## How do you find almost-locked set eliminations?

Advanced methods and approaches for solving Sudoku puzzles

### How do you find almost-locked set eliminations?

In an ordinary Su-Doku puzzle, there are about ninety different ALSs, and if you take two ALSs, there is the possibility that eliminations may be made. But what I really do NOT want to do is find all of them and see every possible combinations of 2 of them to see if they yield eliminations. Is there a more efficient way to find these eliminations by ALS, and if so, what is it?
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Once upon a time I was a teenager who was active on here 2007-2011
999_Springs

Posts: 367
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

Well, it looks like I will have to move ALSs three or four places lower in my solving hierarchy then...
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Once upon a time I was a teenager who was active on here 2007-2011
999_Springs

Posts: 367
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### ALS eliminations

I hear what you're saying, and have been working recently on a possible ALS application strategy. I've only had time to try it (successfully) so far on one relatively tough puzzle (SudoCue Nightmare).
Could you supply a puzzle of your own choice for a second trial? I need the full grid position at the point where one would normally begin to look for ALS-rule or other grouped-AIC applications.
Sudtyro

Posts: 68
Joined: 21 December 2006

Carcul is one of the masters at finding the useful ALS. His favorites were those that had digits that could be isolated in different houses. You could form a strong link between those digits and likely extend both ends of the chain to find a reduction.

Cells which look like they might be a fertile ground for an APE or Sue-de-Coq deduction will also work for ALS as well

If three cells in a house have two digits locked in them, the remaining unsolved cells in that house will form an ALS you might want to check out. This is actually true for any Almost Hidden Set.

Other than that, I usually do not look for an ALS initially. I usually try to find one to extend a simpler chain that I have already started.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Here are a bunch of ALS xz-rule and ALS mutual exclusion rule (mer) solvable puzzles from the zoo. Note they require locked sets and only one ALS to solve the puzzle.

#34 VWXYZ-wing
...97...3.....18..97.8.4.1...97.2.6.13......928..1..7....6.8....61......4......3. #34.5 (from Ruud - A=1 cell ALS xz-rule)
182..............1.7..........36..7..9...5.6..2......8....4..5.8.1.9.7...37.8.2.. #34.1 (6-node XY-chain)
.....56.2.3..98.17..9........1..3...2..4...8....8......1.5.....3...42.5...6...9.1 #34.3~(A=1 cell ALS xz-rule)
.....1.5782..........98.6........93..518.......4..5....1...2.9.6.7.....4...7..... #34.4~(A=1 cell ALS xz-rule)

#39 UVWYXZ-wing
...3.1..7..2...39........5....1...2...3..64..96..2..7.47.....1.6..8.5...2.9...... #39.5
1...4...7........1.....3.258.....4...6.8.9....25..1.......3..5...4.5.7.66...8.... #39.1~(A=1 cell ALS xz-rule)
5..71....1........9.8...5...6...7.9...9.3..1......27.........83....56..4..24....5 #39.2~(A=1 cell ALS xz-rule)
.6.81.95.......8..8..54.........3..76.5...........423...2..6.1...........51.2...4 #39.3>(A=1 cell ALS xz-rule)
.1...53...582...46...7.......75.......2.978.43....6..5....8.9...3.........93....2 #39.4>(A=1 cell ALS xz-rule)

#40 SueDeCoq
9....58..156.....4..2..4..6....57.....12.8.7.......56.........3.7....94.4.3....8. #40.4~(A=1 cell ALS xz-mer)
3.1...6.....3.79...5.92........18.4..6.4.....4.......57.6...3..........852..3...7 #40.3~(A=1 cell ALS xz-mer)
.5..7.2.4.31.....6...63..5..1...5..8..71..4........91.6.54..8......2....78....... #40.1>(A=1 cell ALS xz-mer)
....2.41....19.8..92..5...3..2.....1.........8.7...95..43.7.......6....7.8..32.6. #40.2>(A=1 cell ALS xz-mer)

#41 ALS xz-rule
2.....9....8.49....1....78.79.1..85.....6.......9....282...3.....3.8..47.4.6..1.. #41.1
...6.5........468...3....51.7.38...242....3.....1..8...3..2..........9.67...56... #41.3~

#41 ALS xz-mutual exclusion rule
78..5.1..........9...8...5.57.......6...4...3..83...7132...7............1.596.... #41.7~
9.7..4......6..8.......9.2.......9....3....42.6...25...9.1..3.5.5...81..6......7. #41.6~
...2...8.6....5..9.5....3...2.......4..68..9.....7.5.4.1.96.8...9.....4.376...... #41.5~
..5.6..2....9..48..9...4......2.6..39.35...........91.5.2..81....6......3........ #41.4>(2-element grouped nice loop)
Mike Barker

Posts: 458
Joined: 22 January 2006

sudtyro wrote:grouped-AIC applications

Myth Jellies wrote:APE

What are they?

Also, to start with, are there any puzzles with very large numbers of clues which need an ALS to solve (not counting XYZ-wings)? They will increase the chance of my finding my first ever ALS elimination.
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Once upon a time I was a teenager who was active on here 2007-2011
999_Springs

Posts: 367
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

All of the puzzles I listed require only naked, hidden, and/or locked sets and one ALS move. Here's 41.6 at the point of the ALS (and yes there are other possible techniques). Note there are 6 eliminations possible from the xz-mutual exclusion rule.
Code: Select all
`+----------------------+----------------------+----------------+|     9   1238      7  |  2358   12358     4  | 6   135    13  ||  1345   1234    145  |     6   12357  1357  | 8  1359  1379  ||  1358    138      6  |  3578   13578     9  | 4     2   137  |+----------------------+----------------------+----------------+| 12458      7  12458  |  3458   13458   135  | 9  1368  1368  ||   158     18      3  |   589   15689   156  | 7     4     2  ||   148      6      9  |  3478   13478     2  | 5   138   138  |+----------------------+----------------------+----------------+|  2478      9    248  |     1    2476    67  | 3    68     5  ||  3724      5     24  | 23479  234679     8  | 1    69   469  ||     6   1384    184  |  3459    3459    35  | 2     7   489  |+----------------------+----------------------+----------------+`

Here's 41.1 with more solved cells:
Code: Select all
`+----------------+-------------------+---------------+|   2  357   45  |  378    1   5785  |   9   6  345  ||   6  357    8  |   37    4      9  |  35   2    1  ||  34    1    9  |    23  235     6  |   7   8  345  |+----------------+-------------------+---------------+|   7    9   26  |    1   23      4  |   8   5   36  || 134  358  245  | 2378    6    578  |  34  17    9  || 134  358  456  |    9   35    578  | 346  17    2  |+----------------+-------------------+---------------+|   8    2    1  |    4    7      3  |  56   9   56  ||   9    6    3  |    5    8      1  |   2   4    7  ||   5    4    7  |    6    9      2  |   1   3    8  |+----------------+-------------------+---------------+`

APE is aligned pair exclusion and grouped AIC are Alternating Inference Chains using grouped strong links and/or ALS. I've tried to provide pointers to many techniques here or check out Sudopedia or one of the solving guides listed in the collection.
Mike Barker

Posts: 458
Joined: 22 January 2006

It might be better to start with smaller ALS so here are some XYZ-wings and WXYZ-wings. The beauty of these is that you can use the bivalue cell to help find the ALS. The example is from 30.2.

#13 XYZ-wing
..4....87..38.2..6.6...5....5..9.1.....3.........1.8.2..5....6....1267....1...9.. #13.1
...8..6...1.9..82...56..3...52.....38....4.1.....5.2..2.....1....64.....5...27..4 #13.2
2.9...7...8..49..............41...9..639..4..5.82......3281..........6.5...43...1 #13.3 (BUG+2X)

#30 WXYZ-wing
..96...5......12.7....9.......83....1..9...856..2.4.....3..8...5.......6.17.....4 #30.1 (A=1 cell ALS xz-rule)
.....8..5..7.....1.....286....3..5.7.1...9..46..4..9......5.....4.1..3.68...2.... #30.2 (5-node XY-chain)
7.2....15.1.62.9.....5........38.6....5..7.........327.2.95..83.........1......9. #30.3 (5-node XY-chain)

Code: Select all
`+---------------+-------------+-----------------+| 123   36  46  | 9  134   8  |  2474   474  5  ||  23    8   7  | 5   34   6  |    24     9  1  ||  19    5  49  | 7   14   2  |     8     6  3  |+---------------+-------------+-----------------+|   4    9   8  | 3    6   1  |     5     2  7  ||   7    1   5  | 2    8   9  |     6     3  4  ||   6    2   3  | 4    7   5  |     9     1  8  |+---------------+-------------+-----------------+|  39  367  69  | 8    5  34  |     1    47  2  ||   5    4   2  | 1    9   7  |     3     8  6  ||   8   37   1  | 6    2  34  |    47     5  9  |+---------------+-------------+-----------------+`
Mike Barker

Posts: 458
Joined: 22 January 2006

### Re: How do you find almost-locked set eliminations?

999_Springs wrote:In an ordinary Su-Doku puzzle, there are about ninety different ALSs, and if you take two ALSs, there is the possibility that eliminations may be made. ... Is there a more efficient way to find these eliminations by ALS, and if so, what is it?

In case you'd like to follow up on this topic, I just today posted a discussion of a simplified method to find ALS-based eliminations:
http://www.sudocue.net/forum/viewtopic.php?t=726
It also contains links to reference material that could possibly help with your other questions.

Myth Jellies wrote:Carcul is one of the masters at finding the useful ALS. His favorites were those that had digits that could be isolated in different houses. You could form a strong link between those digits and likely extend both ends of the chain to find a reduction.

I'm not familiar with Carcul's "favorite" ALS's, but they may very well be similar to the "conditional" ALS's I use in my own approach.
Sudtyro

Posts: 68
Joined: 21 December 2006