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[ArkieTech]

Code: Select all

 *-----------*
 |6..|314|...|
 |93.|...|...|
 |5..|.8.|.3.|
 |---+---+---|
 |..5|..1|.7.|
 |.1.|7.6|.9.|
 |.9.|5..|2..|
 |---+---+---|
 |.7.|.5.|..1|
 |...|...|.63|
 |...|438|..9|
 *-----------*


Play/Print this puzzle online

[SteveG48]

Code: Select all

 *--------------------------------------------------------------------*
 | 6     b28     278    | 3      1      4      | 9     b58     2578   |
 | 9      3      1478   | 2      6      5      | 148    48     478    |
 | 5     c24     124    | 9      8      7      | 146    3      246    |
 *----------------------+----------------------+----------------------|
 | 234   c246    5      | 8      9      1      | 346    7      46     |
 | 348    1      348    | 7      2      6      | 3458   9      458    |
 | 7      9      8-6    | 5      4      3      | 2      1      68     |
 *----------------------+----------------------+----------------------|
 | 2348   7      23489  | 6      5      29     | 48     248    1      |
 | 248    2458   2489   | 1      7      29     | 458    6      3      |
 | 1      25-6  a26     | 4      3      8      | 7     a25     9      |
 *--------------------------------------------------------------------*


(6=25)r9c38 - (5=28)r1c28 - (2=46)r34c2 => -6 r6c3,r9c2 ; stte

Typo fixed. Thanks, Bat.

[Leren]

Code: Select all

*--------------------------------------------------------------*
| 6    a28    278    | 3     1     4      | 9    e5-8   2578   |
| 9     3     1478   | 2     6     5      | 148   48    478    |
| 5     24    124    | 9     8     7      | 146   3     246    |
|--------------------+--------------------+--------------------|
| 234   246   5      | 8     9     1      | 346   7     46     |
| 348   1     348    | 7     2     6      | 3458  9     458    |
| 7     9     68     | 5     4     3      | 2     1     68     |
|--------------------+--------------------+--------------------|
| 2348  7     23489  | 6     5     29     | 48    248   1      |
| 248  b2458  2489   | 1     7     29     |c458   6     3      |
| 1     256   26     | 4     3     8      | 7    d25    9      |
*--------------------------------------------------------------*

L2 wing : (8) r1c2 = (8-5) r8c2 = r8c7 - r9c8 = (5) r1c8 => - 8 r1c8; stte

Leren

[pjb]

Code: Select all

 6      b28      278    | 3      1      4      | 9     a58     2578   
 9       3       1478   | 2      6      5      | 148    48     478   
 5       24      124    | 9      8      7      | 146    3      246   
------------------------+----------------------+---------------------
 234     246     5      | 8      9      1      | 346    7      46     
 348     1       348    | 7      2      6      | 3458   9      458   
 7       9       68     | 5      4      3      | 2      1      68     
------------------------+----------------------+---------------------
 2348    7       23489  | 6      5      29     | 48     248    1     
 248    c2458    2489   | 1      7      29     |d458    6      3     
 1       256     26     | 4      3      8      | 7      2-5    9     


(5=8)r1c8 - r1c2 = (8-5)r8c2 = r8c7 => -5 r9c8; stte
This is very similar to above solutions, so
(8=6)r6c3 - (6=8)r134c2 => -8 r12c3; stte

Phil

[Marty R.]

Code: Select all

+-----------------+--------+---------------+
| 6    28   278   | 3 1 4  | 9    58  2578 |
| 9    3    1478  | 2 6 5  | 148  48  478  |
| 5    24   124   | 9 8 7  | 146  3   246  |
+-----------------+--------+---------------+
| 234  246  5     | 8 9 1  | 346  7   46   |
| 348  1    348   | 7 2 6  | 3458 9   458  |
| 7    9    68    | 5 4 3  | 2    1   68   |
+-----------------+--------+---------------+
| 2348 7    23489 | 6 5 29 | 48   248 1    |
| 248  2458 2489  | 1 7 29 | 458  6   3    |
| 1    256  26    | 4 3 8  | 7    25  9    |
+-----------------+--------+---------------+


Play this puzzle online at the Daily Sudoku site

(8=2)r1c2-(2=468)r34c2,r6c3=> -8r12c3

Something seems strange here. I can't see a flaw in my chain, which ends with r6c3=8. Looking at it another way, if r1c2<>8, then r12c3=8 and r6c3 cannot be=8. What's the explanation for this apparent contradiction?

[Leren]

Marty R wrote : What's the explanation for this apparent contradiction?

The "apparent contradiction" stems from the fact that the original premise (that r1c2 <> 8) turns out to be False ie r1c2 = 8 in the solution.

You could have written this as a discontinuous loop that starts and ends on a Strong link (8=2) r1c2 - (2=6) r34c2 - (6=8) r6c3 - r12c3 = (8) r1c2 => r1c2 = 8.

You could call this a contradiction chain because the conclusion "contradicts" the premise. ie if you assume X is 'False and prove that X is True by a chain, then X is True in the solution.

Here is a link to Andrew Stuart's site where he describes this situation under the heading Nice Loops Rule 2. http://www.sudokuwiki.org/Alternating_Inference_Chains

Also your notation was a bit confusing. It looks like there are 2 ALSs in the chain but in fact there are 3. (8=2) r1c2- (2=46) r34c2 - (6=8) r6c3=> - 8 r12c3 would be clearer.

Leren

[bat999]

Code: Select all

.---------------------.----------.------------------.
| 6     d28     278   | 3  1  4  | 9     c58   2578 |
| 9      3      1478  | 2  6  5  | 148    48   478  |
| 5      24     124   | 9  8  7  | 146    3    246  |
:---------------------+----------+------------------:
| 234    246    5     | 8  9  1  | 346    7    46   |
| 348    1      348   | 7  2  6  | 3458   9    458  |
| 7      9      68    | 5  4  3  | 2      1    68   |
:---------------------+----------+------------------:
| 2348   7      23489 | 6  5  29 | 48     248  1    |
| 248   e248-5  2489  | 1  7  29 | 458    6    3    |
| 1     a256    26    | 4  3  8  | 7     b25   9    |
'---------------------'----------'------------------'

(5)r9c2 = r9c8 - (5=8)r1c8 - r1c2 = (8)r8c2 => -5 r8c2; stte

[Marty R.]

Leren

Thank you very much, I appreciate the help.

Also your notation was a bit confusing. It looks like there are 2 ALSs in the chain but in fact there are 3. (8=2) r1c2- (2=46) r34c2 - (6=8) r6c3=> - 8 r12c3 would be clearer.

OK, but I don't understand why you're treating 8=2 and 6=8 as ALS's, even though they are, technically speaking.

(2=468)r34c2,r6c3

I think you're saying this is not a valid ALS because it includes cells from two houses.



Thanks again,
Marty

[Leren]

Hi Marty,

Here are two links to definitions of ALSs that require their cells to be within a single house.

http://sudopedia.enjoysudoku.com/Almost_Locked_Set.html

http://hodoku.sourceforge.net/en/tech_als.php

Leren

[ronk]

Marty R wrote : What's the explanation for this apparent contradiction?

The "apparent contradiction" stems from the fact that the original premise (that r1c2 <> 8) turns out to be False ie r1c2 = 8 in the solution.

You could have written this as a discontinuous loop that starts and ends on a Strong link (8=2) r1c2 - (2=6) r34c2 - (6=8) r6c3 - r12c3 = (8) r1c2 => r1c2 = 8.

You could call this a contradiction chain because the conclusion "contradicts" the premise. ie if you assume X is 'False and prove that X is True by a chain, then X is True in the solution.

Can you argue the case without knowing anything about ultimate values in the solution?

[Leren]

ronk wrote : Can you argue the case without knowing anything about ultimate values in the solution?

Yes. The reason is that discontinuous AICs fall within a more general construction that I might call a Kraken candidate, which i might write as

Code: Select all

1. Assume X rAcB is True.  Apply some AIC1   => YrCcD has some status (True or False).

2. Assume X rAcB is False. Apply some AIC2   => YrCcD has some status (True or False).

If the status of Y is the same for 1 and 2 (either True or False) then it must have that status in the solution. This does not depend on any knowledge of what the status of X is in the solution, because 100% of its possible statuses have been considered.

In the case of discontinuous loops, X = Y, A = C and B = D.

For loops starting and ending on Weak links the Kraken candidate construction looks like

Code: Select all

1. Assume X rAcB is True. Apply some AIC1   => X rAcB is False (AIC1 is a "contradiction" chain).

2. Assume X rAcB is False                   => X rAcB is False (AIC2 is trivial, in fact it is a tautology).

For loops starting and ending on Strong links the Kraken candidate construction looks like

Code: Select all

1. Assume X rAcB is True.                    => X rAcB is True (AIC1 is trivial, in fact it is a tautology).

2. Assume X rAcB is False. Apply some AIC2   => X rAcB is True (AIC2 is a "contradiction" chain).

So I think it's valid to conclude that in the case of a "contradiction" chain, the status of X in the solution is the opposite of the assumed status at the start of the chain, but this conclusion is not based on knowledge of what the status of X really is in the solution, although the language one uses may appear to suggest that, because in practice, the trivial leg of the Kraken candidate construction is usually left out in the notation (but not in the underlying Kraken candidate logic).

Leren

[DonM]

Code: Select all

+-----------------+--------+---------------+
| 6    28   278   | 3 1 4  | 9    58  2578 |
| 9    3    1478  | 2 6 5  | 148  48  478  |
| 5  [2]4*  124   | 9 8 7  | 146  3   246  |
+-----------------+--------+---------------+
| 234[2]46* 5     | 8 9 1  | 346  7   46   |
| 348  1    348   | 7 2 6  | 3458 9   458  |
| 7    9    68*   | 5 4 3  | 2    1   68   |
+-----------------+--------+---------------+
| 2348 7    23489 | 6 5 29 | 48   248 1    |
| 248  2458 2489  | 1 7 29 | 458  6   3    |
| 1    256  26    | 4 3 8  | 7    25  9    |
+-----------------+--------+---------------+


(8=2)r1c2-(2=468)r34c2,r6c3=> -8r12c3

Okay, having gone through a discussion of this same issue several years ago, I'll bite. Regardless of past definitions of an ALS (same house etc.) and regardless of the (IMO) somewhat obscure reasons given in above posts why Marty's solution is flawed: In what way does 468 in r34c2, r6c3 not operate exactly the same as a locked set? The 4, 6, and 8 can't go anywhere else. To me, this is the only subject at hand.

If purists can't bring themselves to call this construct an ALS, then they can give it another name, but IMO, unless it can be proved that the above does not operate as a locked set, then Marty's solution, as expressed, is valid.

Fwiw: I have always had the highest respect for Andrew Stuart, but unfortunately he has caused unnecessary confusion by using the term 'Nice Loops Rules' while using AIC examples. Having said that, since Stuart's rules were referred to, one can also look at Marty's solution as an example of the so-called Nice Loops Rule 2 whereby a strong link starts and ends at (8)r1c2 thus proving r1c2=8.

[Luke]

Careful, Don and Marty, you are flirting with heresy!

[DonM]

Careful, Don and Marty, you are flirting with heresy!

L-u-k-e, beware the dark side of The Force! (And you have a good memory of the days of the so-called 'bent' ALS. )

[ronk]

ronk wrote : Can you argue the case without knowing anything about ultimate values in the solution?

Yes. The reason is that discontinuous AICs fall within a more general construction that I might call a Kraken candidate, ...

Thanks for your very complete answer, a little lengthy but complete. I rarely ask questions that require a lengthy response.

Moreover, I apologize for not providing the correct quote, which happened because I didn't realize the topic changed between here and here. My bad.

Code: Select all

+-----------------+--------+---------------+
| 6    28   278   | 3 1 4  | 9    58  2578 |
| 9    3    1478  | 2 6 5  | 148  48  478  |
| 5    24   124   | 9 8 7  | 146  3   246  |
+-----------------+--------+---------------+
| 234  246  5     | 8 9 1  | 346  7   46   |
| 348  1    348   | 7 2 6  | 3458 9   458  |
| 7    9    68    | 5 4 3  | 2    1   68   |
+-----------------+--------+---------------+
| 2348 7    23489 | 6 5 29 | 48   248 1    |
| 248  2458 2489  | 1 7 29 | 458  6   3    |
| 1    256  26    | 4 3 8  | 7    25  9    |
+-----------------+--------+---------------+


(8=2)r1c2-(2=468)r34c2,r6c3=> -8r12c3

Something seems strange here. I can't see a flaw in my chain, which ends with r6c3=8. Looking at it another way, if r1c2<>8, then r12c3=8 and r6c3 cannot be=8. What's the explanation for this apparent contradiction?

It looked to me like Marty R. was not only using the chain left-to-right to get (8)r1c2 = (8)r6c3, but also right-to-left through the discontinuity (8)r12c3 to get the contradictory (8)r1c2 - (8)r6c3. Even for Marty, that's probably too crazy to be true. Again my bad.