Empty space needs help

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Empty space needs help

Postby JPF » Mon May 29, 2006 11:28 am

In trying to find a Sudoku with a maximum 'emtpy space' ( thread proposed by tso), my generator produced this puzzle :

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


I was wondering how it can be solved.
Thanks.

JPF
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Postby MCC » Mon May 29, 2006 12:10 pm

For a start, the only number that can go in r1c9 is 7.

In column 9, r2c9 is the only place for a 3.

You can then place a 3 in r3c3.

The only place in box 8 for a 3 is r9c4.

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



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Postby RW » Mon May 29, 2006 12:56 pm

I think the problem is here, after applying basic technique:

Code: Select all
 *--------------------------------------------------------------------*
 | 12     268    268    | 9      5      3      | 16     4      7      |
 | 15     9      7      | 4      6      2      | 158    158    3      |
 | 4      56     3      | 17     8      17     | 56     2      9      |
 |----------------------+----------------------+----------------------|
 | 8      2456   2456   | 567    2349   479    | 23459  5679   1      |
 | 9      3      12456  | 1567   24     8      | 245    567    246    |
 | 7      12456  12456  | 156    2349   149    | 23459  569    8      |
 |----------------------+----------------------+----------------------|
 | 6      7      12     | 8      49     49     | 12     3      5      |
 | 3      1458   14589  | 2      7      56     | 1489   1689   46     |
 | 25     2458   24589  | 3      1      56     | 7      689    246    |
 *--------------------------------------------------------------------*


Coloring eliminates '2' from r5c3, but that doesn't help. Possible move that solves the puzzle:

Code: Select all
If r1c1=1 => r9c1=2 => r89c9=46
             r1c7=6 => r3c7=5 => r5c57=24
   => r5c9=empty cell => r1c1<>1


RW
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Postby Mike Barker » Mon May 29, 2006 1:30 pm

Here's another solution (not all may be necessary):
    Hidden Single: r9c4 => r9c4=3,r46c4<>3
    Hidden Single: r2c6 => r2c6=2,r2c123<>2,r456c6<>2
    Hidden Single: r2c4 => r2c4=4,r4567c4<>4
    Hidden Single: r1c4 => r1c4=9,r1c23<>9,r467c4<>9
    Naked Single: r7c4 => r7c3567<>8,r35c4<>8,r8c56<>8,r9c6<>8
    Hidden Single: r2c9 => r2c9=3,r23c7<>3,r2c3<>3
    Hidden Single: r3c3 => r3c3=3
    Naked Row Pair: r7c56 => r7c3<>49,r7c7<>49
    Naked Block Pair: r7c56 => r8c5<>49,r8c6<>49,r9c6<>49
    Naked Single: r8c5 => r8c6<>7,r345c5<>7
    Naked Single: r3c5 => r3c267<>8,r5c5<>8
    Hidden Single: r5c6 => r5c6=8
    Naked Row Pair: r3c46 => r3c27<>1
    Naked Column Pair: r89c6 => r4c6<>56,r6c6<>56
    Locked Column box/box: r68c2|r5678c3 => r12c2<>1,r12c3<>1
    Column Swordfish Fillet-o-Fish: r19c1|r1469c2|r59c9 => r5c3<>2
    UVWXYZ-wing: r5c789|r46c8, r5c5 => r5c3<>4
    ALS xz-mer with A=1 cells: r5c5-24-r5c789|r46c8 => r46c7<>5,r46c7<>6,r6c7<>9,r89c8<>9,r4c7<>9
    Hidden Single: r8c7 => r8c7=9,r8c23<>9
    Locked Column line/box: r89c8 => r2c8<>8
    Locked Column line/box: r89c9 => r5c9<>4
    WXYZ-wing: r2c8|r3c7|r1c9, r2c1 => r2c7<>5
    ALS xz-rule with A=1 cells: r8c9-4-r9c169 => r9c8<>6
    Naked Single: r9c8 => r9c23<>8,r8c8<>8
    Nice Loop: r1c7=1=r1c1=2=r9c1-2-r7c3=2=r7c7~2~r1c7 => r7c7<>1

    The Solution is completed with singles
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Posts: 458
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Postby Sped » Mon May 29, 2006 2:13 pm

Here is my ugly solution, which includes one questionable UR step. I hope a UR expert will tell me if I did it right.

Code: Select all
 *-----------*
 |...|.53|.4.|
 |...|.6.|...|
 |4..|...|.29|
 |---+---+---|
 |8..|...|..1|
 |93.|...|...|
 |7..|...|..8|
 |---+---+---|
 |67.|...|.35|
 |3..|2..|...|
 |...|.1.|7..|
 *-----------*


After basic moves, including removing the 2 in r5c3 on colors, we get:

Code: Select all
 *-----------------------------------------------------------------------------*
 | 12      268     2678    | 9       5       3       | 168     4       67      |
 | 15      589     5789    | 4       6       2       | 158     1578    3       |
 | 4       56      3       | 17      8       17      | 56      2       9       |
 |-------------------------+-------------------------+-------------------------|
 | 8       2456    2456    | 567     2349*   479*    | 234569  5679    1       |
 | 9       3       1456    | 1567    24      8       | 2456    567     2467    |
 | 7       12456   12456   | 156     2349*   149*    | 234569  569     8       |
 |-------------------------+-------------------------+-------------------------|
 | 6       7       12      | 8       49*     49*     | 12      3       5       |
 | 3       14589   14589   | 2       7       56      | 14689   1689    46      |
 | 25      24589   24589   | 3       1       56      | 7       689     246     |
 *-----------------------------------------------------------------------------*


I'm not sure if this is valid or not, but the URs in r7c56 and r6c56/r4c56 tell me that there has to be a 4 or 9 in box 5 outside of r6c56/r4c56. The only possibility is to set r5c5=4.

From there we advance to:

Code: Select all
 *--------------------------------------------------------------------*
 | 12     268    2678   | 9      5      3      | 168    4      67     |
 | 15     589    5789   | 4      6      2      | 158    157    3      |
 | 4      56     3      | 17     8      17     | 56     2      9      |
 |----------------------+----------------------+----------------------|
 | 8      2456   2456   | 567    23     79     | 34     5679   1      |
 | 9      3      156    | 1567   4      8      | 256    567    267    |
 | 7      12456  12456  | 156    23     19     | 34     569    8      |
 |----------------------+----------------------+----------------------|
 | 6      7      12     | 8      9      4      | 12     3      5      |
 | 3      1458   1458   | 2      7      56     | 9      168    46     |
 | 25     24589  24589  | 3      1      56     | 7      68     246    |
 *--------------------------------------------------------------------*


There is a simple nice loop that sets r1c9=7:

[r1c9]=7=[r2c8]=1=[r8c8]-1-[r7c7]-2-[r5c7]=2=[r5c9]=7=[r1c9] => r1c9=7

Simple steps later:

Code: Select all
 *--------------------------------------------------------------------*
 | 12     268    268    | 9      5      3      | 16     4      7      |
 | 15     9      7      | 4      6      2      | 8      15     3      |
 | 4      56     3      | 17     8      17     | 56     2      9      |
 |----------------------+----------------------+----------------------|
 | 8      2456   2456   | 567    23     79     | 34     5679   1      |
 | 9      3      156    | 1567   4      8      | 25     567    26     |
 | 7      12456  12456  | 156    23     19     | 34     569    8      |
 |----------------------+----------------------+----------------------|
 | 6      7      12     | 8      9      4      | 12     3      5      |
 | 3      1458   1458   | 2      7      56     | 9      168    46     |
 | 25     2458   9      | 3      1      56     | 7      68     246    |
 *--------------------------------------------------------------------*


From there the XY chain:

5-(r5c7)-2-(r7c7)-1-(r7c3)-2-(r9c1)-5-(r2c1)-1-(r2c8)-5
eliminates the 5s in r3c7, r4c8, r5c8, and r6c8, which solves the puzzle.

In nice loop notation:

[r3c7]-5-[r5c7]-2-[r7c7]-1-[r7c3]-2-[r9c1]-5-[r2c1]-1-[r2c8]-5-[r3c7] => r3c7<>5
[r456c8]-5-[r5c7]-2-[r7c7]-1-[r7c3]-2-[r9c1]-5-[r2c1]-1-[r2c8]-5-[r456c8] => r456c8<>5
Sped
 
Posts: 126
Joined: 26 March 2006

Postby RW » Mon May 29, 2006 2:50 pm

sped wrote:
Code: Select all
 *-----------------------------------------------------------------------------*
 | 12      268     2678    | 9       5       3       | 168     4       67      |
 | 15      589     5789    | 4       6       2       | 158     1578    3       |
 | 4       56      3       | 17      8       17      | 56      2       9       |
 |-------------------------+-------------------------+-------------------------|
 | 8       2456    2456    | 567     2349*   479*    | 234569  5679    1       |
 | 9       3       1456    | 1567    24      8       | 2456    567     2467    |
 | 7       12456   12456   | 156     2349*   149*    | 234569  569     8       |
 |-------------------------+-------------------------+-------------------------|
 | 6       7       12      | 8       49*     49*     | 12      3       5       |
 | 3       14589   14589   | 2       7       56      | 14689   1689    46      |
 | 25      24589   24589   | 3       1       56      | 7       689     246     |
 *-----------------------------------------------------------------------------*


I'm not sure if this is valid or not, but the URs in r7c56 and r6c56/r4c56 tell me that there has to be a 4 or 9 in box 5 outside of r6c56/r4c56. The only possibility is to set r5c5=4.


I'm afraid this exclusion is not valid. The four cell square can be used with uniqueness reductions Only if it functions as a corner - the pattern continues both vertically and horizontally. This is a deadly pattern:

Code: Select all
ab  .  ab|
---------+--
ab  .  ab|ab
.   .  . |.
ab  .  ab|ab


This is not:
Code: Select all
ab  .  ab
---------
ab  .  ab
.   .  .
ab  .  ab


possible unique solution to the second:
Code: Select all
a   .  b
--------
.   .  a
.   .  .
b   .  .


An example of a valid reduction with four candidate cells within a block can be found here.

RW
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Postby JPF » Mon May 29, 2006 5:16 pm

Thanks for the quick answer.

JPF
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Postby daj95376 » Mon May 29, 2006 6:02 pm

One more answer, if it's okay. (my chain notation probably isn't accurate.)

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

r9c4    =  3     Hidden Single
r2c9    =  3     Hidden Single
r3c3    =  3     Hidden Single
r2c6    =  2     Hidden Single
r2c4    =  4     Hidden Single
r1c4    =  9     Hidden Single
r7c4    =  8     Naked  Single
r7      -  49    Naked  Pair
    b8  -  49    Naked  Pair
r8c5    =  7     Naked  Single
r3c5    =  8     Naked  Single
r5c6    =  8     Hidden Single
r3      -  17    Naked  Pair
  c6    -  56    Naked  Pair
  c1    -  1     Locked Candidate (2)
 
*-----------------------------------------------------------------------------*
| 12      268     2678    | 9       5       3       | 168     4       67      |
| 15      589     5789    | 4       6       2       | 158     1578    3       |
| 4       56      3       | 17      8       17      | 56      2       9       |
|-------------------------+-------------------------+-------------------------|
| 8       2456    2456    | 567     2349    479     | 234569  5679    1       |
| 9       3       12456   | 1567    24      8       | 2456    567     2467    |
| 7       12456   12456   | 156     2349    149     | 234569  569     8       |
|-------------------------+-------------------------+-------------------------|
| 6       7       12      | 8       49      49      | 12      3       5       |
| 3       14589   14589   | 2       7       56      | 14689   1689    46      |
| 25      24589   24589   | 3       1       56      | 7       689     246     |
*-----------------------------------------------------------------------------*

[r1c1]=1=[r2c1]=5=[r3c2]=6=[r3c7]=5
                 =[r9c1]=2=[r89c9]=46=[r1c9]=7=[r1c7]=6
                                     =[r5c9]=2=[r5c57]=4 => [r1c1]<>1

r1c1    =  2     [r1c1]<>1

Naked/Hidden Singles complete puzzle!
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Postby Sped » Mon May 29, 2006 6:14 pm

RW wrote:I'm afraid this exclusion is not valid. The four cell square can be used with uniqueness reductions Only if it functions as a corner - the pattern continues both vertically and horizontally.

Thanks for straightening me out.

Without setting r5c5=4 I can re-work the loop that sets r1c9=7, but that XY chain to finish it is no longer there.

Back to the drawing board.
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Posts: 126
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