Almost locked rules (for now)

Advanced methods and approaches for solving Sudoku puzzles

Almost locked rules (for now)

Postby bennys » Wed Dec 14, 2005 7:58 am

Code: Select all
Almost locked rules
-------------------

ALS - almost locked set
For two sets A,B that have common candidate x we will say that x is restricted common if it can't be
in A and B in the same time.(being in A exclude it from B)

Almost locked sets xz rule
--------------------------
If
1)A,B ALS
2) x is restricted common
Then Any other common candidate (lets say z) cant appear outside of A and B if it can see all the z candidates in both A and B.
The reason?
If z appear then both A and B are locked but only one can get the x.

 +-------------+-------------+-------------+
 | 24  7   8   | 24  6   5   | 1   9   3   |
 | 9   3   24  | 248 1   48  | 56  7   56  |
 | 5   1   6   | 7   3   9   | 8   4   2   |
 +-------------+-------------+-------------+
 | 28  9   23  | 458 7   6   |#35  1  #45  |
 |*17  6   5   | 3   9   14  | 2   8  #47  |
 | 178 4  *13  | 58  2   18  | 357 6   9   |
 +-------------+-------------+-------------+
 | 6   5   7   | 1   4   2   | 9   3   8   |
 | 14  2   14  | 9   8   3   | 67  5   67  |
 | 3   8   9   | 6   5   7   | 4   2   1   |
 +-------------+-------------+-------------+
A={R5C1,R6C3}
B={R4C7,R4C9,R5C9}
x=7
z=3


Almost locked sets xy wing rule
-------------------------------

If A B C almost locked sets
x common to A,B
y restricted common to B,C
z restricted common to A,C

then  a cell that can 'see' all the x candidates of both A and B can't be x.



+-------------------+-------------------+-------------------+
| 459   1    %479   |%78    3    %89    | 2    %58    6     |
| 59    569   679   | 278   289   4     |^35    1358  13    |
| 8     3     2     | 5     6     1     | 7     9     4     |
+-------------------+-------------------+-------------------+
|*139   7    *139   | 6     4     5     | 8     123   1239  |
| 6     28   *134   | 9     28    7     |^34    13    5     |
| 24    289   5     | 238   1     238   |^49    6     7     |
+-------------------+-------------------+-------------------+
| 7     4     39    | 23    259   6     | 1     235   8     |
| 12359 2569  1369  | 4     2589  2389  | 3569  7     239   |
| 2359  2569  8     | 1     7     239   | 3569  4     239   |
+-------------------+-------------------+-------------------+

A={R4C1,R4C3,R5C3}
B={R2C7,R5C7,R6C7}
C={R1C3,R1C4,R1C6,R1C8}
x=9
y=5
z=4

2 ALS 2 restricted common rule
------------------------------
If A have degrees of freedom of 2
and B and C are ALS
with
x restricted common to A and B
y restricted common to A and C
and z common to A B C
then you cant have z in a cell that can see all the z candidates in A B C

n ALS n restricted common rule
-----------------------------
If A have degrees of freedom of n
and we have a collection S of n  disjoint ALS (that are also disjoint with A) 
each of the ALS has different restricted common with A
and z common to A and all the members of S
then we cant have z in a cell that can see all the z candidates in A and all the members of S.

+----------------------+----------------------+----------------------+
| 139   *1379   2579   |$17     1247   8      | 1459   123459 6      |
|#16     4      257    | 3      1267   9      | 8      125    12     |
| 8     *1369   29     | 5      1246   46     | 7      12349  1234   |
+----------------------+----------------------+----------------------+
| 39     58     1      | 679    34679  34567  | 2      4679   478    |
| 7     *39     4      | 2      8      36     | 169    169    5      |
| 2      58     6      | 179    1479   457    | 3      479    478    |
+----------------------+----------------------+----------------------+
| 4      267    3      | 8      567    1      | 56     2567   9      |
| 169    1679   79     | 4      35679  2      | 156    8      137    |
| 5      12679  8      | 679    3679   367    | 146    123467 12347  |
+----------------------+----------------------+----------------------+
A={R1C2,R3C2,R5C}
B={R2C1}
C={R1C4}
X=6
Y=7
Z=1
bennys
 
Posts: 156
Joined: 28 September 2005

Postby Jeff » Wed Dec 14, 2005 8:05 am

bennys, I salute you for your contribution to manual sudoku solving.

from Jeff:D:D:D
Jeff
 
Posts: 708
Joined: 01 August 2005

Postby bennys » Wed Dec 14, 2005 8:12 am

Thanks
bennys
 
Posts: 156
Joined: 28 September 2005

Re: Almost locked rules (for now)

Postby ronk » Wed Dec 14, 2005 11:32 am

bennys wrote:Almost locked rules
-------------------
Almost locked sets xy wing rule
-------------------------------

Although the following variation (of your xy wing rule example) doesn't fit your definition, it still appears like an xy-wing with two Almost Locked Sets.
Code: Select all
+-------------------+-------------------+-------------------+
| 459   1    %479   |%78    3    %89    | 2    %58    6     |
| 59    569   679   | 278   289   4     |^35    1358  13    |
| 8     3     2     | 5     6     1     | 7     9     4     |
+-------------------+-------------------+-------------------+
| 139   7     139   | 6     4     5     | 8     123   1239  |
| 6     28    134   | 9     28    7     |^34    13    5     |
| 24    289   5     | 238   1     238   | 49    6     7     |
+-------------------+-------------------+-------------------+
| 7     4     39    | 23    259   6     | 1     235   8     |
| 12359 2569  1369  | 4     2589  2389  | 3569  7     239   |
| 2359  2569  8     | 1     7     239   | 3569  4     239   |
+-------------------+-------------------+-------------------+

B={R2C7,R5C7}
C={R1C3,R1C4,R1C6,R1C8}
x=?
y=5
z=4

... letting us eliminate candidate 4 from r5c3.


Where does this pattern fit in your thoughts for Almost Locked Sets?
ronk
2012 Supporter
 
Posts: 4691
Joined: 02 November 2005
Location: Southeastern USA

Postby Myth Jellies » Wed Dec 14, 2005 11:53 am

Ronk, that looks like a simple ALS xz rule to me.
Myth Jellies
 
Posts: 593
Joined: 19 September 2005

Postby Bob Hanson » Fri Dec 16, 2005 11:59 pm

bennys,

I'm going to try to write the equivalent in "Medusa" graphics and see what comes of it.

bennys wrote:I. Almost locked sets xz rule
--------------------------
If
1)A,B ALS
2) x is restricted common
Then any other common candidate (lets say z) can't appear outside of A
and B if it can see all the z candidates in both A and B.

Code: Select all
         x.......x
        /         \
   a---A           B---b
        \         /
         z...*...z
 
   * can be eliminated

 +-------------+-------------+-------------+
 | 24  7   8   | 24  6   5   | 1   9   3   |
 | 9   3   24  | 248 1   48  | 56  7   56  |
 | 5   1   6   | 7   3   9   | 8   4   2   |
 +-------------+-------------+-------------+
 | 28  9   23  | 458 7   6   |#35  1  #45  |
 |*17  6   5   | 3   9   14  | 2   8  #47  |
 | 178 4  *13  | 58  2   18  | 357 6   9   |
 +-------------+-------------+-------------+
 | 6   5   7   | 1   4   2   | 9   3   8   |
 | 14  2   14  | 9   8   3   | 67  5   67  |
 | 3   8   9   | 6   5   7   | 4   2   1   |
 +-------------+-------------+-------------+
A={R5C1,R6C3}
B={R4C7,R4C9,R5C9}
x=7
z=3

         7.......7
        /         \
   1---A           B---4,5
        \         /
         3...*...3

              *=r6c7#3 can be eliminated


bennys wrote:II. Almost locked sets xy wing rule
-------------------------------

If A B C almost locked sets
x common to A,B
y restricted common to B,C
z restricted common to A,C

then a cell that can 'see' all the x candidates of both A and B can't be x.

Code: Select all
             c
             |
         z---C---y
         :       :
         :       :
         z       y
        /         \
   a---A           B---b
        \         /
         x...*...x
 
   * can be eliminated

+-------------------+-------------------+-------------------+
| 459   1    %479   |%78    3    %89    | 2    %58    6     |
| 59    569   679   | 278   289   4     |^35    1358  13    |
| 8     3     2     | 5     6     1     | 7     9     4     |
+-------------------+-------------------+-------------------+
|*139   7    *139   | 6     4     5     | 8     123   1239  |
| 6     28   *134   | 9     28    7     |^34    13    5     |
| 24    289   5     | 238   1     238   |^49    6     7     |
+-------------------+-------------------+-------------------+
| 7     4     39    | 23    259   6     | 1     235   8     |
| 12359 2569  1369  | 4     2589  2389  | 3569  7     239   |
| 2359  2569  8     | 1     7     239   | 3569  4     239   |
+-------------------+-------------------+-------------------+

A={R4C1,R4C3,R5C3}
B={R2C7,R5C7,R6C7}
C={R1C3,R1C4,R1C6,R1C8}
x=9
y=5
z=4

             7,8,9
             |
         4---C---5
         :       :
         :       :
         4       5
        /         \
 1,3---A           B---3,4
        \         /
         9...*...9

   *=r4c9#9 can be eliminated

bennys wrote:III. 2 ALS 2 restricted common rule
------------------------------
If A have degrees of freedom of 2
and B and C are ALS
with
x restricted common to A and B
y restricted common to A and C
and z common to A B C
then you cant have z in a cell that can see all the z candidates in A B C


Code: Select all
        y---C--c
        :   |
        :   z
        :   :   
        y   *...z
        ||  :   |
    a===A===z   B--b
        ||      |
        x.......x

   * can be eliminated

+----------------------+----------------------+----------------------+
| 139   *1379   2579   |$17     1247   8      | 1459   123459 6      |
|#16     4      257    | 3      1267   9      | 8      125    12     |
| 8     *1369   29     | 5      1246   46     | 7      12349  1234   |
+----------------------+----------------------+----------------------+
| 39     58     1      | 679    34679  34567  | 2      4679   478    |
| 7     *39     4      | 2      8      36     | 169    169    5      |
| 2      58     6      | 179    1479   457    | 3      479    478    |
+----------------------+----------------------+----------------------+
| 4      267    3      | 8      567    1      | 56     2567   9      |
| 169    1679   79     | 4      35679  2      | 156    8      137    |
| 5      12679  8      | 679    3679   367    | 146    123467 12347  |
+----------------------+----------------------+----------------------+
A={R1C2,R3C2,R5C}
B={R2C1}
C={R1C4}
X=6
Y=7
Z=1
        7---C
        :   |
        :   1
        :   :   
        7   *...1
        ||  :   |
   39===A===1   B
        ||      |
        6.......6

   *=r1c1#1 can be eliminated

(A can only accommodate up to two FALSE components, but *=TRUE
forces A7, A6, and A1 all to be FALSE. Thus * cannot be TRUE.)

bennys wrote:III. (generalized) n ALS n restricted common rule
-----------------------------
If A have degrees of freedom of n
and we have a collection S of n disjoint ALS (that are also disjoint with A)
each of the ALS has different restricted common with A
and z common to A and all the members of S
then we cant have z in a cell that can see all the z candidates in A and all the members of S.


You will have to imagine this one!

OK, to these I add:

IV. Almost-locked sets mutual exclusion rule
-------------------------------

If A and B are almost-locked sets
x,y restricted common to A,B

any z common to any other candidates of A OR B may be eliminated

and

any x or y common to BOTH A AND B may be eliminated

(note that when the number of candidates in both A and B = 2, then
this is the naked pair / X-Wing Rule)

Code: Select all
            *'
           . .
          .   .       
         X.....X
        /       \
*..z---A        B---z'...*
        \       /
         Y.....Y
          .   .       
           . .
            *'

     any * or *' may be eliminated

   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |   459 |     1 |   479 ||    78 |     3 |   A89 ||     2 |    58 |     6
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |    59 |   569 |   679 ||   278 |   289 |     4 ||    35 |  1358 |    13
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |     8 |     3 |     2 ||     5 |     6 |     1 ||     7 |     9 |     4
===========================||=======================||=======================
r4 |   139 |     7 |   139 ||     6 |     4 |     5 ||     8 |   123 |  1239
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |     6 |    28 |   134 ||     9 |    28 |     7 ||    34 |    13 |     5
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |    24 |   289 |     5 ||   238 |     1 |   238 ||    49 |     6 |     7
===========================||=======================||=======================
r7 |     7 |     4 |    39 ||   B23 |   259 |     6 ||     1 |   235 |     8
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 | 12359 |  2569 |  1369 ||     4 |  2589 | B2389 ||  3569 |     7 |   239
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |  2359 |  2569 |     8 ||     1 |     7 |  B239 ||  3569 |     4 |   239
-----------------------------------------------------------------------------

A=r1c6

B=r7c4 r8c6 r9c6
x=8
y=9
      9.....9   2...*
      |      \ /
      A       B
      |      / \
      8.....8   3
       .   .
        . .
         *'

     *=(r7c5#2 and r8c5#2) and *'=r6c6#8 may be eliminated

--edited 12/18/05


I/II. (generalized) Almost locked sets multiple weak link rule
-------------------------------

If A, B, C, D,...,N are are some number of almost locked sets
x restricted common to A and B
y restricted common to B and C
etc.

and

z common to N and A

then a cell that can 'see' all the z candidates of both A and N can't be z
Code: Select all

         x....(...x
        /         \
   a---A           B---b    )n=1,2,3,...
        \         /
         z...*.(..z

   * can be eliminated

This I would call "Full Almost-Locked Medusa".
To my knowledge, it has not been implemented.
YOUR NAME HERE. Sudoku Assistant implements this
only in the case where n=1 or no two almost-locked sets
with candidate number N>2 are adjacent.

IV. (generalized) Almost-locked sets mutual exclusion rule
-------------------------------

If A, B, C, D,...,N are are some number of almost locked sets
x restricted common to A and B
y restricted common to B and C
etc. including
z restricted common to N and A

any k common to any other candidates of any of these sets may be eliminated

Code: Select all
            *'
           . .
          .   .       
          X..(.X
         /      \
*...z---A        B---z'...*   ) n=1,2,3,...
        \       /
         Y...(.Y
          .   .       
           . .
            *'

   all  * and *' may be eliminated



V. Almost-locked X-generalization rule
-------------------------------

All statements regarding almost-locked sets in relation to
N cells with, collectively, N+m candidates, where m is the
"degree of freedom" of the set (usually m=1) can be recast
in terms of N rows/columns and N+m columns/rows,
in which case "x", "y", and "z" (above) refer to the column/row
number of a cell possible for a specific candidate k.

Basically everything said about "cells and candidates" in Sudoku can
also be said about "rows and columns" or "columns and rows".

Code: Select all
   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |   459 |     1 |   479 ||    78 |     3 |    89 ||     2 |    58 |     6
   |     A |       |       ||       |       |    fA ||       |       |     
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |    59 |   569 |   679 ||   278 |   289 |     4 ||    35 |  1358 |    13
   |     A |     A |       ||       |     f |       ||       |       |     
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |     8 |     3 |     2 ||     5 |     6 |     1 ||     7 |     9 |     4
   |       |       |       ||       |       |       ||       |       |     
===========================||=======================||=======================
r4 |   139 |     7 |   139 ||     6 |     4 |     5 ||     8 |   123 |  1239
   |     A |       |       ||       |       |       ||       |       |     A
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |     6 |    28 |   134 ||     9 |    28 |     7 ||    34 |    13 |     5
   |       |       |       ||       |       |       ||       |       |     
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |    24 |   289 |     5 ||   238 |     1 |   238 ||    49 |     6 |     7
   |       |     A |       ||       |       |       ||     A |       |     
===========================||=======================||=======================
r7 |     7 |     4 |    39 ||    23 |   259 |     6 ||     1 |   235 |     8
   |       |       |       ||       |       |       ||       |       |     
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 | 12359 |  2569 |  1369 ||     4 |  2589 |  2389 ||  3569 |     7 |   239
   |     A |     A |       ||       |       |     A ||     A |       |     A
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |  2359 |  2569 |     8 ||     1 |     7 |   239 ||  3569 |     4 |   239
   |     A |     A |       ||       |       |     A ||     A |       |     A
-----------------------------------------------------------------------------

A=(rows for candidate 9):{12489 2689 189 689 489}
B=Chain 6(f) -- r1c6#9 and r2c5#9
*=r1c3#9

r1c3 ISN'T 9: r1c3#9 is incompatibly weakly linked to 6(f) involving nodes r1c6#9 chain 6(F) and
r2c5#9 chain 6(f) via ALS 9-Col { r1c1 r2c1 r4c1 r8c1 r9c1 } { r2c2 r6c2 r8c2 r9c2 }
{ r1c6 r8c6 r9c6 } { r6c7 r8c7 r9c7 } { r4c9 r8c9 r9c9 }

Interpretation: If r1c6=9, then r1c3 cannot be 9, but if r1c6 is not 9, then r2c5 is 9,
which "locks" Set A to include row 1, which forces r1c3 to again NOT be 9.

Another way to look at it: r1c3#9 and r2c5#9 are mutually exclusive, since
the first is weakly linked to row 1 of the set, and the second is weakly
linked to row 2. But, in addition, there is a chain r2c5#9--r1c6#9, which
r1c3#9 is ALSO weakly linked to. This linkage demands that if
r1c3 is 9 then r2c5#9 is ALSO 9. There's the contradiction.

Whew! This is tricky! I'm a little uneasy about the way these are NOT
disjoint -- but I think that arises because they are two independent TYPES
of almost-locked sets -- single candidate row/column-based and block
cell-candidate based.
Code: Select all
            9(r1c3)
           . .
          .   .       
         r1    9(r1c6)
        /       \
r3689--A         B---z'...*
        \       /
         r2....9(r2c5)

Oddly enough, the fact that r1c6#9 is part of r1 has no bearing on the issue!
Last edited by Bob Hanson on Sun Dec 18, 2005 2:58 pm, edited 1 time in total.
Bob Hanson
 
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Re: Almost locked rules (for now)

Postby Bob Hanson » Sat Dec 17, 2005 12:06 am

ronk wrote:
bennys wrote:Almost locked rules
-------------------
Almost locked sets xy wing rule
-------------------------------

Although the following variation (of your xy wing rule example) doesn't fit your definition, it still appears like an xy-wing with two Almost Locked Sets.
Code: Select all
+-------------------+-------------------+-------------------+
| 459   1    %479   |%78    3    %89    | 2    %58    6     |
| 59    569   679   | 278   289   4     |^35    1358  13    |
| 8     3     2     | 5     6     1     | 7     9     4     |
+-------------------+-------------------+-------------------+
| 139   7     139   | 6     4     5     | 8     123   1239  |
| 6     28    134   | 9     28    7     |^34    13    5     |
| 24    289   5     | 238   1     238   | 49    6     7     |
+-------------------+-------------------+-------------------+
| 7     4     39    | 23    259   6     | 1     235   8     |
| 12359 2569  1369  | 4     2589  2389  | 3569  7     239   |
| 2359  2569  8     | 1     7     239   | 3569  4     239   |
+-------------------+-------------------+-------------------+

B={R2C7,R5C7}
C={R1C3,R1C4,R1C6,R1C8}
x=?
y=5
z=4

... letting us eliminate candidate 4 from r5c3.


Where does this pattern fit in your thoughts for Almost Locked Sets?

ronk, that's rule I.:

Code: Select all

         5.......5
        /         \
   3---A           B---7,8,9
        \         /
         4...*...4

              *=r5c3#4 can be eliminated
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Postby Myth Jellies » Sat Dec 17, 2005 7:32 am

Kudos to bennys, and all the others who have worked on ALS and similar schemes under different names.

Some nifty additions by Bob, too. However, consider this grid, which is "almost identical" to ones presented by bennys and ronk earlier...
Bob Hanson wrote:
Code: Select all
   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |   459 |     1 |   479 ||    78 |     3 |   A89 ||     2 |    58 |     6
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |    59 |   569 |   679 ||   278 |   289 |     4 ||    35 |  1358 |    13
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |     8 |     3 |     2 ||     5 |     6 |     1 ||     7 |     9 |     4
===========================||=======================||=======================
r4 |   139 |     7 |   139 ||     6 |     4 |     5 ||     8 |   123 |  1239
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |     6 |    28 |   134 ||     9 |    28 |     7 ||    34 |    13 |     5
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |    24 |   289 |     5 ||   238 |     1 |   238 ||    49 |     6 |     7
===========================||=======================||=======================
r7 |     7 |     4 |    39 ||   B23 |   259 |    69 ||     1 |   235 |     8
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 | 12359 |  2569 |  1369 ||     4 |  2589 | B2389 ||  3569 |     7 |   239
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |  2359 |  2569 |     8 ||     1 |     7 |  B239 ||  3569 |     4 |   239
-----------------------------------------------------------------------------

A=r1c6

B=r7c4 r8c6 r9c6
x=8
y=9
      8.....8   2...*
      |      \ /
      A       B
      |      / \
      9.....9   3
       .   .
        . .
         *'

     *=(r7c5#2 and r8c5#2) and *'=r7c6#9 may be eliminated



...inquiring minds want to know, Bob, why you added and then deleted a false candidate 9 in cell r7c6 when you had a legitimate 8 in r6c6 already that could be removed using the exact same rule:)

Keep up the good work.
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Postby ronk » Sat Dec 17, 2005 1:08 pm

Bob Hanson wrote:
Code: Select all
A=r1c6

B=r7c4 r8c6 r9c6
x=8
y=9
      8.....8   2...*
      |      \ /
      A       B
      |      / \
      9.....9   3
       .   .
        . .
         *'

     *=(r7c5#2 and r8c5#2) and *'=r7c6#9 may be eliminated

Those candidates are a "Sue De Coq", a configuration she termed a Two-Sector Disjoint Subset this October. In the terms of this thread, it has one set with two degrees of freedom (almost almost locked set), and two sets with one degree of freedom (almost locked sets).

Using bennys' rule style and your diagraming style, that configuration can be represented as follows:

Code: Select all
If A has two degrees of freedom (almost almost locked set),
and B and C have one degree of freedom (almost locked sets),
and w and x are restricted common to A and B,
and y and z are restricted common to A and C,
then w and x may be eliminated from cells of unit J containing A and B
then y and z may be eliminated from cells of unit K containing A and C

If A contains a fifth candidate a ... different from w, x, y, and z ... then candidate a may be eliminated from cells of both units J and K

               w*               y*   
             .   .            .    . 
            .      .         .      .
           w........w       y........y
           |         \\   //         |
           B            A            C       
           |         // ||\\         |
           x........x   ||  z........z
            .      .    ||   .      .
             .    .     a     .    . 
               x*       .       z*   
                        .             
                        a*           

------------------------+------------------------+-----------------------
   459       1     479  |    78       3      89C |     2      58       6 
    59     569     679  |   278     289       4  |    35    1358      13 
     8       3       2  |     5       6       1  |     7       9       4 
------------------------+------------------------+-----------------------
   139       7     139  |     6       4       5  |     8     123    1239 
     6      28     134  |     9      28       7  |    34      13       5 
    24     289       5  |   238       1     238  |    49       6       7 
------------------------+------------------------+-----------------------
     7       4      39  |    23B    259      69  |     1     235       8 
 12359    2569    1369  |     4    2589    2389A |  3569       7     239 
  2359    2569       8  |     1       7     239A |  3569       4     239 
------------------------+------------------------+-----------------------

A = {r8c6,r9c6} = {2389}
B = {r7c4} = {23}
C = {r1c6} = {89}
    {wx} = {23}
    {yz} = {89}

Then eliminations are r7c5#2, r8c5#2, r6c6#8, and r7c6#9
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Postby ronk » Sun Dec 18, 2005 4:38 pm

It would likely be difficult to find a better example for my previous illustration than Sue De Coq's exercise #1. This one has a 5th candidate in the Almost Almost Locked Set.
Code: Select all
 *--------------------------------------------------------------------*
 | 567    8      47     | 46     2      1      | 379   A49     3459   |
 | 1      459    24     | 7      89     3      | 6     A248    45     |
 | 267    469    3      | 468    689    5      | 278   A1248  B14     |
 |----------------------+----------------------+----------------------|
 | 8      3      6      | 9      4      7      | 1      5      2      |
 | 9      7      1      | 258    58     28     | 4      3      6      |
 | 4      2      5      | 3      1      6      | 89    C89     7      |
 |----------------------+----------------------+----------------------|
 | 2367   46     2478   | 1      678    289    | 5      2469   349    |
 | 23567  156    9      | 256    567    4      | 23     126    8      |
 | 256    1456   248    | 2568   3      289    | 29     7      149    |
 *--------------------------------------------------------------------*

 A = {r1c8,r2c8,r3c8} = {12489}
 B = {r3c9} = {14}
 C = {r6c8} = {89}
     {wx} = {14}
     {yz} = {89}
      {a} = {2}

Then eliminations are r1c9#4, r2c9#4, r7c8#9, r3c7#2, r7c8#2, and r8c8#2
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Postby Bob Hanson » Sun Dec 18, 2005 6:52 pm

Myth Jellies wrote:...inquiring minds want to know, Bob, why you added and then deleted a false candidate 9 in cell r7c6 when you had a legitimate 8 in r6c6 already that could be removed using the exact same rule:)


I didn't see it! Thanks. Super -- then there's no need to improvise. I will edit that.
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Postby Bob Hanson » Sun Dec 18, 2005 7:08 pm

ronk wrote:Then eliminations are r7c5#2, r8c5#2, r6c6#8, and r7c6#9


Yes, even better!

I think the point is going to be that these aren't really "rules" so
much as specific configurations of a general idea -- that these almost-
locked-sets or (almost)n-locked sets can be combined in any number of
ways to generate interesting "manifestations" that could be easily
findable.

And, it is fascinating that there may be several different combinations of
sets that result in similar or the same eliminations.

Excellent!
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Postby Bob Hanson » Sun Dec 18, 2005 7:13 pm

By the way, the next step is to consider strong chains of almost-locked sets.
I've got the idea written down, just looking for examples:
Code: Select all
     x
     |
  w--A--y--B--z
           |
           etc.

The idea here is that A and B are NOT disjoint -- they share some number
of candidates. This is the Medusa idea when the almost-locked sets are
restricted to 2-values, but it generalizes in some interesting ways.

If anyone wants to work on this, feel free!
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Postby ronk » Wed Dec 21, 2005 5:19 pm

Bob Hanson wrote:By the way, the next step is to consider strong chains of almost-locked sets.

I was obviously incorrect in thinking that is in your present implementation of Almost Locked Sets. Since it's not, would you please explain (most likely again) what you *did* implement?

TIA, Ron
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Postby ronk » Wed Dec 21, 2005 5:31 pm

ronk wrote:It would likely be difficult to find a better example for my previous illustration than Sue De Coq's exercise #1.

I ran across puzzle #121 of the top870 with an Almost Almost Locked Set that eliminated 10 candidates. I posted that and 21 other top870 puzzles that can be advanced (beyond the capability of basic techniques) with this method on the Two-Sector Disjoint Subset thread.
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