The New Sudoku Players' Forum

[Mino21]

Let's see if I understand: in practice two consecutive group-nodes can have a cell in common only if connected by a strong link, right?

So can all these 3 cases



be part of a valid cycle?

[yzfwsf]

Strong and weak must alternate in the AIC or Cycle. Cases 1 and 2 are not possible in AIC or Cycle.Case 3 is valid.
For overlapping groupped links, the previous is weak link and the next is weak link , it itself acts as a strong link.

[Mino21]

Strong and weak must alternate in the AIC or Cycle.

Why? I think a cycle can have any number of odd subsequences of consecutive strong links and only 1 even subsequence of consecutive strong links.

Sorry if I insist, but the following statement

in practice two consecutive group-nodes can have a cell in common only if connected by a strong link, right?

is to be considered true or false?

[eleven]

"(r8c8 r9c8)--(r8c8 r8c9)"
This is not a weak link, which would mean, that if a digit is on one side, it cannot be on the other side. But r8c8 is on both sides.

[Mino21]

Okay, and I assume that the same reasoning would have applied even if the link had been strong ... so we can conclude that two consecutive group-nodes cannot under any circumstances have a cell in common, right?

And does this possibly apply (I don't know if from a practical point of view it can happen) even if the two group-nodes are not consecutive?

Sorry if my questions may sound stupid!

[yzfwsf]

In a word, the state of overlapping cells is indeterminate, and this uncertainty cannot affect subsequent logical reasoning.

[Mino21]

In a word, the state of overlapping cells is indeterminate, and this uncertainty cannot affect subsequent logical reasoning.

Why?
Isn't this:

two consecutive group-nodes cannot under any circumstances have a cell in common

a consequence of this:

"(r8c8 r9c8)--(r8c8 r8c9)"
This is not a weak link, which would mean, that if a digit is on one side, it cannot be on the other side. But r8c8 is on both sides.

?

Alternatively, could you give me an example of a cycle with two consecutive group nodes sharing a cell?

[Mino21]

_UP_

[Pat]

• AIC
  
• Grouped x-cycle
  
  does this help?

[Mino21]

does this help?

I don't think...

Is it so difficult to answer this question?

Isn't this:

two consecutive group-nodes cannot under any circumstances have a cell in common

a consequence of this:

"(r8c8 r9c8)--(r8c8 r8c9)"
This is not a weak link, which would mean, that if a digit is on one side, it cannot be on the other side. But r8c8 is on both sides.

?

Alternatively, could you give me an example of a cycle with two consecutive group-nodes sharing a cell?

EDIT:

To be more specific, if what I have assumed is true, then in the following case:



ignoring singles-nodes cases, we can only have the following 2 cases:

[Pat]

you recently asked:

—so we can conclude that two consecutive group-nodes cannot under any circumstances have a cell in common, right?

and got a response from yzfwsf;
if this response needs clarification,
i do hope someone finds the right words.

[Mino21]

Maybe I don't understand that response cause I'm stupid, but believing something without a logical explanation is equivalent to believing a religious dogma.

I hope someone can answer the questions posed in my previous message.

[Cenoman]

Definition of a strong link:
A, B are (sudoku) logical variables (i.e. they can be True or False), typically candidates, groups of candidates, sets of candidates (pairs, triples, quads,...)

They are in a strong link if at least one is TRUE.
They are in a weak link if at most one is TRUE.
A strong link is equivalent to the logical equation A OR B = TRUE
A weak link is equivalent to the logical equation A NAND B = TRUE

In Eureka notation, a strong link is written A = B, a weak link is written A - B (symbols '=' and '-' are used in a fully different way than in maths)

Code: Select all

     c1  c2  c3 
   +---+---+---+
r1 | x | x | x |
r2 |   | x |   |
r3 |   | x |   |
   +---+---+---+

The configuration of five candidates of the same digit in a box (I assume it to be box 1) that you consider repeatedly in this thread is that of the Empty Rectangle: in this configuration, you can write three strong links (encompassing the five candidates)
Xr1c123 = Xr123c2
Xr1c13 = Xr123c2
Xr1c123 = Xr23c2

but only two weak links (encompassing the five candidates)
Xr1c123 - Xr23c2
Xr1c13 - Xr123c2

The assertion

two consecutive group-nodes cannot under any circumstances have a cell in common

is false. As shown above, two consecutive group-nodes that are in a STRONG LINK can have a cell in common, but two consecutive group-nodes that are in a WEAK LINK must be disjoint.

[Mino21]

A strong link is equivalent to the logical equation A OR B = TRUE
A weak link is equivalent to the logical equation A NAND B = TRUE

About strong link, why OR and not XOR?
In x-cycles (not grouped) a strong link means only 2 candidates in a unit, and this implies that one is true and the other is false, they cannot be both true.

[P.O.]

hi Mino21,
in a weak group link the value considered is eliminated, you have full knowledge, you can do what you want with another value;
in a strong group link, one of the value considered is true, you don't known which one, you don't have full knowledge, you can't do anything with it.