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[ArkieTech]

Code: Select all

 *-----------*
 |...|...|5..|
 |...|...|.64|
 |81.|...|9..|
 |---+---+---|
 |.2.|..8|..9|
 |...|7.1|...|
 |.34|.5.|...|
 |---+---+---|
 |..1|.37|...|
 |..6|8..|.2.|
 |...|...|148|
 *-----------*


Play/Print this puzzle online

[SteveG48]

Code: Select all

 *--------------------------------------------------------------------*
 | 46     46     2379   | 29     2789   239    | 5      1378   1237   |
 | 23579  579    23579  | 1      2789   2359   | 2378   6      4      |
 | 8      1      2357   | 2456   267    23456  | 9      37     237    |
 *----------------------+----------------------+----------------------|
 | 157-6  2     a57     | 3     a46     8      |a47     157    9      |
 |b569   b569    8      | 7      2469   1      | 234    35     236    |
 | 1679   3      4      | 269    5      269    | 278    178    1267   |
 *----------------------+----------------------+----------------------|
 | 24     8      1      | 24     3      7      | 6      9      5      |
 | 459    459    6      | 8      1      459    | 37     2      37     |
 | 23579  579    23579  | 2569   269    2569   | 1      4      8      |
 *--------------------------------------------------------------------*


(6=457)r4c357 - (5=69)r5c12 => -6 r4c1 ; lclste

[pjb]

Code: Select all

  46     46     2379   | 29     2789   239    | 5      1378   1237   
  23579  579    23579  | 1      2789   2359   | 2378   6      4     
  8      1      2357   | 2456   267    23456  | 9      37     237   
  ---------------------+----------------------+---------------------
bd1567   2     b57     | 3     e6-4    8      |a47     157    9     
  569    569    8      | 7     f2469   1      | 23-4   35     236   
 c1679   3      4      | 269    5      269    | 278    178    1267   
  ---------------------+----------------------+---------------------
  24     8      1      | 24     3      7      | 6      9      5     
  459    459    6      | 8      1      459    | 37     2      37     
  23579  579    23579  | 2569   269    2569   | 1      4      8     


(4=7)r4c7 - r4c13 = (7-1)r6c1 = (1-6)r4c1 - (6-4)r4c5 = r5c5 => -4 r4c5, r6c7; lclste

Phil

[Leren]

Code: Select all

*--------------------------------------------------------------*
| 46    46    2379   | 29    2789  239    | 5     1378  1237   |
| 23579 579   23579  | 1     2789  2359   | 2378  6     4      |
| 8     1     2357   | 2456  267   23456  | 9     37    237    |
|--------------------+--------------------+--------------------|
| 1567  2    a57     | 3     46    8      | 4-7  a157   9      |
| 569   569   8      | 7     2469  1      | 234   35    236    |
| 1679  3     4      |b269   5    b269    |b278  b178  b1267   |
|--------------------+--------------------+--------------------|
| 24    8     1      | 24    3     7      | 6     9     5      |
| 459   459   6      | 8     1     459    | 37    2     37     |
| 23579 579   23579  | 2569  269   2569   | 1     4     8      |
*--------------------------------------------------------------*

ALS XZ Rule: X = 1, Z = 7: (7=1) r4c38 - (1=7) r6c46789 => - 7 r4c7; lclste

Leren

[Marty R.]

Code: Select all

+-----------------+-----------------+----------------+
| 46    46  2379  | 29   2789 239   | 5    1378 1237 |
| 23579 579 23579 | 1    2789 2359  | 2378 6    4    |
| 8     1   2357  | 2456 267  23456 | 9    37   237  |
+-----------------+-----------------+----------------+
| 1567  2   57    | 3    46   8     | 47   157  9    |
| 569   569 8     | 7    2469 1     | 234  35   236  |
| 1679  3   4     | 269  5    269   | 278  178  1267 |
+-----------------+-----------------+----------------+
| 24    8   1     | 24   3    7     | 6    9    5    |
| 459   459 6     | 8    1    459   | 37   2    37   |
| 23579 579 23579 | 2569 269  2569  | 1    4    8    |
+-----------------+-----------------+----------------+


Play this puzzle online at the Daily Sudoku site

DP (29) r16c46: 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r6c46<>6

[daj95376]

_

FWIW: two more [band 2] variants (from my solver).

Code: Select all

 +-----------------------------------------------------------------------+
 |  46     46     2379   |  29     2789   239    |  5      1378   1237   |
 |  23579  579    23579  |  1      2789   2359   |  2378   6      4      |
 |  8      1      2357   |  2456   267    23456  |  9      37     237    |
 |-----------------------+-----------------------+-----------------------|
 |  1567   2      57     |  3      46     8      |  47     157    9      |
 |  569    569    8      |  7      2469   1      |  234    35     236    |
 |  1679   3      4      |  269    5      269    |  278    178    1267   |
 |-----------------------+-----------------------+-----------------------|
 |  24     8      1      |  24     3      7      |  6      9      5      |
 |  459    459    6      |  8      1      459    |  37     2      37     |
 |  23579  579    23579  |  2569   269    2569   |  1      4      8      |
 +-----------------------------------------------------------------------+
 # 113 eliminations remain

 non-ALS:  4r4c7 = (4-6)r4c5 = (6-1)r4c1 = (1-7)r6c1 = 7r6c789  =>  -7 r4c7

     ALS:  (57=1)r4c38 - (1=ALS=7)r4c13,r5c12                   =>  -7 r4c7


Marty, your conclusion doesn't follow from your assumption. However, you came close to having a discontinuous chain.

6r4c5 = r4c1 - (659=7)r5c12,r4c3 - (7=4)r4c7 - (4=6)r4c5

[Marty R.]

Marty, your conclusion doesn't follow from your assumption. However, you came close to having a discontinuous chain.

6r4c5 = r4c1 - (659=7)r5c12,r4c3 - (7=4)r4c7 - (4=6)r4c5

Danny,

Surely you know me well enough to realize that I have no idea of what a discontinuous chain is.

And I don't understand about the conclusion not following the assumption. I think my assumption is 6s in r6c46 which lead to an invalidity, ergo those two cells are not 6. Can you clarify?

[daj95376]

And I don't understand about the conclusion not following the assumption. I think my assumption is 6s in r6c46 which lead to an invalidity, ergo those two cells are not 6. Can you clarify?

DP (29) r16c46: 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r6c46<>6

Your initial assumption is r1c6<>3. I don't see any way this can lead to your conclusion of r6c46<>6.

At best, the presence of a contradiction would lead to a conclusion of r1c6=3. Unfortunately, this doesn't go very far.

BTW: I should have said "discontinuous loop" instead of "discontinuous chain". Sorry for the poor wording!

[Marty R.]

Your initial assumption is r1c6<>3. I don't see any way this can lead to your conclusion of r6c46<>6.

At best, the presence of a contradiction would lead to a conclusion of r1c6=3. Unfortunately, this doesn't go very far.

Where did I go wrong? That's a pretty common start for a DP, either r1c6=3 or r6c46=6. What should I have done if I wanted to use that DP?

P.S. Instead of 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r6c46<>6, could I have changed the conclusion to r1c6=3, as in 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r1c6=3?

[daj95376]

Where did I go wrong? That's a pretty common start for a DP, either r1c6=3 or r6c46=6. What should I have done if I wanted to use that DP?

P.S. Instead of 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r6c46<>6, could I have changed the conclusion to r1c6=3, as in 3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7 contradiction=>r1c6=3?

I'm not sure where you went wrong. Your blue logic works, but most conclusions derived from a contradiction are frowned upon.

You can turn your contradiction into a lasso that uses the UR twice and an ALS once.

Code: Select all

3r1c6=6r6c46-r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7-(4=6)r4c5-6r6c46=3r1c6
*****************                                    ###################


A lasso is just a special form of discontinuous loop. The (*) linkage is repeated in reverse order in the (#) linkage. Unfortunately, the UR logic is a hindrance in this case. Remove the URs and you get the discontinuous loop that I mentioned in a previous message.

Code: Select all

6r4c5=r4c1-(659=7)r5c12r4c3-(7=4)r4c7-(4=6)r4c5


[Marty R.]

Danny,

Thank you for your time and patience. I think I have a better understanding of that situation now.

I realize that conclusions derived from contradictions don't have the cachet that other conclusions do, but sometime one does what he needs to do in order to solve the puzzle.

Thanks again,
Marty

[ArkieTech]

Surely you know me well enough to realize that I have no idea of what a discontinuous chain is.

A chain consists of alternating inferences (aic) if you can make a loop and keep them alternating then you have a continuous loop. If in connecting a loop you have to connect two weak or two strong inferences together you have a discontinuous loop -- remove the candidate if the inferences are weak and set the candidate if they are strong.

[pjb]

I have thought that the distinction between AICs and discontinuous loops is unnecessary. AICs are often considered good logic but discontinuous loops "contradiction" and less acceptable.
In an AIC, if the starting number is true, then any same number it sees is false. If starting number is false and a remote same number at the far end of an AIC is true, then any number it sees is false, and when this number is false in both instances, it must be false.

In the case if a type II discontinuous loop, if number A is true, then as above all same numbers A it sees are false, but also other numbers B in the same call are false. If number A is false, and a number B at the end of an AIC is true, and it sees the starting cell, then the B is again false. So whether A is true or false, B is false. Why is this inferior logic?

The type I (two weak links to same number) and type III (two strong links to same number) are a bit trickier.
For type I, if it's false it's false, but if it's true, and the same number at the end of an AIC is true, and it sees the starting cell, then it's false again.
For type III, if it's true it's true, but if it's false, and the same number at the end of an AIC is false, but makes a strong link back to the starting cell, then it's true again.

Phil

[daj95376]

I have thought that the distinction between AICs and discontinuous loops is unnecessary. AICs are often considered good logic but discontinuous loops "contradiction" and less acceptable.
In an AIC, if the starting number is true, then any same number it sees is false. If starting number is false and a remote same number at the far end of an AIC is true, then any number it sees is false, and when this number is false in both instances, it must be false.

In the case if a type II discontinuous loop, if number A is true, then as above all same numbers A it sees are false, but also other numbers B in the same call are false. If number A is false, and a number B at the end of an AIC is true, and it sees the starting cell, then the B is again false. So whether A is true or false, B is false. Why is this inferior logic?

The type I (two weak links to same number) and type III (two strong links to same number) are a bit trickier.
For type I, if it's false it's false, but if it's true, and the same number at the end of an AIC is true, and it sees the starting cell, then it's false again.
For type III, if it's true it's true, but if it's false, and the same number at the end of an AIC is false, but makes a strong link back to the starting cell, then it's true again.

Hmmm!!! It appears that you are working from information on paulspages.co.uk on Nice Loops. Ouch!

First off, discontinuous loops are encompassed by AIC.

It turns out that all chains found so far which qualify as theoretical can be described as Alternating Inference Chains. XY-Wings, X-Cycles, Bivalue XY-Chains, Bilocation XY-Chains, Mixed XY-Chains, Continuous and Discontinuous Nice Loops, Dual Implication Chains, chains employing Unique Rectangles, XYZ-Wings, even the ALS XZ-Rule deductions are all Alternating Inference Chains (AICs). Furthermore, AIC's are all guaranteed to be pattern-based, theoretical, and not brute force.

As for paulspages three example types for a discontinuous loop:

Code: Select all

Type I:  assume that a candidate is true in a cell ...
         and deduce that the candidate must be false in that cell


NL     notation:  [r1c2]-1-[r3c1]=1=[r9c1]=2=[r9c8]=1=[r1c8]-1-[r1c2] => r1c2<>1

Eureka notation:  1r1c2 - r3c1 = (1-2)r9c1 = (2-1)r9c8 = r1c8 - 1r1c2 => -1 r1c2 (literal/looping format)


Eureka notation:         1r3c1 = (1-2)r9c1 = (2-1)r9c8 = 1r1c8        => -1 r1c2 (non-looping AIC format)


Code: Select all

Type II:  assume that a candidate is false in a cell ...
          and deduce that the candidate must be true in that cell


NL     notation:  [r4c2]=8=[r6c2]=6=[r6c8]-6-[r5c8]-2-[r5c5]=2=[r4c5]=8=[r4c2]   => r4c2=8

Eureka notation:  8r4c2 = (8-6)r6c1 = r6c8 - (6=2)r5c8 - r5c5 = r4c5 - (2=8)r4c2 => =8 r4c2


Eureka notation:  8r4c2 = (8-6)r6c1 = r6c8 - (6=2)r5c8 - r5c5 = r4c5 - (2=8)r4c2            (daj format)


Code: Select all

Type III:  assume that a candidate is false in a cell ...
           and deduce that another candidate must be false in that cell


NL     notation:  [r1c1]=5=[r1c5]-5-[r6c5]-9-[r6c3]-6-[r2c3]-7-[r1c1]      => r1c1<>7

Eureka notation:  5r1c1 = r1c5 - (5=9)r6c5 - (9=6)r6c3 - (6=7)r2c3 - 7r1c1 => -7 r1c1 (literal/looping format)


Eureka notation:  5r1c1 = r1c5 - (5=9)r6c5 - (9=6)r6c3 - (6=7)r2c3         => -7 r1c1 (non-looping AIC format)


The Type II scenario matches the discontinuous loop type that I use. I don't treat the Type I and Type III scenarios as discontinuous loops because I use the AIC format ... and it (technically) never loops back to the starting candidate/cell. Note: I do make an exception when discussing the presence of a lasso.

IIRC, someone uses the literal (Eureka) format for the Type i and Type III scenarios. You?

[DonM]

Surely you know me well enough to realize that I have no idea of what a discontinuous chain is.

A chain consists of alternating inferences (aic) if you can make a loop and keep them alternating then you have a continuous loop. If in connecting a loop you have to connect two weak or two strong inferences together you have a discontinuous loop -- remove the candidate if the inferences are weak and set the candidate if they are strong.

When it comes to AICs, although one could change a word here or there, the above is pretty much all one has to know except that I would add (as SteveK always did) that when you have a continuous loop, all weak links are proven strong. I especially hope that nobody gets caught up in Type this and Type that- I'm sorry, but Phil's post made my head spin. If one wants to understand AICs, best to stick with Myth Jellies original threads on the subject.