yzfwsf wrote:denis_berthier wrote:z-chain[5]: c6n5{r5 r3} - r1c5{n5 n2} - r1c9{n2 n8} - b9n8{r7c9 r9c7} - c7n5{r9 .} ==> r5c9 ≠ 5

Your chain symbol will hide the details.

Absolutely correct. You can freely ignore Denis' denials and distortions. They have nothing to do with the reality.

I have the same chain in my solver, which looks very complicated.

The pattern you displayed is essentially the same, but not exactly. Since Denis can't or won't accept such essential equivalences, you should stick to exact ones in order to preempt his false claims. It won't stop him, of course, but at least it removes any last hint of credibility from those claims (at least for those who are capable of checking and understanding the facts).

Cell Forcing Chain: Each candidate in r1c9 true in turn will all lead r5c9<>5

2r1c9 - r1c5 = (2-5)r3c6 = r5c6 - 5r5c9

5r1c9 - 5r5c9

8r1c9 - r7c9 = (8-5)r9c7 = r4c7 - 5r5c9

There are two slight problems with that if you want to compare it with Denis' z-chain fully accurately. First, one of its csp-variables (SIS) is different, so it's not depicting exactly the same pattern (even though it effectively does). Second, it's written as a kraken (verity) instead of a contradiction net (which is what Denis' z-chains are), which of course is just a different perspective for the same thing. (Interestingly, Denis thinks contradictions are more elegant than "reasoning by cases", while most of the rest of us see it the other way around in the sudoku context.)

Both are inconsequential differences that shouldn't make any difference to anyone who understands such equivalences, but in this case that can't be assumed, because anything you say can and will be used against you (or at least against me).

Your kraken translates to this triangular matrix (5x5 TM):

- Code: Select all
`c6n5| 5r5c6 5r3c6`

b2n2| . . . 2r3c6 2r1c5

r1c9| 5r1c9 . . . 2r1c9 8r1c9

b9n8| . . . . . . . . . 8r7c9 8r9c7

c7n5| 5r4c7 . . . . . . . . . 5r9c7

===================================

-5r5c9

The only difference between that and Denis' z-chain is the second SIS (csp-variable) which should be r1c5 instead of b2n2. The exact matrix (5x5 TM) for his z-chain:

- Code: Select all
`c6n5| 5r5c6 5r3c6`

r1c5| . . . 5r1c5 2r1c5

r1c9| 5r1c9 . . . 2r1c9 8r1c9

b9n8| . . . . . . . . . 8r7c9 8r9c7

c7n5| 5r4c7 . . . . . . . . . 5r9c7

===================================

-5r5c9

There are several perspectives to see and write the logic contained in that TM. Denis' z-chain is one of them, depicting it as a contradiction net (but like you said, hiding essential details making it look simpler than it is). Your kraken-cell is another, if the one csp-variable is changed:

2r1c9 - (2=5)r1c5 - r3c6 = r5c6 - 5r5c9

5r1c9 - 5r5c9

8r1c9 - r7c9 = (8-5)r9c7 = r4c7 - 5r5c9

However, to make it match more closely the logic of the z-chain, it should be written as a contradiction net. Simply flipping it around won't do, though, because that would put the contradiction in the cell r1c9 (which of course works too, but it's not the same):

- Code: Select all
`[!]r1c9`

||

.- (5)r5c6 = r3c6 - (5=2)r1c5 - (2)r1c9

/ ||

(5)r5c9 -------------------------------- (5)r1c9

\ ||

`- (5)r4c7 = (5-8)r9c7 = r7c9 - (8)r1c9

=> -5r5c9

In the z-chain the contradiction is in the column 5c7. Thus, the exactly equivalent contradiction net is this:

- Code: Select all
`.------------------------- (5)r1c9`

/ ||

(5)r5c9 - r5c6 = r3c6 - (5=2)r1c5 - (2)r1c9 (5)r[!]c7

\ || ||

\ (8)r1c9 - r7c9 = (8-5)r9c7

\ ||

'-------------------------------------------(5)r4c7

=> -5r5c9

In any case, the TM is the most neutral way to depict the pattern, allowing many different interpretations. Yet no matter how it's read, the result is a net of some kind. It doesn't change anything that Denis refuses to accept that fact and insists on calling it a simple chain that is "in essence" equivalent to a basic AIC. That's ridiculous, of course, even in the case of of such a simple z-chain with just one z-candidate. The simplest ways to write it as an AIC require nesting:

(5)r1c9 = [(5)r5c6 = r3c6 - (5=2)r1c5 - (2=8)r1c9 - r7c9 = (8-5)r9c7 = (5)r4c7] => -5 r5c9

or:

(5)r5c6 = r3c6 - (5=2)r1c5 - r1c9 = [(5=8)r1c9 - r7c9 = (8-5)r9c7 = (5)r4c7] => -5 r5c9

That's not trivial logic, and writing it as a z-chain doesn't change that fact in any way. It only hides the complexity by abstracting away the z-candidate (5r1c9).

PS. Denis, you can freely spread your ridiculous lies and red warnings about me as much as you like. Anyone with a few brain cells should easily see who's speaking the truth and who's gaslighting. With that latest stunt you lost the last bit of respect I had for you. I don't think I'm the only one either. (I don't much care about which way the leadership of this forum sees it. I'm fully prepared to get kicked out, if that's what it takes to defend the truth and take a stand against your all-encompassing and toxic narcissism.)