different links?

Moreover I didn’t understand the list of the conjugate pairs that you wrote down.
I am not able to understand what you mean. Again, just a single explanation and then I will carry on.

Thank you for your patience and your help
Stefano


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PostPosted: Thu Jan 14, 2010 8:50 pm 
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Hello Stefano,

stef71 wrote:
Where are the two different links?
That something is forced has nothing to do with strong or weak links.


Here the definition of a strong link:

A strong link exists between 2 candidates when one of them must be true. Strong links can be present in bivalue cells, in bilocal units and in Almost Locked Sets.
For bivalue cells and bilocal units, a strong link also enforces a weak link.

If the following logical deductions can be made for candidates A and B, they have a strong link:
  • If A is false, B must be true.
  • If B is false, A must be true.


Take a closer look at "Third link on 3 between the cells r1c6 and r6c6".
If r1c6 = 3 -> r6c6 <> 3 -> r6c6 = 9. This since r6c6 is a bivalue cell, i.e. there is strong link between the digits 3 and 9 in r6c6.
If rc6c = 3 -> r1c6 <> 3 -> r1c6 = 4. This since r1c6 is a bivalue cell, i.e. there is strong link between the digits 3 and 4 in r1c6.

But this does not mean that there automatically is a strong link between the digits 3 in r1c6 and r6c6:
Assume A is digit 3 in r1c6, and B is digit 3 in r6c6.

A=false (r1c6 <> 3) does not imply that B=true (r6c6 = 3).
B=false (r6c6 <> 3) does not imply that A=true (r1c6 = 3).
This since r4c6 can be 3, i.e. there is more than two cells that can be 3 in column 6.


For "First link on 6 between the cells r5c4 and r1c4" this is different:
Assume A is digit 6 in r1c4, and B is digit 6 in r5c4.

A=false (r1c4 <> 6) => B=true (r5c4 = 6). This since there now are no other cells in column 4, which can be = 6.
B=false (r5c4 <> 6) => A=true (r1c4 = 6). This since there now are no other cells in column 4, which can be = 6.


Hopefully you can now spot the second weak link in the XY-Chain in example 1.




stef71 wrote:
Moreover I didn’t understand the list of the conjugate pairs that you wrote down.
I am not able to understand what you mean. Again, just a single explanation and then I will carry on.
Here the definition of a Conjugate pair:
A conjugate pair is a pair of candidates with a strong link. They are the last 2 candidates for a single digit in a house they share.

Børge wrote:
Digit 1: r5, r6, c3, c5, n4, n5.
For digit 1 there is a conjugate pair in row 5, in row 6, in column 3, in column 5, in nonet 4 and in nonet 5,
i.e. there is a strong link for digit 1 in row 5, in row 6, in column 3, in column 5, in nonet 4 and in nonet 5,
i.e. thare are 6 different conjugate pairs for digit 1.

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