The New Sudoku Players' Forum

by **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

by **StrmCkr** » Tue Nov 28, 2006 7:48 am

removed

by **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.

by **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...

Havard

by **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.

by **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.]

by **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.

by **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

The New Sudoku Players' Forum

Combined Overlapping Almost Locked Sets Rule (CoALS?)

check this out.