Combined Overlapping Almost Locked Sets Rule (CoALS?)

Advanced methods and approaches for solving Sudoku puzzles

Postby Steve R » Mon Nov 27, 2006 3:39 pm

Let’s go back to basics.

In terms of chains, nice or not, no difficulty arises if two nodes share cells. The same applies if the same set of cells is visited several times by the same chain. As long as each link is sound and the links are properly joined, the logic of the chain is unaffected. Programmers will evidently need to avoid the opportunity to circle endlessly in some loop.

In terms of links such as A -x- B, Myth Jellies pointed the way. All that matters is what the link means. In this case:
- if the set of cells, A, has x as an entry, B does not and
- if the set of cells, B, has x as an entry, A does not.
Suppose A and B are ALSs. Then the link represents the statement “x is a restricted common candidate of A and B.” Havard, I think, was effectively saying that a candidate for A∩B cannot be a restricted common candidate. That is true though it is probably more efficient to target the restricted common candidates then pursue a subset of those that are not.

If a nice chain starts at A and comes back to A again, you have a nice loop. Allowing for groups requires minor changes to the definition of continuity. For example, continuity at x= A =y demands |A| = 1 as well as x ≠ y. In other respects I believe the traditional pattern carries through.

I have an uneasy feeling that some of the difficulty may arise from allowing improper “links” to slip through the net. At any rate, if someone can explain where they see problems with overlap, perhaps we shall both become little wiser.

Steve
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check this out.

Postby StrmCkr » Tue Nov 28, 2006 7:48 am

removed
Last edited by StrmCkr on Sat Dec 13, 2014 6:39 am, edited 1 time in total.
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Postby Myth Jellies » Wed Nov 29, 2006 6:10 am

Mike, I gave a CoALS vs ALS AIC comparison up above. It sort of got buried, so this is a heads up in case you missed it.
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Postby Havard » Thu Dec 07, 2006 1:48 am

Found an interesting example of overlapping als:
Code: Select all
 3569     5689     68   |   79       2        1379  |  369      4        15679 
 359      7        24   |   489      1348     6     |  2389     1235     1259   
 369      1        24   |   5        3478     34789 |  23689    2367     2679   
--------------------------------------------------------------------------------
 2        356      7    |  B49       1456     149   |  3469     8        14569 
AC156    AC568    AC168 |   3       AC4568   C2489  |  7       C256     -24569 
 4        3568     9    |   278      15678    278   |  236      12356    1256   
--------------------------------------------------------------------------------
 79       249      5    |   6        478      2478  |  1        29       3     
 8        26       3    |   1        9        5     |  246      267      2467   
 1679     2469     16   |   247      347      2347  |  5        269      8     

A{[r5c125]-56-[r5c1235](56|18|4)-4-[r5c5]} -4- B{[r4c4](4|9)} -9- C{[r5c6]-9-[r5c123568](9|1248|56)-56-[r5c1258]}

A: Almost Locked Set in the cells [r5c1235] with the numbers (14568).
B: Almost Locked Set in the cells [r4c4] with the numbers (49).
C: Almost Locked Set in the cells [r5c123568] with the numbers (1245689).

[r5c9]<>5.
[r5c9]<>6.

 3569     5689     68   |   79       2        1379  |  369      4        15679 
 359      7        24   |   489      1348     6     |  2389     1235     1259   
 369      1        24   |   5        3478     34789 |  23689    2367     2679   
--------------------------------------------------------------------------------
 2        356      7    |  B49       1456     149   |  3469     8        14569 
AC156    AC568    AC168 |   3       AC4568   C2489  |  7       AC256    -249   
 4        3568     9    |   278      15678    278   |  236      12356    1256   
--------------------------------------------------------------------------------
 79       249      5    |   6        478      2478  |  1        29       3     
 8        26       3    |   1        9        5     |  246      267      2467   
 1679     2469     16   |   247      347      2347  |  5        269      8     

A{[r5c8]-2-[r5c12358](2|1568|4)-4-[r5c5]} -4- B{[r4c4](4|9)} -9- C{[r5c6]-9-[r5c123568](9|14568|2)-2-[r5c68]}

A: Almost Locked Set in the cells [r5c12358] with the numbers (124568).
B: Almost Locked Set in the cells [r4c4] with the numbers (49).
C: Almost Locked Set in the cells [r5c123568] with the numbers (1245689).

[r5c9]<>2.

All those eliminations can really be performed in one step, but the solver found the "least" complex one first...:)

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Postby Havard » Thu Dec 07, 2006 1:55 am

another one (from the same sudoku!)
Code: Select all
E3569    E5689    E68  |   AE79      2       E139   | E69       4       -1567   
 359      7        24  |    489      13       6     |  289      1235     1259   
 369      1        24  |    5       -3478    -34789 |  2689     2367     2679   
--------------------------------------------------------------------------------
 2        356      7   |   BD49      1456    D149   |  3469     8        14569 
C156     C568     C168 |    3       C4568    C2489  |  7       C256      49     
 4        3568     9   |   -278      15678    278   |  236      15       1256   
--------------------------------------------------------------------------------
 79       249      5   |    6        478      2478  |  1        29       3     
 8        26       3   |    1        9        5     |  246      267      2467   
 1679     246      16  |   -247      347      2347  |  5        69       8     

A{[r1c4](7|9)} -9- B{[r4c4](9|4)} -4- C{[r5c56]-4-[r5c123568](4|12568|9)-9-[r5c6]}
-9- D{[r4c46]-9-[r4c46](9|4|1)-1-[r4c6]} -1- E{[r1c6]-1-[r1c123467](1|35689|7)-7-[r1c4]}

A: Almost Locked Set in the cells [r1c4] with the numbers (79).
B: Almost Locked Set in the cells [r4c4] with the numbers (49).
C: Almost Locked Set in the cells [r5c123568] with the numbers (1245689).
D: Almost Locked Set in the cells [r4c46] with the numbers (149).
E: Almost Locked Set in the cells [r1c123467] with the numbers (1356789).

[r1c9 r3c56 r69c4]<>7.
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Postby ronk » Thu Dec 07, 2006 6:21 am

Havard wrote:Found an interesting example of overlapping als:
Code: Select all
 3569     5689     68   |   79       2        1379  |  369      4        15679 
 359      7        24   |   489      1348     6     |  2389     1235     1259   
 369      1        24   |   5        3478     34789 |  23689    2367     2679   
--------------------------------------------------------------------------------
 2        356      7    |  B49       1456     149   |  3469     8        14569 
AC156    AC568    AC168 |   3       AC4568   C2489  |  7       C256     -24569 
 4        3568     9    |   278      15678    278   |  236      12356    1256   
--------------------------------------------------------------------------------
 79       249      5    |   6        478      2478  |  1        29       3     
 8        26       3    |   1        9        5     |  246      267      2467   
 1679     2469     16   |   247      347      2347  |  5        269      8     

A{[r5c125]-56-[r5c1235](56|18|4)-4-[r5c5]} -4- B{[r4c4](4|9)} -9- C{[r5c6]-9-[r5c123568](9|1248|56)-56-[r5c1258]}

A: Almost Locked Set in the cells [r5c1235] with the numbers (14568).
B: Almost Locked Set in the cells [r4c4] with the numbers (49).
C: Almost Locked Set in the cells [r5c123568] with the numbers (1245689).

[r5c9]<>5.
[r5c9]<>6.
(...)
[r5c9]<>2.

Sets B and C are an excellent example of the "doubly-linked" ALS xz-rule -- meaning either x=4 and z=9 or x=9 and z=4 -- for exclusions r4c5<>4, r4c6<>4 and r4c6<>9.

Sets B and C form a continuous ALS loop which may be expressed (perhaps too cryptically) ...

-B-4-C(12568)-9-B-

... denoting digits 12568 as locked into set C for the three additional exclusions you show above. No overlap required.

[edit: That puzzle is #21 from the top1465.]
Last edited by ronk on Thu Dec 07, 2006 9:15 pm, edited 1 time in total.
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Postby Myth Jellies » Thu Dec 07, 2006 9:09 am

(1&2&5&6&8&9=4)C - (4=9)B - (9=1&2&4&5&6&8)C...AIC loop (endpoints exclude each other)

Perhaps a little less cryptic.

Nice finds, Havard. That second one is a real humdinger.
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Postby Havard » Thu Dec 07, 2006 2:33 pm

ronk wrote:Sets B and C are an excellent example of the "doubly-linked" ALS xz-rule -- meaning either x=4 and z=9 or x=9 and z=4 -- for exclusions r4c5<>4, r4c6<>4 and r4c6<>9.

Sets B and C form a continuous ALS loop which may be expressed (perhaps too cryptically) ...

-B-4-C(12568)-9-B-

... denoting digits 12568 as locked into set C for the three additional exclusions you show above. No overlap required.


I definitely agree with your POV ronk, it is a much better choice to do it that way. You get a lot more eliminations as well:
Code: Select all
 3569     5689     68  |    79       2        1379  |  369      4        15679 
 359      7        24  |    489      1348     6     |  2389     1235     1259   
 369      1        24  |    5        3478     34789 |  23689    2367     2679   
--------------------------------------------------------------------------------
 2        356      7   |   A49      -1456    -149   |  3469     8        14569 
B156     B568     B168 |    3       B4568    B2489  |  7       B256     -24569 
 4        3568     9   |    278      15678    278   |  236      12356    1256   
--------------------------------------------------------------------------------
 79       249      5   |    6        478      2478  |  1        29       3     
 8        26       3   |    1        9        5     |  246      267      2467   
 1679     2469     16  |    247      347      2347  |  5        269      8     

A{[r4c4](49|49)} -49- B{[r5c56]-49-[r5c123568](49|12568|49)-49-[r5c56]} -49- A{[r4c4](49|49)}

A: Almost Locked Set in the cells [r4c4] with the numbers (49).
B: Almost Locked Set in the cells [r5c123568] with the numbers (1245689).

[r5c9]<>2.
[r5c9]<>5.
[r5c9]<>6.
[r4c56]<>4.
[r4c6]<>9.


I was just trying to show an example of valid overlap.:)

Havard
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