The New Sudoku Players' Forum

[Jeff]

I use the notation introduced to this forum by, I believe, Wolfgang, i.e. '-' for a strong link (exactly two candidates in a sector) and '~' for a weak link (more than two candidates in a sector).

In terms of nice loops, a weak link of a digit has 2 or more candidates of that digit in a sector. Does this apply to your chains too?

[jphamel]

Getting back to your original question, your colouring in the top 4 cells are correct. But that's almost as far as you can go. A 6 in r3c7 means the 6s can be excluded from r7c7 and r8c7, so must go in r7c9. So that's correct. But a 6 in r1c9 does not lead you to a potential placement in the bottom cells - you know it has to go in c7, but you don't know which row. If there were only 2 cells in box 9 with candidate 6 it'd be a different matter.

Thank you very much Shazbot, this is the explanation I was looking for.

From there I have been able to understand more clearly the links between conjugates pairs where a strong link can only have 2 cells in a group. and here with the 6s it seems I have 2 separate chains if I understand correctly, which did not allow me to deduce wich 6 had to be removed.

JPHamel
Let love rule thy heart, logic rule thy mind and faith rule your soul.

[rubylips]

In terms of nice loops, a weak link of a digit has 2 or more candidates of that digit in a sector.

In that case, I don't see what a strong link is. When a link isn't weak according to the definition given above, there must be just a single candidate position for that digit, i.e. its position must be known, so clearly that isn't a strong link. Is the term used to refer to a bivalue node?

Just to clarify my own definition, a strong link occurs when a sector (i.e. a row, column or box) has exactly two candidate positions for a value or when a cell has exactly two candidate values. Of course, the inference is that when one candidate is known to be false (or true), the other candidate must be true (or false), whereas with a weak link we know only that when one candidate is true, the other must be false.

Thanks.

[Jeff]

In terms of nice loops, a weak link of a digit has 2 or more candidates of that digit in a sector.

In that case, I don't see what a strong link is.

In nice loops, the definition of a strong link is exactly the same, ie bilocational conjugate. However, a strong link is considered as a subset of a weak link which is unconditional as defined above. These definitions allow each implication in a nice loop to be precisely expressed. To avoid confusion, I prefer to use the terms 'conjugate link' and 'unconditional link' instead.

[ronk]

To avoid confusion, I prefer to use the terms 'conjugate link' and 'unconditional link' instead.

Instead of what? Instead of 'strong link' and 'weak link', respectively?

If so, and using the terms interchangeably, for an unconditional link ... If A true, then B false

... and additionally, for the conjugate link subset of the unconditional link ... If A false, then B true

When the underlying logic is the conditional 'if this .... then that', to then (pun intended) have a link termed unconditional is confusing to me.

[Jeff]

[Instead of what? Instead of 'strong link' and 'weak link', respectively?

Yes Ronk, I prefer to use the terms 'conjugate link' and 'unconditional link' instead of 'strong link' and 'weak link', respectively.

If so, and using the terms interchangeably, for an unconditional link ... If A true, then B false

... and additionally, for the conjugate link subset of the unconditional link ... If A false, then B true

When the underlying logic is the conditional 'if this .... then that', to then (pun intended) have a link termed unconditional is confusing to me.

You are right, it's confusing when you consider the implications as the set elements.

When I said conjugate link was a subset of unconditional link, the set elements are the number of candidates in a unit, hence
Conjugate link = {exactly 2} this is what I consider as conditional because the number of candidate has to be exactly 2
Unconditional link = {exactly 2, 3, 4,5,......}

In terms of implications:
Unconditional link implies If A true, then B false.
Conjugate link implies If A false, then B true as well as If A true, then B false, since a unit with exactly 2 candidates can be a conjugate link as well as an unconditional link.

[ronk]

When I said conjugate link was a subset of unconditional link, the set elements are the number of candidates in a unit, hence
Conjugate link = {exactly 2} this is what I consider as conditional because the number of candidate has to be exactly 2

I usually think of a 'unit' as a row, column, or 3x3 box ... but you must consider a cell to be a 'unit' too. In hindsight, that seems eminently reasonable.

But I have a lot more trouble with the notion of a bivalued cell being a 'conjugate link'.

[Jeff]

I usually think of a 'unit' as a row, column, or 3x3 box ... but you must consider a cell to be a 'unit' too. In hindsight, that seems eminently reasonable.

But I have a lot more trouble with the notion of a bivalued cell being a 'conjugate link'.

Hi ronk, I don't think of a 'unit' as anything else other than a row, column, or 3x3 box. I have trouble with the notion of a bivalued cell being a 'conjugate link' too because there is no linkage within a cell (in nice loops al least). Instead, a cell can be of bivalue, trivalue or polyvalue. In nice loops, links and values are counterparts.

Code: Select all

                                   Link - Value
          Within a unit between 2 cells - Within one Cell
                         Conjugate link - Bivalued cell
                     Unconditional link - Polyvalued cell

       A unit with a conjugate link has _ A bivalued cell has exactly 2 candidates
      exactly 2 candidates in that unit   in that cell
                                               
  A unit with an unconditional link has _ A polyvalued cell has 2 or more candidates
      2 or more candidates in that unit   in that cell

             Conjugate link is a subset _ Bivalued cell is a subset of
                  of unconditional link   polyvalued cell

  A conjugate link can be implied as if _ A bivalued cell can be implied as if it is
            it is an unconditional link   a polyvalued cell