The New Sudoku Players' Forum

[Ajò Dimonios]

and the answer if it is permissible to delete the 7 in r6c4

[mith]

The strong link shows either 2r4c6 or (16)r4c56 is true; in either case, -7r4c6

[Ajò Dimonios]

Hi Mith

What does not convince me of all this speech are two things:

1) The insertion of 6 in r4c5 and 1 in r4c9 (which are the real candidates in cells r4c9 and r4c5) does not lead to the immediate elimination of 7 in r4c6 but you must enter several singles before deleting the 7 in r4c6. Why?
2) What is the logical step that allows (4,5,1,6) r5c3479- (1 | 6) r4c789? If I start from (4,5,1,6) r5c3479 true I can only write (16) r4c789 false. In order to write (4,5,1,6) r5c3479- (1 | 6) r4c789 I have to assume that (4,5,1,6) r5c3479 is false and returning to the beginning of the chain assume that R4c6 = 2 is true .

However, thinking about this I think I understand that there are two completely different methods to solve a puzzle. In one case it is assumed that a candidate is eliminated and an AIC is built around it
that it determines this process (this is always possible), the logic of this construction is not as it appears at the end of the operation. Another system is the one I adopt and I prefer that of looking for a possible AIC, building it with sequential logic in which the result of a step depends on the previous one and seeing at the end which candidate can be removed.

Paolo

[mith]

1) The insertion of 6 in r4c5 and 1 in r4c9 (which are the real candidates in cells r4c9 and r4c5) does not lead to the immediate elimination of 7 in r4c6 but you must enter several singles before deleting the 7 in r4c6. Why?

Because you're looking at the wrong end of the chain. If 6r4c5 and 1r4c9 (as we know to be true from the solution), then (16)r4c56 is false. Since there is a strong link between the ends of the chain, NOT((16)r4c56) => 2r4c6, which is what eliminates the 7.

What is the logical step that allows (4,5,1,6) r5c3479- (1 | 6) r4c789? If I start from (4,5,1,6) r5c3479 true I can only write (16) r4c789 false.

The commas show the order of the digits in row 5. What the link is saying is that the following cannot both be true:

4r5c3 AND 5r5c4 AND 1r5c7 AND 6r5c9
1r4c789 OR 6r4c789

If the first is true, both the 1 and 6 are both in box 6 row 5, so neither can be in box 6 row 4. If the second is true, then at least one of 1 and 6 is in box 6 row 4, so they cannot both be in box 6 row 5.

[Ajò Dimonios]

If the first is true, both the 1 and 6 are both in box 6 row 5, so neither can be in box 6 row 4. If the second is true, then at least one of 1 and 6 is in box 6 row 4, so they cannot both be in box 6 row 5.

Of course it is true because the negation (1 | 6) r4c789 has three possibilities that answer true. What is the logical reason to also enter options where 1 is true 6 is false and 1 is false 6 is true in r4c789 if it is not needed in this implication, maybe because it is needed later? This type of choice is not sequential but is used to make it useful in the last inference. I'm not saying it's false but the reason for inserting it is hidden. This denotes that the logic is not sequential.

[Ajò Dimonios]

Because you're looking at the wrong end of the chain. If 6r4c5 and 1r4c9 (as we know to be true from the solution), then (16)r4c56 is false. Since there is a strong link between the ends of the chain, NOT((16)r4c56) => 2r4c6, which is what eliminates the 7.

I'm not saying that the AIC chain is false but that the mechanism that leads to 2 in r4c6 by eliminating 7 in r4c6 is not immediate if you look at it from outside the AIC. This is indisputable if sequential entries are made after entering 6 in r4c5 and 1 in r4c9.

[mith]

Of course it is true because the negation (1 | 6) r4c789 has three possibilities that answer true. What is the logical reason to also enter options where 1 is true 6 is false and 1 is false 6 is true in r4c789 if it is not needed in this implication, maybe because it is needed later? This type of choice is not sequential but is used to make it useful in the last inference. I'm not saying it's false but the reason for inserting it is hidden. This denotes that the logic is not sequential.

I'm not entirely clear what you are trying to say here.

(1|6)r4c789 is the same as
(1r4c789 AND -6r4c789) OR (-1r4c789 AND 6r4c789) OR (1r4c789 AND 6r4c789)

NOT((1|6)r4c789) then is the same as
NOT((1r4c789 AND -6r4c789) OR (-1r4c789 AND 6r4c789) OR (1r4c789 AND 6r4c789)) which is the same as:
NOT(1r4c789 AND -6r4c789) AND NOT(-1r4c789 AND 6r4c789) AND NOT(1r4c789 AND 6r4c789) which is the same as:
(-1r4c789 OR 6rc4c789) AND (1r4c789 OR -6r4c789) AND (-1r4c789 OR -6r4c789) =>
(-1r4c789 AND -6r4c789) =>
(1r4c56 AND 6r4c56) =>
(16)r4c56

whereas:

NOT((16)r4c789) is the same as
NOT(1r4c789 AND 6r4c789) which is the same as:
(-1r4c789 OR -6r4c789), which is not sufficient to imply (16)r4c56.

the mechanism that leads to 2 in r4c6 by eliminating 7 in r4c6 is not immediate if you look at it from outside the AIC

(6r4c5 AND 1r4c9) =>
NOT((16)r4c56) =>
2r4c6 (by the strong link) =>
-7r3c6

The elimination doesn't have to immediately lead a conclusion about which end of the chain is true. All that matters is that we know at least one end of the chain is true, and that both ends of the chain imply the elimination. Either the 2 is in r4c6 or the 1 is in r4c6 (and the 6 in r4c5), but whichever is true 7 is not in r4c6.

Code: Select all

Strong link: (A OR B)
Conditional elimination: (A => C) AND (B => C)
Antidistributive law: (A OR B) => C
Therefore: C

where here A is 2r4c6, B is (16)r4c56, and C is -7r4c6. The strong link already proves NOT(B) => A, we don't need to prove it another way to reach a conclusion about C.

[Ajò Dimonios]

I believe that the mechanism is correct but it is not sequential. In the logical implication (4,5,1,6) r5c3479- (1 | 6) r4c789 of the AIC the introduction of "|" it is neither useful nor necessary. Its use is instead necessary for the final step. By the time you add this operator you already know why it will be used later. This is the only objection I place on the resolution.

[eleven]

Btw, if that is "the official definition" and should be common knowledge, can you tell me where it's actually written like that? I'd like to have a link to point to when things like these are discussed the next time. [

Sorry, could not find something (99.99 % of AIC stuff is about unique puzzles, where it is simpler).
[Added:] but i guess you can ask anyone, who likes mathematics, and s/he will give you the same answer.

[mith]

I believe that the mechanism is correct but it is not sequential. In the logical implication (4,5,1,6) r5c3479- (1 | 6) r4c789 of the AIC the introduction of "|" it is neither useful nor necessary. Its use is instead necessary for the final step. By the time you add this operator you already know why it will be used later. This is the only objection I place on the resolution.

Ok, I understand.

Yes, both of the following are valid:

(4,5,1,6)r5c3479 - (1|6)r4c789
(4,5,1,6)r5c3479 - (16)r4c789

There is nothing wrong with the latter, as long as it then leads to a strong link that completes the AIC and provides eliminations. In this case, the first immediately leads to the valid chain (and the second does not). When going through the chain,

the first link says: "If 4, 5, 1, 6 are placed in r5c3479 in that order, neither 1 nor 6 can be placed in r4c789."
the second link says: "If 4, 5, 1, 6 are placed in r5c3479 in that order, 1 and 6 can not be placed together in r4c789."

There is nothing wrong with the second link. It's just not enough of a restriction to complete the chain with (16)r4c56 - "1 and 6 can not be placed together in r4c789" does not imply "1 and 6 are both placed in r4c56". (There may well be some other way to complete the chain that uses (16)r4c789.)

So yes, you need "|" for the strong link to be valid, and therefore you use it in the chain. There's nothing dirty or objectionable about that. The only goal of an AIC is to find two ends that lead to eliminations and that can be made to meet up through alternating links. As long as the links are valid, you can rely on the conclusion of a strong link between the ends (and therefore any resulting eliminations).

[mith]

Btw, if that is "the official definition" and should be common knowledge, can you tell me where it's actually written like that? I'd like to have a link to point to when things like these are discussed the next time. [

Sorry, could not find something (99.99 % of AIC stuff is about unique puzzles, where it is simpler).
[Added:] but i guess you can ask anyone, who likes mathematics, and s/he will give you the same answer.

I'm inclined to agree with eleven on this one - my personal definition of a strong link would be something along the lines of...

But a mathematician also writes: I would in no way guarantee that any mathematician would give the same answer, we're an ornery bunch.

[SpAce]

I am happy to weigh in on anything I understand (or think i understand).

I'm glad to hear that! I see that you already weighed in on the Aug 1 puzzle, and I appreciate that. It was clearly one of the debates begging for closure. The other two are:

May 17
Another one Trick SE 9.0 Puzzle

As you can see, those disputes are with eleven. I can understand if you don't want to get involved in them.

[mith]

Not wanting to bump a bunch of old threads:

For the May 17 one, the strong link appears to be valid, it's just not as obvious? I can't really comment on whether the notation is fine or needs something extra, but the logic is sound.
The other one looks like it will give me a headache right now. Care to summarize the dispute?

[Ajò Dimonios]

Hi Mith

Mith wrote:
There is nothing wrong with the second link. It's just not enough of a restriction to complete the chain with (16)r4c56 - "1 and 6 can not be placed together in r4c789" does not imply "1 and 6 are both placed in r4c56". (There may well be some other way to complete the chain that uses (16)r4c789.)

I would like to ask you a clarification, what are the other possibilities that could be valid when both 16 are not present in r4c789 in addition to 16r4c56? It seems to me that the only restriction is that I cannot write 16r4c789 = 16r4c56 but 16r4c789-16r4c56 because the other strong link options are prohibited by the Sudoku rules.

Paolo

[eleven]

For the May 17 one, the strong link appears to be valid, it's just not as obvious?

Oh my god, he can't stop it. The link is as valid as a link (backdoor)=(any candidate).
[Added: It's always true, even if you find a chain (backdoor false) => (candidate false).]