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Almost Locked Sets

 
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Thu Aug 31, 2006 3:12 pm    Post subject: Almost Locked Sets Reply with quote

Ruud is running another theme week on his excellent Nightmare web site. This time the theme is Almost Locked Sets.

I've read quite a bit about the "ALS" technique, but I still don't think I really know how to use it. But in working these puzzles I've noticed that the double-implication chain technique works very well. I'm hoping somebody on this forum can help me understand why that is so. Maybe there's some connection between "ALS" and "DIC" that can be made more explicit.

Anyway, here's the Nightmare from Tuesday, August 29. I got through this one with one short chain that ended in a contradiction, plus one double-implication chain, and a little bit of coloring at the very end. I'll post my solution in a day or two.
Code:
 *-----------*
 |...|...|54.|
 |.41|.9.|..2|
 |..2|4..|...|
 |---+---+---|
 |..3|..1|2..|
 |...|5..|..1|
 |.9.|.7.|..3|
 |---+---+---|
 |9..|...|4.8|
 |83.|.2.|.7.|
 |.16|...|...|
 *-----------*
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Marty R.



Joined: 12 Feb 2006
Posts: 5157
Location: Rochester, NY, USA

PostPosted: Thu Aug 31, 2006 11:22 pm    Post subject: Reply with quote

Quote:
But in working these puzzles I've noticed that the double-implication chain technique works very well. I'm hoping somebody on this forum can help me understand why that is so.


Well, that somebody certainly isn't me, but I thought I'd relate what I did with this very interesting puzzle. By the way, I don't know what an Almost Locked Set is, even though I looked at an explanation that someone pointed me to a few weeks ago.

The first non-basic thing was an X-Wing. I was stuck after that, so I solved a cell via a chain which resulted in a contradiction. After some more basic moves, a Type 4 rectangle showed up, which in turn led to a Type 1. The latter cleared things out to a point where all unsolved cells contained two candidates, except for one cell with three. Thus, killing the BUG finished things off.

To repeat, a very interesting puzzle.
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Fri Sep 01, 2006 9:13 am    Post subject: What it is Reply with quote

Marty R wrote:
... I don't know what an Almost Locked Set is, ...

Well, I can define it, Marty. My problem is how to spot it, and get something useful out of it, even when I know it's there.

A "locked set" is a pair, or a triplet, or a quad -- a set of n values that must be contained in exactly n cells, in some order.

An "almost locked set" is similar, except that there are n + 1 values that must be contained in exactly n cells. So it could be a single bi-valued cell, a pair of cells that must contain two of three possible values, and so forth.

For instance, if the possibilities in r1c1-3 are {1, 2}, {2, 3}, {1, 3} then we have a locked set -- the triplet {1, 2, 3}. But if the possibilities are {1, 2, 3}, {2, 3, 4}, {1, 2, 4} then we have the ALS {1, 2, 3, 4} in those three cells. dcb
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Fri Sep 01, 2006 10:14 am    Post subject: Reply with quote

As i said, i never found an ALS directly, but i found a chain and then constructed an ALS. For me it is a nice notation, but very hard to spot as 2 sets of almost locked sets. I give an example in this puzzle:

Code:
367    678   9    | 123678  1368   23678  | 5      4      67         
3567   4     1    | 3678    9      35678  | 368    368    2         
3567   5678  2    | 4       3568   35678  | 13689  1368   679 
----------------------------------------------------------     
567    567   3    | 689     68     1      | 2      5689   4         
24     26    8    | 5       346    369    | 7      69     1         
1      9     45   | 268     7      2468   | 68     568    3         
----------------------------------------------------------     
9      25    7    |B136    B1356  B356    | 4     B1236   8         
8      3     45   | 169     2      4569   | 169    7      569       
24     1     6    | 3789    34-58  3-5789 |A39    A23    A59         

First i found a chain to eliminate 5 from r9c5:
r9c5=5 => r8c6=4 => r8c3=5 => r8c9<>5 => r9c9=5.

When i looked around, if this can be expressed as ALS, i found this, where also 5 in r9c6 can be aliminated:
r9c56=5 => r7c456=136 => r7c8=2 => r9c8=3 => r9c7=9 => r9c9=5
=> r9c56<>5

Written as ALS:
A={2359} in r9c789, B={12356} in r7c4568, x=2, z=5
I.e. a 5 in r9c5 or r9c6 locks 2 in A to r9c8 and in B to r7c8.
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Ruud



Joined: 18 Jan 2006
Posts: 31

PostPosted: Fri Sep 01, 2006 12:30 pm    Post subject: Reply with quote

Almost Locked Sets are more abundant in a sudoku than anything else. I was amazed to see how many could be found.

This is the grid in the state in which SudoCue reports the first ALS step:

Code:
.---------------------.---------------------.---------------------.
| 367    678    9     | 123678 1368   23678 | 5      4      67    |
| 3567   4      1     | 3678   9      35678 | 368    368    2     |
| 3567   5678   2     | 4      3568   35678 | 13689  1368   679   |
:---------------------+---------------------+---------------------:
| 567    567    3     | 689    68     1     | 2      5689   4     |
| 24     26     8     | 5      346    369   | 7      69     1     |
| 1      9      45    | 268    7      2468  | 68     568    3     |
:---------------------+---------------------+---------------------:
| 9      25     7     | 136    1356   356   | 4      1236   8     |
| 8      3      45    | 169    2      4569  | 169    7      569   |
| 24     1      6     | 3789   3458   35789 | 39     23     59    |
'---------------------'---------------------'---------------------'


The question is: Where are these ALS' ?

This is the list of ALL sets found in this grid:

Code:
R1C1 R1C2 R1C4 R1C5 R1C6 v123678
R1C1 R1C2 R1C4 R1C5 R1C9 v123678
R1C1 R1C2 R1C4 R1C6 R1C9 v123678
R1C1 R1C2 R1C5 R1C6 R1C9 v123678
R1C1 R1C2 R1C5 R1C9 v13678
R1C1 R1C2 R1C6 R1C9 v23678
R1C1 R1C2 R1C9 v3678
R1C1 R1C4 R1C5 R1C6 R1C9 v123678
R1C1 R1C9 v367
R1C2 R1C4 R1C5 R1C6 R1C9 v123678
R1C2 R1C9 v678
R1C9 v67
R2C1 R2C4 R2C6 R2C7 v35678
R2C1 R2C4 R2C6 R2C8 v35678
R2C1 R2C4 R2C7 R2C8 v35678
R2C1 R2C6 R2C7 R2C8 v35678
R2C4 R2C6 R2C7 R2C8 v35678
R2C4 R2C7 R2C8 v3678
R2C7 R2C8 v368
R3C1 R3C2 R3C5 R3C6 v35678
R3C1 R3C2 R3C5 R3C6 R3C7 R3C8 v1356789
R3C1 R3C2 R3C5 R3C6 R3C7 R3C9 v1356789
R3C1 R3C2 R3C5 R3C6 R3C8 v135678
R3C1 R3C2 R3C5 R3C6 R3C8 R3C9 v1356789
R3C1 R3C2 R3C5 R3C6 R3C9 v356789
R3C1 R3C2 R3C5 R3C7 R3C8 R3C9 v1356789
R3C1 R3C2 R3C6 R3C7 R3C8 R3C9 v1356789
R3C1 R3C5 R3C6 R3C7 R3C8 R3C9 v1356789
R3C2 R3C5 R3C6 R3C7 R3C8 R3C9 v1356789
R4C1 R4C2 v567
R4C1 R4C2 R4C4 R4C5 v56789
R4C1 R4C2 R4C4 R4C8 v56789
R4C1 R4C2 R4C5 v5678
R4C1 R4C2 R4C5 R4C8 v56789
R4C1 R4C4 R4C5 R4C8 v56789
R4C2 R4C4 R4C5 R4C8 v56789
R4C4 R4C5 v689
R4C4 R4C5 R4C8 v5689
R4C5 v68
R5C1 v24
R5C1 R5C2 v246
R5C1 R5C2 R5C5 v2346
R5C1 R5C2 R5C5 R5C6 v23469
R5C1 R5C2 R5C5 R5C8 v23469
R5C1 R5C2 R5C6 R5C8 v23469
R5C1 R5C2 R5C8 v2469
R5C1 R5C5 R5C6 R5C8 v23469
R5C2 v26
R5C2 R5C5 R5C6 R5C8 v23469
R5C2 R5C6 R5C8 v2369
R5C2 R5C8 v269
R5C5 R5C6 R5C8 v3469
R5C6 R5C8 v369
R5C8 v69
R6C3 v45
R6C3 R6C4 R6C6 R6C7 v24568
R6C3 R6C4 R6C6 R6C8 v24568
R6C3 R6C4 R6C7 R6C8 v24568
R6C3 R6C6 R6C7 R6C8 v24568
R6C3 R6C7 R6C8 v4568
R6C4 R6C6 R6C7 v2468
R6C4 R6C6 R6C7 R6C8 v24568
R6C4 R6C7 v268
R6C4 R6C7 R6C8 v2568
R6C7 v68
R6C7 R6C8 v568
R7C2 v25
R7C2 R7C4 R7C5 R7C6 v12356
R7C2 R7C4 R7C5 R7C8 v12356
R7C2 R7C4 R7C6 R7C8 v12356
R7C2 R7C5 R7C6 R7C8 v12356
R7C4 R7C5 R7C6 v1356
R7C4 R7C5 R7C6 R7C8 v12356
R8C3 v45
R8C3 R8C4 R8C6 R8C7 v14569
R8C3 R8C4 R8C6 R8C9 v14569
R8C3 R8C4 R8C7 R8C9 v14569
R8C3 R8C6 R8C7 R8C9 v14569
R8C3 R8C6 R8C9 v4569
R8C4 R8C6 R8C7 R8C9 v14569
R8C4 R8C7 v169
R8C4 R8C7 R8C9 v1569
R9C1 v24
R9C1 R9C4 R9C5 R9C6 R9C7 R9C8 v2345789
R9C1 R9C4 R9C5 R9C6 R9C7 R9C9 v2345789
R9C1 R9C4 R9C5 R9C6 R9C8 R9C9 v2345789
R9C1 R9C4 R9C5 R9C7 R9C8 R9C9 v2345789
R9C1 R9C4 R9C6 R9C7 R9C8 R9C9 v2345789
R9C1 R9C5 R9C6 R9C7 R9C8 R9C9 v2345789
R9C1 R9C5 R9C7 R9C8 R9C9 v234589
R9C1 R9C7 R9C8 v2349
R9C1 R9C7 R9C8 R9C9 v23459
R9C1 R9C8 v234
R9C4 R9C5 R9C6 R9C7 R9C8 R9C9 v2345789
R9C4 R9C5 R9C6 R9C7 R9C9 v345789
R9C4 R9C6 R9C7 R9C8 R9C9 v235789
R9C4 R9C6 R9C7 R9C9 v35789
R9C7 v39
R9C7 R9C8 v239
R9C7 R9C8 R9C9 v2359
R9C7 R9C9 v359
R9C8 v23
R9C9 v59
R1C1 R2C1 R3C1 v3567
R1C1 R2C1 R3C1 R4C1 R5C1 v234567
R1C1 R2C1 R3C1 R4C1 R9C1 v234567
R1C1 R2C1 R3C1 R5C1 R9C1 v234567
R1C1 R2C1 R4C1 v3567
R1C1 R2C1 R4C1 R5C1 R9C1 v234567
R1C1 R3C1 R4C1 v3567
R1C1 R3C1 R4C1 R5C1 R9C1 v234567
R2C1 R3C1 R4C1 v3567
R2C1 R3C1 R4C1 R5C1 R9C1 v234567
R1C2 R3C2 R4C2 v5678
R1C2 R3C2 R4C2 R5C2 v25678
R1C2 R3C2 R4C2 R7C2 v25678
R1C2 R3C2 R5C2 R7C2 v25678
R1C2 R4C2 R5C2 R7C2 v25678
R3C2 R4C2 R5C2 R7C2 v25678
R4C2 R5C2 R7C2 v2567
R5C2 R7C2 v256
R1C4 R2C4 R4C4 R6C4 R7C4 R8C4 v1236789
R1C4 R2C4 R4C4 R6C4 R7C4 R9C4 v1236789
R1C4 R2C4 R4C4 R6C4 R8C4 R9C4 v1236789
R1C4 R2C4 R4C4 R7C4 R8C4 R9C4 v1236789
R1C4 R2C4 R6C4 R7C4 R8C4 R9C4 v1236789
R1C4 R4C4 R6C4 R7C4 R8C4 R9C4 v1236789
R2C4 R4C4 R6C4 R7C4 R8C4 R9C4 v1236789
R2C4 R4C4 R7C4 R8C4 R9C4 v136789
R1C5 R3C5 R4C5 R5C5 R7C5 v134568
R1C5 R3C5 R4C5 R5C5 R9C5 v134568
R1C5 R3C5 R4C5 R7C5 v13568
R1C5 R3C5 R4C5 R7C5 R9C5 v134568
R1C5 R3C5 R5C5 R7C5 R9C5 v134568
R1C5 R4C5 R5C5 R7C5 R9C5 v134568
R3C5 R4C5 R5C5 R7C5 R9C5 v134568
R3C5 R4C5 R5C5 R9C5 v34568
R1C6 R2C6 R3C6 R5C6 R6C6 R7C6 R8C6 v23456789
R1C6 R2C6 R3C6 R5C6 R6C6 R7C6 R9C6 v23456789
R1C6 R2C6 R3C6 R5C6 R6C6 R8C6 R9C6 v23456789
R1C6 R2C6 R3C6 R5C6 R7C6 R8C6 R9C6 v23456789
R1C6 R2C6 R3C6 R5C6 R7C6 R9C6 v2356789
R1C6 R2C6 R3C6 R6C6 R7C6 R8C6 R9C6 v23456789
R1C6 R2C6 R5C6 R6C6 R7C6 R8C6 R9C6 v23456789
R1C6 R3C6 R5C6 R6C6 R7C6 R8C6 R9C6 v23456789
R2C6 R3C6 R5C6 R6C6 R7C6 R8C6 R9C6 v23456789
R2C6 R3C6 R5C6 R7C6 R8C6 R9C6 v3456789
R2C6 R3C6 R5C6 R7C6 R9C6 v356789
R2C7 R3C7 R6C7 R8C7 v13689
R2C7 R3C7 R6C7 R9C7 v13689
R2C7 R3C7 R8C7 R9C7 v13689
R2C7 R6C7 v368
R2C7 R6C7 R8C7 R9C7 v13689
R2C7 R6C7 R9C7 v3689
R3C7 R6C7 R8C7 R9C7 v13689
R2C8 R3C8 R4C8 R5C8 R6C8 v135689
R2C8 R3C8 R4C8 R5C8 R6C8 R7C8 v1235689
R2C8 R3C8 R4C8 R5C8 R6C8 R9C8 v1235689
R2C8 R3C8 R4C8 R5C8 R7C8 R9C8 v1235689
R2C8 R3C8 R4C8 R6C8 R7C8 R9C8 v1235689
R2C8 R3C8 R5C8 R6C8 R7C8 R9C8 v1235689
R2C8 R3C8 R5C8 R7C8 R9C8 v123689
R2C8 R3C8 R6C8 R7C8 R9C8 v123568
R2C8 R3C8 R7C8 R9C8 v12368
R2C8 R4C8 R5C8 R6C8 v35689
R2C8 R4C8 R5C8 R6C8 R7C8 R9C8 v1235689
R2C8 R4C8 R5C8 R6C8 R9C8 v235689
R3C8 R4C8 R5C8 R6C8 R7C8 R9C8 v1235689
R4C8 R5C8 R6C8 v5689
R1C9 R3C9 v679
R1C9 R3C9 R8C9 v5679
R1C9 R3C9 R9C9 v5679
R1C9 R8C9 R9C9 v5679
R3C9 R8C9 R9C9 v5679
R8C9 R9C9 v569
R1C1 R1C2 R2C1 R3C1 v35678
R1C1 R1C2 R2C1 R3C2 v35678
R1C1 R1C2 R3C1 R3C2 v35678
R1C1 R2C1 R3C1 R3C2 v35678
R1C2 R2C1 R3C1 R3C2 v35678
R1C4 R1C5 R1C6 R2C4 R2C6 R3C5 v1235678
R1C4 R1C5 R1C6 R2C4 R2C6 R3C6 v1235678
R1C4 R1C5 R1C6 R2C4 R3C5 R3C6 v1235678
R1C4 R1C5 R1C6 R2C6 R3C5 R3C6 v1235678
R1C4 R1C5 R2C4 R2C6 R3C5 R3C6 v1235678
R1C4 R1C6 R2C4 R2C6 R3C5 R3C6 v1235678
R1C5 R1C6 R2C4 R2C6 R3C5 R3C6 v1235678
R1C5 R2C4 R2C6 R3C5 R3C6 v135678
R1C6 R2C4 R2C6 R3C5 R3C6 v235678
R2C4 R2C6 R3C5 R3C6 v35678
R1C9 R2C7 R2C8 v3678
R1C9 R2C7 R2C8 R3C7 R3C8 v136789
R1C9 R2C7 R2C8 R3C7 R3C9 v136789
R1C9 R2C7 R2C8 R3C8 v13678
R1C9 R2C7 R2C8 R3C8 R3C9 v136789
R1C9 R2C7 R2C8 R3C9 v36789
R1C9 R2C7 R3C7 R3C8 R3C9 v136789
R1C9 R2C8 R3C7 R3C8 R3C9 v136789
R2C7 R2C8 R3C7 R3C8 v13689
R2C7 R2C8 R3C7 R3C8 R3C9 v136789
R2C7 R2C8 R3C8 v1368
R4C1 R4C2 R5C1 R5C2 v24567
R4C1 R4C2 R5C1 R6C3 v24567
R4C1 R4C2 R5C2 v2567
R4C1 R4C2 R5C2 R6C3 v24567
R4C1 R4C2 R6C3 v4567
R4C1 R5C1 R5C2 R6C3 v24567
R4C2 R5C1 R5C2 R6C3 v24567
R5C1 R5C2 R6C3 v2456
R5C1 R6C3 v245
R4C4 R4C5 R5C5 R5C6 v34689
R4C4 R4C5 R5C5 R5C6 R6C4 v234689
R4C4 R4C5 R5C5 R5C6 R6C6 v234689
R4C4 R4C5 R5C5 R6C4 R6C6 v234689
R4C4 R4C5 R5C6 v3689
R4C4 R4C5 R5C6 R6C4 v23689
R4C4 R4C5 R5C6 R6C4 R6C6 v234689
R4C4 R4C5 R6C4 v2689
R4C4 R4C5 R6C4 R6C6 v24689
R4C4 R5C5 R5C6 R6C4 R6C6 v234689
R4C5 R5C5 R5C6 R6C4 R6C6 v234689
R4C5 R5C5 R6C4 R6C6 v23468
R4C5 R6C4 v268
R4C5 R6C4 R6C6 v2468
R4C8 R5C8 R6C7 v5689
R4C8 R6C7 R6C8 v5689
R5C8 R6C7 v689
R5C8 R6C7 R6C8 v5689
R7C2 R8C3 v245
R7C2 R9C1 v245
R8C3 R9C1 v245
R7C4 R7C5 R7C6 R8C4 v13569
R7C4 R7C5 R7C6 R8C4 R8C6 v134569
R7C4 R7C5 R7C6 R8C4 R8C6 R9C4 R9C5 v13456789
R7C4 R7C5 R7C6 R8C4 R8C6 R9C4 R9C6 v13456789
R7C4 R7C5 R7C6 R8C4 R8C6 R9C5 v1345689
R7C4 R7C5 R7C6 R8C4 R8C6 R9C5 R9C6 v13456789
R7C4 R7C5 R7C6 R8C4 R9C4 R9C5 R9C6 v13456789
R7C4 R7C5 R7C6 R8C4 R9C4 R9C6 v1356789
R7C4 R7C5 R7C6 R8C6 R9C4 R9C5 R9C6 v13456789
R7C4 R7C5 R8C4 R8C6 R9C4 R9C5 R9C6 v13456789
R7C4 R7C6 R8C4 R8C6 R9C4 R9C5 R9C6 v13456789
R7C5 R7C6 R8C4 R8C6 R9C4 R9C5 R9C6 v13456789
R7C8 R8C7 R8C9 R9C7 R9C8 v123569
R7C8 R8C7 R8C9 R9C7 R9C9 v123569
R7C8 R8C7 R8C9 R9C8 R9C9 v123569
R7C8 R8C7 R9C7 R9C8 v12369
R7C8 R8C7 R9C7 R9C8 R9C9 v123569
R7C8 R8C9 R9C7 R9C8 R9C9 v123569
R8C7 R8C9 R9C7 R9C8 R9C9 v123569
R8C7 R8C9 R9C7 R9C9 v13569
R8C7 R8C9 R9C9 v1569
R8C9 R9C7 R9C8 R9C9 v23569
R8C9 R9C7 R9C9 v3569


The same cells can be used in various configurations to form new sets. Once you know what to look for, it's harder NOT to see them than to find them.

These are all the possible ALS-XZ moves in this grid:

Code:
A=R4C1 R4C2 R6C3 v4567, B=R4C5 R6C4 R6C6 v2468, X=V4, Z=V6
A=R6C7 R6C8 v568, B=R5C1 R5C2 R6C3 v2456, X=V5, Z=V6
A=R2C8 R3C8 R6C8 R7C8 R9C8 v123568, B=R5C1 R5C2 R6C3 v2456, X=V5, Z=V6
A=R9C7 R9C8 R9C9 v2359, B=R8C3 R9C1 v245, X=V2, Z=V5
A=R5C1 R5C2 R5C6 R5C8 v23469, B=R9C1 R9C7 R9C8 v2349, X=V4, Z=V3
A=R5C1 R5C2 R5C6 R5C8 v23469, B=R9C1 R9C7 R9C8 R9C9 v23459, X=V4, Z=V3
A=R5C1 R5C2 R5C6 R5C8 v23469, B=R9C1 R9C8 v234, X=V4, Z=V3
A=R5C2 R5C6 R5C8 v2369, B=R7C2 R7C4 R7C5 R7C6 v12356, X=V2, Z=V3
A=R1C1 R1C2 R1C4 R1C5 R1C9 v123678, B=R4C4 R4C5 R5C6 R6C4 v23689, X=V2, Z=V3
A=R1C4 R1C5 R2C4 R2C6 R3C5 R3C6 v1235678, B=R4C4 R4C5 R5C6 R6C4 v23689, X=V2, Z=V3
A=R4C4 R4C5 R5C5 R6C4 R6C6 v234689, B=R7C4 R7C5 R7C6 R8C4 v13569, X=V9, Z=V3
A=R4C4 R4C5 R5C6 R6C4 R6C6 v234689, B=R7C4 R7C5 R7C6 R8C4 R8C6 v134569, X=V4, Z=V3
A=R4C4 R4C5 R6C4 R6C6 v24689, B=R7C4 R7C5 R7C6 R8C4 R8C6 v134569, X=V4, Z=V9
A=R2C7 R6C7 v368, B=R8C9 R9C7 R9C9 v3569, X=V3, Z=V6
A=R7C4 R7C5 R7C6 R7C8 v12356, B=R9C4 R9C6 R9C7 R9C8 R9C9 v235789, X=V2, Z=V5
A=R7C4 R7C5 R7C6 R7C8 v12356, B=R9C7 R9C8 R9C9 v2359, X=V2, Z=V5
A=R5C1 R5C2 v246, B=R6C3 R6C7 R6C8 v4568, X=V4, Z=V6
A=R8C3 v45, B=R9C1 R9C7 R9C8 R9C9 v23459, X=V4, Z=V5
A=R7C4 R7C5 R7C6 R7C8 v12356, B=R9C4 R9C6 R9C7 R9C8 R9C9 v235789, X=V2, Z=V5
A=R7C4 R7C5 R7C6 R7C8 v12356, B=R9C7 R9C8 R9C9 v2359, X=V2, Z=V5
A=R2C7 R6C7 R9C7 v3689, B=R8C9 R9C9 v569, X=V9, Z=V6


A and B are the two Almost Locked Sets.

X is a digit which cannot be in both sets at the same time, because all the candidates X in set A can see all the candidates X in set B. As a result, at least one of these sets will be locked for the remaining digits.

Z is one of these remaining digits. It is present in both sets. When there are candidates for this digit outside these 2 sets which can see ALL candidates for Z in both sets, it can be eliminated.

The listed combinations can all perform such eliminations.

With some practice, you should be able to find at least some of them, especially the shorter ones.

Ruud
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AZ Matt



Joined: 03 Nov 2005
Posts: 63
Location: Hiding under my desk in Phoenix AZ USA

PostPosted: Fri Sep 01, 2006 4:51 pm    Post subject: Fascinating stuff Reply with quote

I found this thread from this very forum helpful:

http://www.sudoku.com/forums/viewtopic.php?t=2510

This is great stuff...
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Fri Sep 01, 2006 10:25 pm    Post subject: Finding the right pairs of "ALS"s Reply with quote

Thanks for the long list of almost locked sets, Ruud. Smile

In the list Ruud posted, there are 254 distinct Almost Locked Sets. So there are 127 x 253 = 32,131 distinct pairs of "ALS"s. Ruud also shows us 21 different pairs that allow the XZ exclusion rule to be applied.

Perhaps that's why this technique is hard to master. In this puzzle, there are roughly 1,500 "duds" for every ALS pair that actually helps one make an exclusion. (i"m just guessing, but I suppose the ratio of useful ALS pairs to all ALS pairs is on the same order of magnitude in most puzzles where this technique -- or something similar -- is necessary.) I can see that many of the theoretically possible ALS pairs aren't logically connected (by lying in the same row / column, or by intersecting in an interesting way within one 3x3 box). Still, it seems that separating the wheat from the chaff is a fairly tall order. Or am I missing something obvious? dcb
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David Bryant



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Posts: 559
Location: Denver, Colorado

PostPosted: Sat Sep 02, 2006 10:46 pm    Post subject: Solving the 8/29/06 Nightmare with DIC's Reply with quote

Here's how I used the DIC technique to solve this "Nightmare." I'm starting from the position Ruud posted above.
Code:
.---------------------.---------------------.---------------------.
| 367    678    9     | 123678 1368   23678 | 5      4      67    |
| 3567   4      1     | 3678   9      35678 | 368    368    2     |
| 3567   5678   2     | 4      3568   35678 | 13689  1368   679B  |
:---------------------+---------------------+---------------------:
| 567    567    3     | 689    68     1     | 2      5689B  4     |
| 24A    26A    8     | 5      346    369B  | 7      69     1     |
| 1      9      45    | 268    7      2468  | 68     568    3     |
:---------------------+---------------------+---------------------:
| 9      25     7     | 136    1356   356   | 4      1236   8     |
| 8      3      45    | 169B   2      4569  | 169B   7      569B  |
| 24A    1      6     | 3789   3458   35789 | 39*    23A    59B   |
'---------------------'---------------------'---------------------'

Starting from the "alpha star" in r9c7 we can reason as follows.

A. r9c7 = 3 ==> r9c8 = 2 ==> r9c1 = 4 ==> r5c1 = 2 ==> r5c2 = 6 ==> {5, 7} pair in r4c12
B1. r9c7 = 9 ==> r3c9 = 9
B2. r9c7 = 9 ==> r9c9 = 5 ==> r8c9 = 6 ==> r8c7 = 1 ==> r8c4 = 9 ==> r5c6 = 9 ==> r4c8 = 9

From this we can draw two simple inferences.

1. Either r5c3 = 6 or r5c6 = 9; therefore r5c6 <> 6
2. Either {5, 7} pair in r4c12 or r4c8 = 9; therefore r4c8 <> 5

Eliminating "5" at r4c8 reveals the {6, 8, 9} triplet in row 4 and the hidden pair {5, 7} in r4c12 -- now the grid looks like this.
Code:
.---------------------.---------------------.---------------------.
| 367    678    9     | 123678 1368   23678 | 5      4      67    |
| 3567   4      1     | 3678   9      35678 | 368    368    2     |
| 3567   5678   2     | 4      3568   35678 | 13689  1368   679   |
:---------------------+---------------------+---------------------:
| 57     57     3     | 689    68     1     | 2      689    4     |
| 24     26     8     | 5      346    39    | 7      69     1     |
| 1      9      45    | 268    7      2468  | 68     568    3     |
:---------------------+---------------------+---------------------:
| 9      25     7     | 136    1356   356   | 4      1236   8     |
| 8      3      45    | 169    2      4569  | 169    7      569   |
| 24     1      6     | 3789   3458   35789 | 39     23     59    |
'---------------------'---------------------'---------------------'

Quite a few simple moves follow from this point, beginning with r6c3 = 4, r5c1 = 2, r5c2 = 6, r8c3 = 5, etc. If you play it out you'll hit a point where you can either rely on a "UR" in r16c46 that will reveal the pair {6, 7} in r1c69, or else you can find a fork on the digit "6" that allows you to set r2c4 = 3. In other words, it's fairly simple from here on out.

When I went through this puzzle the first time I noticed the forcing chain that Ravel mentioned above, but in a different form.
Code:
.---------------------.---------------------.---------------------.
| 367    678    9     | 123678 1368   23678 | 5      4      67    |
| 3567   4      1     | 3678   9      35678 | 368    368    2     |
| 3567   5678   2     | 4      3568   35678 | 13689  1368   679   |
:---------------------+---------------------+---------------------:
| 567    567    3     | 689    68     1     | 2      5689   4     |
| 24     26     8     | 5      346    369   | 7      69     1     |
| 1      9      45    | 268    7      2468  | 68     568    3     |
:---------------------+---------------------+---------------------:
| 9      25     7     | 136    1356   356   | 4      1236   8     |
| 8      3      45A   | 169    2      4569  | 169    7      569A  |
| 24B    1      6     | 3789   3458   35789 | 39B    23B    59*   |
'---------------------'---------------------'---------------------'

A. r9c9 = 9 ==> r8c9 = 5 ==> r8c3 = 4
B. r9c9 = 9 ==> r9c7 = 3 ==> r9c8 = 2 ==> r9c1 = 4

So we must have r9c9 = 5, which is the same ellimination Ravel illustrated (including an application of the ALS XZ rule). After this move the DIC from r9c7 is a little easier to spot. But that DIC still works, even without first setting r9c9 = 5.

Someone -- I think it was Keith -- was recently asking how to spot a good cell to serve as the "alpha star" for a double-implication chain. I've been practicing this technique for quite a while, and here's the best advice I have right now.

1. I examine the patterns of missing digits looking for a single-digit spiral; that is, I look for a setup like the one with the "9"s above, where setting say a "9" in a single cell will force all the rest of the "9"s in the puzzle.

2. If I can find such a cell that only has two values left in it, I start tracing out what happens with the other value.

3. If I can't find such a happy pattern, I start lowering my sights a little, and look for a cell that will force most of the missing "9"s (or some other digit ... it depends on the patterns I can spot).

4. On a really tough puzzle the first DIC I use may not allow me to solve any particular cell. But it often allows me to make one or two eliminations from cells with 3 possibilities, and that opens up more bi-valued cells that can serve as the "alpha star" in a new DIC. dcb

PS Oh -- I went through the list of Almost Locked Sets that Ruud kindly posted for this point in this puzzle, and I found several that led to the conclusion r9c9 = 5. I didn't find any that allowed me to eliminate "5" from r4c8. The best one I found (in terms of making progress) eliminates "6" at r5c8. There may be another DIC in here somewhere that allows one to set r5c8 = 9, but I haven't found it yet.
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