Variation of Locked Candidates

Advanced methods and approaches for solving Sudoku puzzles

Variation of Locked Candidates

Postby Ruud » Mon Jan 01, 2007 1:25 pm

Here is an interesting move:

Code: Select all
 XY+ XY+ XY+ | XY  /   / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .

Row 1 has a bivalue cell with candidates XY, and all other candidates for X & Y are located in the intersection with box 1.
Box 1 also has a bivalue cell with candidates XY. Because the intersection must contain an X or an Y, we can eliminate candidates for X & Y from the remaining cells in box 1.

It should be possible to generalize this to larger ALS sizes in either of the intersecting sectors.

Here is a real-life example. It is a Sudoku-X (with diagonal constraints), but the move does not use the diagonals. It is the only example I have, so far.

Original puzzle:
Code: Select all
. . .|. . .|. . .
3 . .|5 . 7|. . 2
. . .|. . .|. . .
-----+-----+-----
4 6 .|2 . 9|. 7 8
. 9 .|1 . 8|. 4 .
1 2 .|7 . 5|. 6 3
-----+-----+-----
. . .|. . .|. . .
8 . .|9 . 4|. . 7
. . .|. . .|. . .

Here is the point where this technique can be used, bypassing several other steps:
Code: Select all
.------------------.------------------.------------------.
| 9     1458  1246 | 346   18    236  | 7     135   14   |
| 3     148   146  | 5     9     7    | 46    18    2    |
| 256   1458  7    | 346   18    236  | 3458  1359  69   |
:------------------+------------------+------------------:
| 4     6     5    | 2     3     9    | 1     7     8    |
| 7     9     3    | 1     6     8    | 2     4     5    |
| 1     2     8    | 7     4     5    | 9     6     3    |
:------------------+------------------+------------------:
|#56   -13457 124  | 368   57    136  | 348   12589 69   |
| 8    %135  %16   | 9     2     4    |#56    13    7    |
|-25   -13457 9    | 368   57    136  | 4568  1258  14   |
'------------------'------------------'------------------'

I assume this is a rare move, as nobody has mentioned it before, but the generalized version may be more frequent, yet undetected.

Should I be embarrassed for not spotting a simple alternative?:)

Ruud
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Postby Mike Barker » Mon Jan 01, 2007 1:38 pm

I think this is the same as http://forum.enjoysudoku.com/viewtopic.php?t=4477 . If so we should agree on a name and I'll post it in the collection.
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variation of Locked Candidates ~~ law of leftovers

Postby Pat » Mon Jan 01, 2007 1:56 pm

Ruud wrote:Here is an interesting move:

Code: Select all
 XY+ XY+ XY+ | XY  /   / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .



Row 1 has a bivalue cell with candidates XY, and all other candidates for X & Y are located in the intersection with box 1.

Box 1 also has a bivalue cell with candidates XY.

Because the intersection must contain an X or an Y,
we can eliminate candidates for X & Y from the remaining cells in box 1.



        r1 \ b1 has 5 cells of not-XY;
        therefore,
        b1 \ r1 must also have 5 such cells!
        knowing that r2c1 is XY,
        the other 5 cells in b1 \ r1 must all be not-XY.
r.e.s. (2005.Nov.5) wrote:Bob Harris's laws of leftovers (explained in his tutorials on sudoku with irregular boxes) can also be applied here -- a different way of looking at the picture:
    If two units intersect, in each of these units the set of "leftovers" (cells in the unit but not in the intersection) contains the same set of digits.

~ Pat
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Re: Variation of Locked Candidates

Postby ronk » Mon Jan 01, 2007 2:03 pm

Ruud wrote:
Code: Select all
.------------------.------------------.------------------.
| 9     1458  1246 | 346   18    236  | 7     135   14   |
| 3     148   146  | 5     9     7    | 46    18    2    |
| 256   1458  7    | 346   18    236  | 3458  1359  69   |
:------------------+------------------+------------------:
| 4     6     5    | 2     3     9    | 1     7     8    |
| 7     9     3    | 1     6     8    | 2     4     5    |
| 1     2     8    | 7     4     5    | 9     6     3    |
:------------------+------------------+------------------:
|#56   -13457 124  | 368   57    136  | 348   12589 69   |
| 8    %135  %16   | 9     2     4    |#56    13    7    |
|-25   -13457 9    | 368   57    136  | 4568  1258  14   |
'------------------'------------------'------------------'

Based on just the four tagged cells (r7c1, r8c237), that move may not be valid.

I think the correct sets of cells are A={r7c2}={56} and B={r8c238}={1356}. Then one can apply the "doubly-linked" ALS xz-rule to eliminate the doubly-linked candidates z1=5 and z2=6 from box 7. You might also recognize this as a "Sue de Coq."

[edit: I see Pat used the "law of leftovers -- cool rule.]
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Postby Ruud » Tue Jan 02, 2007 1:06 am

Thanks to all for the fast replies.

Mike:

I completely missed that thread. It is the same technique, including the generalization. "Advanced Locked Candidates" sounds fine as a name for this technique, using the same acronym.

Pat

Law of Leftovers is a versatile instrument, which can be used to replicate many techniques, including all unfinned fish. It shows a lot of similarities with constraint subsets, which may even be a higher level of abstraction.

Ron

I should have spotted that Sue-De-Coq move. However, this technique uses r8c7 in stead of r8c8, which would be part of the Sue-De-Coq move.

Here is the way to describe it in ALC terms:

In row 8, the bivalue cell r8c7 forces r8c23 to contain either a 5 or a 6.
In box 7, the bivalue cell r7c1 also contains either 5 or 6.
As a result, digits 5 & 6 in box 7 can only appear in r7c1 and r8c23. We can eliminate digit(s) 5 (& 6) from the cells marked with a minus sign.

[off topic]

I noticed that simple tricks like these do not receive a lot of attention. When a move can be replicated by an ALS move or an AIC/Nice Loop, one does not see any added value and moves on to the next subject. However, there is a significant gap between SSTS and highly advanced techniques like AIC, Nice Loops, ALS, etc. This gap is only filled with Uniqueness-based techniques which are unacceptable to some players and fish techniques which only operate on single digits. A few simple techniques like these could really help a lot of players to learn to use advanced logic in solving Sudokus and connect to the more advanced techniques.

[/off topic]
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Re: Variation of Locked Candidates

Postby udosuk » Tue Jan 02, 2007 1:36 am

Ruud wrote:Here is an interesting move:

Code: Select all
 XY+ XY+ XY+ | XY  /   / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .

Row 1 has a bivalue cell with candidates XY, and all other candidates for X & Y are located in the intersection with box 1.
Box 1 also has a bivalue cell with candidates XY. Because the intersection must contain an X or an Y, we can eliminate candidates for X & Y from the remaining cells in box 1.

Like Pat said, I think LOL pretty much covers this technique...

r1c456789 and r23c123 must contain exactly the same set of 6 numbers... Since r1c4 is the only cell which can contain the candidates X,Y in the first region, We know there is only 1 X or 1 Y in these 6 numbers... Which must be in r2c1 in the second region... Therefore no other cells in r23c123 can contain X,Y...

Incidentally, we don't need to restrict the possible candidates for r1c1234 to apply this technique...
As long as X,Y are locked in r1c1234:
Code: Select all
"/" cells: neither X nor Y

 ##  ##  ## | ##  /   / | /   /   /
 XY  **  ** | .   .   . | .   .   .
 **  **  ** | .   .   . | .   .   .
 -----------+-----------+----------
 .   .   .  | .   .   . | .   .   .
 .   .   .  | .   .   . | .   .   .
 .   .   .  | .   .   . | .   .   .
 -----------+-----------+----------
 .   .   .  | .   .   . | .   .   .
 .   .   .  | .   .   . | .   .   .
 .   .   .  | .   .   . | .   .   .

You can then eliminate both X,Y from all the "**" cells... Also, r1c4 must be either X or Y...

To generalise it to an "almost triples" scenario:
Code: Select all
"/" cells: no X,Y,Z

 #    #    # | # / / | / / # 
 XYZ  *    * | . . . | . . . 
 *    XYZ  * | . . . | . . . 
-------------+-------+-------
 .    .    . | . . . | . . . 
 .    .    . | . . . | . . . 
 .    .    . | . . . | . . . 
-------------+-------+-------
 .    .    . | . . . | . . . 
 .    .    . | . . . | . . . 
 .    .    . | . . . | . . . 

Once again, on LOL terms, there are 4 cells in r1c456789 which don't contain X,Y,Z, and we must find matching cells in r23c123, which are the 4 "*" cells... Moreover, r1c49 must be from X,Y,Z only...

The original candidates of r1c12349 are irrelevant...

Such is the amazing power of LOL...:!:
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Postby ronk » Tue Jan 02, 2007 6:02 am

Ruud wrote:Here is the way to describe it in ALC terms:

In row 8, the bivalue cell r8c7 forces r8c23 to contain either a 5 or a 6.
In box 7, the bivalue cell r7c1 also contains either 5 or 6.
As a result, digits 5 & 6 in box 7 can only appear in r7c1 and r8c23.

Here is the way to describe it in ALS terms:

In row 8, the bivalue cell r8c8 forces a naked triple (either 356 or 156) in box 7 (r7c1 and r8c23).

End of comparable description, and then you actually have a "stepping stone" to ALS techniques. To what is ALC is a stepping stone?
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Postby Steve R » Tue Jan 02, 2007 8:40 pm

I am not comfortable with the proposal “advanced locked candidates.” Mike’s original suggestion of “almost locked candidates” is more apt in my opinion.

A couple of questions:
- has anyone discovered an example of ALC which is not a Sue de Coq?
- SudoCue says Ruud’s puzzle is invalid and suggests it is part of a more complete original. If so, can we see it?

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Postby daj95376 » Tue Jan 02, 2007 10:31 pm

Steve R wrote:- SudoCue says Ruud’s puzzle is invalid and suggests it is part of a more complete original. If so, can we see it?

Ruud wrote:Here is a real-life example. It is a Sudoku-X (with diagonal constraints), but the move does not use the diagonals. It is the only example I have, so far.
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Postby Ruud » Tue Jan 02, 2007 10:31 pm

udosuk

ALC moves can be replicated by LoL when the supporting ALS sizes are equal. The following move with different ALS sizes may not be replicable with LoL:

Code: Select all
 XY+ XY+ XY+ | XZ  YZ  / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .


ronk wrote:To what is ALC is a stepping stone?

To Sue-De-Coq, which is slightly harder to spot.:)

Steve R:

I do not see this as an "almost" pattern. The candidates are definitely locked, but you use 2 ALS in the process. Maybe we should call it the "ALS-LC rule", a name that is more in line with the other ALS rules.

has anyone discovered an example of ALC which is not a Sue de Coq?

I have not found an example yet, but I will check my collection of puzzles that allow Sue-De-Coq moves to see if these moves always complement each other.

SudoCue says Ruud’s puzzle is invalid and suggests it is part of a more complete original. If so, can we see it?

This example is a Sudoku-X, as I mentioned in my opening post.

para, a member of the SudoCue forum, has found an ALC move in a recent Nightmare:

Code: Select all
8 . .|. . .|. . .
. 7 .|. . 2|. 9 .
. 1 2|3 . .|. 6 .
-----+-----+-----
. . 6|. . 7|2 . 5
. . .|. . .|. . .
5 . 7|4 . .|6 . .
-----+-----+-----
. 6 .|. . 5|9 4 .
. 9 .|8 . .|. 1 .
. . .|. . .|. . 3


Code: Select all
.------------------------.------------------------.------------------------.
| 8       345     349    | 1679    14679   1469   | 1347    25      1247   |
| 6       7       345    | 15      1458    2      | 13458   9       148    |
| 49      1       2      | 3       45789   489    | 4578    6       478    |
:------------------------+------------------------+------------------------:
| 1349    348     6      | 19*     1389    7      | 2       38      5      |
| 139     238     1389   | 12569   1235689 13689  | 14      7       149    |
| 5       238     7      | 4       12389*  1389*  | 6       38      19*    |
:------------------------+------------------------+------------------------:
| 1237    6       138    | 127     1237    5      | 9       4       278    |
| 2347    9       345    | 8       2347    34     | 57      1       6      |
| 1247    458     148    | 12679   124679  1469   | 78      25      3      |
'------------------------'------------------------'------------------------'

r6c28 are used in the complementary Sue-De-Coq move.

Ruud
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Postby udosuk » Wed Jan 03, 2007 1:35 am

Ruud wrote:udosuk

ALC moves can be replicated by LoL when the supporting ALS sizes are equal. The following move with different ALS sizes may not be replicable with LoL:

Code: Select all
 XY+ XY+ XY+ | XZ  YZ  / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .

"Replicable with LoL" is not easy to define here... For me LoL is very powerful and it can replicate all LC moves plus these "ALC" (whatever A stands for) moves... Moreover, I think the "XY+" notations in r1c123 are redundant because we only need the info from r1c456789 and r23c123 to make those eliminations (provided the original puzzle is a valid one)...

I suppose in your example a Z is locked in r1c45, otherwise r1c45 could be [XY] and you cannot eliminate X,Y from the "**" cells...

So using LoL: r1c456789=r23c123

Since there must be exactly one of {XY} and exactly one Z in r1c456789 (r1c45 in particular), r23c123 must have the same content, i.e. r2c1 contains the representative of {XY} and one of the "**" cells must be Z...

Therefore we can eliminate X,Y from the "**" cells as well as Z from r1c123...

For me it's just advanced application of the LoL principle...
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Postby ronk » Wed Jan 03, 2007 1:51 am

Ruud wrote:
Steve R wrote:has anyone discovered an example of ALC which is not a Sue de Coq?

I have not found an example yet, but I will check my collection of puzzles that allow Sue-De-Coq moves to see if these moves always complement each other.

I examined the first 10 Sue de Coq moves in my collection, and couldn't find an ALC. Here are two of them.

Code: Select all
.3..8...4......36..6...1..2..3..2.......3815...8....2.2...64..9.862......7.......

After exhausting SSTS:
 159     3       2       | 5679    8       5679    | 579     179     4
 8      -1459    157     | 4579    2       579     | 3       6       15
 459     6       57      | 3       4579    1       | 5789    789     2
-------------------------+-------------------------+--------------------
-145679 A1459    3       | 145679  14579   2       | 469     49      8
-469     2      B49      | 469     3       8       | 1       5       7
-145679 A1459    8       | 145679  14579   5679    | 469     2       3
-------------------------+-------------------------+--------------------
 2      C15      15      | 78      6       4       | 78      3       9
 349     8       6       | 2       179     379     | 457     147     15
 349     7       49      | 1589    159     359     | 2       148     6

 A = {r46c2} = {1459}
 B = {r5c3} = {49}
 C = {r7c2} = {15}
 Eight exclusions r456c1<>4,9 and r2c2<>1,5

Perhaps I don't know what to look for. Here I was looking for bivalue 49 in column 2. If a "hidden bivalue (hidden ALS)" is permissible, it's there in r2c2.

Code: Select all
.3....2..9.4.7.5.11...5.3.....4.5.........7.96...3...5.9....6...8...2...4..1.7..3
 
After exhausting SSTS:
 578    3      5678   | 689    14689  14689  | 2     -46789 A4678
 9      26     4      | 2368   7      368    | 5     B68     1
 1      267    2678   | 2689   5      4689   | 3     -46789 A4678
----------------------+----------------------+--------------------
 2378   17     3789   | 4      12689  5      | 18     1368   268
 2358   145    358    | 68     1268   168    | 7      13468  9
 6      124    289    | 7      3      189    | 148    1248   5
----------------------+----------------------+--------------------
 2357   9      1357   | 358    48     348    | 6      14578 -2478
 357    8      13567  | 3569   469    2      | 149    1457  C47
 4      256    256    | 1      689    7      | 89     258    3

 A = {r13c9} = {4678}
 B = {r2c8} = {68}
 C = {r8c9} = {47}
 Six exclusions r13c8<>6,8 and r7c9<>4,7

Here I was looking for bivalue 68 in column 9. There is none, but note the hidden trivalue (hidden ALS) 268 in r47c9.

[edit: clarify & correct typos]
Last edited by ronk on Tue Jan 02, 2007 10:51 pm, edited 1 time in total.
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Postby udosuk » Wed Jan 03, 2007 2:07 am

ronk wrote:
Code: Select all
.3..8...4......36..6...1..2..3..2.......3815...8....2.2...64..9.862......7.......

After exhausing SSTS:
 159     3       2       | 5679    8       5679    | 579     179     4
 8      -1459    157     | 4579    2       579     | 3       6       15
 459     6       57      | 3       4579    1       | 5789    789     2
-------------------------+-------------------------+--------------------
-145679 A1459    3       | 145679  14579   2       | 469     49      8
-469     2      B49      | 469     3       8       | 1       5       7
-145679 A1459    8       | 145679  14579   5679    | 469     2       3
-------------------------+-------------------------+--------------------
 2      C15      15      | 78      6       4       | 78      3       9
 349     8       6       | 2       179     379     | 457     147     15
 349     7       49      | 1589    159     359     | 2       148     6

 A = {r46c2} = {1459}
 B = {r5c3} = {49}
 C = {r7c2} = {15}
 Eight exclusions r456c1<>4,9 and r2c2<>1,5

Perhaps I don't know what to look for. Here I was looking for bivalue 49 in column 2. If a "hidden bivalue (ALS)" is permissible, it's there in r2c2.

ALC/LoL are both applicable here... I'll take the LoL approach:

r123789c2=r456c13

In r456c13 there is a bivalue cell r5c3={49}, which in r123789c2 can only appear in r2c2, therefore r2c2={49}.

Because r13789c2 do not contain {49}, we must have 5 cells which do not contain {49} in r456c13, which must be the 5 cells excluding r5c3... So {49} are eliminated from r456c1+r46c3...

ronk wrote:
Code: Select all
.3....2..9.4.7.5.11...5.3.....4.5.........7.96...3...5.9....6...8...2...4..1.7..3
 
After exhausing SSTS:
 578    3      5678   | 689    14689  14689  | 2     -46789 A4678
 9      26     4      | 2368   7      368    | 5     B68     1
 1      267    2678   | 2689   5      4689   | 3     -46789 A4678
----------------------+----------------------+--------------------
 2378   17     3789   | 4      12689  5      | 18     1368   268
 2358   145    358    | 68     1268   168    | 7      13468  9
 6      124    289    | 7      3      189    | 148    1248   5
----------------------+----------------------+--------------------
 2357   9      1357   | 358    48     348    | 6      14578 -2478
 357    8      13567  | 3569   469    2      | 149    1457  C47
 4      256    256    | 1      689    7      | 89     258    3

 A = {r13c9} = {4678}
 B = {r2c8} = {68}
 C = {r8c9} = {47}
 Six exclusions r13c8<>6,8 and r7c9<>4,7

Here I was looking for bivalue 68 in column 9. There is none, but note the hidden trivalue (ALS) 268 in r47c9.

More complicated here...

r123c78=r456789c9

In r123c78 we have 2 cells taking the values 2 and {68} respectively, and these values can only appear in 2 cells in r456789c9, namely r47c9, therefore r47c9 cannot contain other values => r7c9={28}

Now r5689c9 have 3 fixed cells and 1 bivalue cell (r569c9=[953] and r8c9={47}), and these must be the content in r23c7+r13c8... Therefore one of r13c8 must be 9 and the other {47} => {68} are eliminated from r13c8

:idea:
udosuk
 
Posts: 2698
Joined: 17 July 2005

Postby Steve R » Wed Jan 03, 2007 3:09 am

Apologies to Ruud for missing the reference to SudokuX.

Interesting stuff, Ron. I think both examples contain ALCs however. The first:
Code: Select all
 159     3       2       | 5679    8       5679    | 579     179     4
 8      -1459    157     | 4579    2       579     | 3       6       15
 459     6       57      | 3       4579    1       | 5789    789     2
-------------------------+-------------------------+--------------------
-145679 A1459    3       | 145679  14579   2       | 469     49      8
-469     2      B49      | 469     3       8       | 1       5       7
-145679 A1459    8       | 145679  14579   5679    | 469     2       3
-------------------------+-------------------------+--------------------
 2      C15      15      | 78      6       4       | 78      3       9
 349     8       6       | 2       179     379     | 457     147     15
 349     7       49      | 1589    159     359     | 2       148     6

 A = {r46c2} = {1459}
 B = {r5c3} = {49}
 C = {r7c2} = {15}
 Eight exclusions r456c1<>4,9 and r2c2<>1,5


has almost locked candidates {4, 9} in column 2: they either lie in box 4 or, if outside the box, in the single cell r2c2. As there is an ALS, r5c3, with candidates {4, 9} in the box, the eliminations follow.

The second example is:

Code: Select all
 578    3      5678   | 689    14689  14689  | 2     -46789 A4678
 9      26     4      | 2368   7      368    | 5     B68     1
 1      267    2678   | 2689   5      4689   | 3     -46789 A4678
----------------------+----------------------+--------------------
 2378   17     3789   | 4      12689  5      | 18     1368   268
 2358   145    358    | 68     1268   168    | 7      13468  9
 6      124    289    | 7      3      189    | 148    1248   5
----------------------+----------------------+--------------------
 2357   9      1357   | 358    48     348    | 6      14578 -2478
 357    8      13567  | 3569   469    2      | 149    1457  C47
 4      256    256    | 1      689    7      | 89     258    3

 A = {r13c9} = {4678}
 B = {r2c8} = {68}
 C = {r8c9} = {47}
 Six exclusions r13c8<>6,8 and r7c9<>4,7

Here the almost locked candidates are {4, 7, 9} in box 3. If not in column 9, they lie in the two cells r13c8. The ALS {r5c9, r8c9} with candidates {4, 7, 9} in the column completes the ALC picture.

Steve
Steve R
 
Posts: 74
Joined: 03 April 2006

Postby ronk » Wed Jan 03, 2007 3:39 am

udosuk wrote:
Ruud wrote:udosuk

ALC moves can be replicated by LoL when the supporting ALS sizes are equal. The following move with different ALS sizes may not be replicable with LoL:

Code: Select all
 XY+ XY+ XY+ | XZ  YZ  / | /   /   /
 XY  **  **  | .   .   . | .   .   .
 **  **  **  | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 ------------+-----------+----------
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .
 .   .   .   | .   .   . | .   .   .

"Replicable with LoL" is not easy to define here... For me LoL is very powerful and it can replicate all LC moves plus these "ALC" (whatever A stands for) moves... Moreover, I think the "XY+" notations in r1c123 are redundant ...

I agree, and think exemplars could be ...
Code: Select all
 "/" cells: neither X nor Y

 .  .  .  | XY+  /    /  | /  /  /    .    .  .  | XY ** ** | ** ** **
 XY ** ** | .    .    .  | .  .  .    XY+  /  /  | .  .  .  | .  .  .
 ** ** ** | .    .    .  | .  .  .    /    /  /  | .  .  .  | .  .  .
 ---------+--------------+--------    -----------+----------+--------
 .  .  .  | .    .    .  | .  .  .    .    .  .  | .  .  .  | .  .  .




 "/" cells: neither X nor Y nor Z

 .  .  .  | XYZ+ XYZ+ /  | /  /  /    .    .  .  | XY ** ** | ** ** **
 XY ** ** | .    .    .  | .  .  .    XYZ+ /  /  | .  .  .  | .  .  .
 ** ** ** | .    .    .  | .  .  .    XYZ+ /  /  | .  .  .  | .  .  .
 ---------+--------------+--------    -----------+----------+--------
 .  .  .  | .    .    .  | .  .  .    .    .  .  | .  .  .  | .  .  .

[edit to add: For all four illustrations, both X and Y may be excluded from the ** cells, and the extra candidates (tagged '+') may be excluded from the XY+ and XYZ+ cells. Is that correct:?:

So of four different POVs, I understand Sue-de-Coq, doubly-linked ALS xz-rule and (I think) the Law-of-Leftovers, but not Almost Locked Candidates. I'll read it again.]
Last edited by ronk on Wed Jan 03, 2007 9:44 am, edited 1 time in total.
ronk
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