May 20, 2006 - Colored Box-Box Interactions

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Myth Jellies
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May 20, 2006 - Colored Box-Box Interactions

Post by Myth Jellies »

Code: Select all

*-----------*
 |..5|2.9|3..|
 |...|..7|2..|
 |.2.|.4.|.19|
 |---+---+---|
 |...|...|59.|
 |...|8.1|...|
 |.36|...|...|
 |---+---+---|
 |56.|.2.|.7.|
 |..4|7..|...|
 |..9|6.3|8..|
 *-----------*
Basic methods take us to...

Code: Select all

*-----------------------------------------------------------------------------*
 | 14678   1478    5       | 2       168     9       | 3       468     4678    |
 | 13468   9       138     | 13      1368    7       | 2       4568    4568    |
 | 3678    2       378     | 35      4       568     | 67      1       9       |
 |-------------------------+-------------------------+-------------------------|
 | 1478    1478    1278    | 34      367     246     | 5       9       123678  |
 | 479     5       27      | 8       3679    1       | 467     236     2367    |
 | 14789   3       6       | 459     579     245     | 147     28      1278    |
 |-------------------------+-------------------------+-------------------------|
 | 5       6       138     | 149     2       48      | 19      7       13      |
 | 1238    18      4       | 7       1589    58      | 169     2356    12356   |
 | 127     17      9       | 6       15      3       | 8       245     1245    |
 *-----------------------------------------------------------------------------*
If we focus on the eights, we have...

Code: Select all

 *------------------------------------*
 | 8   8   . | .   8   . | .   8   8 |
 | 8   .   8 | .   8   . | .   8   8 |
 | 8   .   8 | .   .   8 | .   .   . |
 |-----------+-----------+-----------|
 | 8   8   8 | .   .   . | .   .   8 |
 | .   .   . | 8   .   . | .   .   . |
 | 8   .   . | .   .   . | .   8   8 |
 |-----------+-----------+-----------|
 | .   .   8 | .   .   8 | .   .   . |
 | 8   8   . | .   8   8 | .   .   . |
 | .   .   . | .   .   . | 8   .   . |
 *------------------------------------*
If we look at boxes 2 & 3, we note that except for the 8 in r3c6, we would have a box-box reduction that would eliminate all 8's in box 1 in rows 1 & 2. Let's try a little grouped coloring in box 2 that isolates r3c6...

Code: Select all

 *------------------------------------*
 | 8   8   . | .  A8   . | .   8   8 |
 | 8   .   8 | .  A8   . | .   8   8 |
 | 8   .   8 | .   .  a8 | .   .   . |
 |-----------+-----------+-----------|
 | 8   8   8 | .   .   . | .   .   8 |
 | .   .   . | 8   .   . | .   .   . |
 | 8   .   . | .   .   . | .   8   8 |
 |-----------+-----------+-----------|
 | .   .   8 | .   .   8 | .   .   . |
 | 8   8   . | .   8   8 | .   .   . |
 | .   .   . | .   .   . | 8   .   . |
 *------------------------------------*
We can also color the conjugates in row 7, giving us...

Code: Select all

 *------------------------------------*
 | 8   8   . | .  A8   . | .   8   8 |
 | 8   .  -8 | .  A8   . | .   8   8 |
 | 8   .   8 | .   .  a8 | .   .   . |
 |-----------+-----------+-----------|
 | 8   8   8 | .   .   . | .   .   8 |
 | .   .   . | 8   .   . | .   .   . |
 | 8   .   . | .   .   . | .   8   8 |
 |-----------+-----------+-----------|
 | .   .  B8 | .   .  b8 | .   .   . |
 | 8   8   . | .   8   8 | .   .   . |
 | .   .   . | .   .   . | 8   .   . |
 *------------------------------------*
Note that we have a simple multi-coloring case here, where a and b exclude each other, therefore A or B or both A & B must be true. If A is true, the box-box interaction wipes out the 8s in r12c123, and if B is true the remaining 8's in column 3 are killed. In either case, we can eliminate the 8 in r2c3...

Code: Select all

*-----------------------------------------------------------------------------*
 | 14678   1478    5       | 2       168     9       | 3       468     4678    |
 | 13468   9      -13      | 13      1368    7       | 2       4568    4568    |
 | 3678    2       378     | 35      4       568     | 67      1       9       |
 |-------------------------+-------------------------+-------------------------|
 | 1478    1478    1278    | 34      367     246     | 5       9       123678  |
 | 479     5       27      | 8       3679    1       | 467     236     2367    |
 | 14789   3       6       | 459     579     245     | 147     28      1278    |
 |-------------------------+-------------------------+-------------------------|
 | 5       6       138     | 149     2       48      | 19      7       13      |
 | 1238    18      4       | 7       1589    58      | 169     2356    12356   |
 | 127     17      9       | 6       15      3       | 8       245     1245    |
 *-----------------------------------------------------------------------------*
This exposes a naked pair which takes us to...

Code: Select all

 *-----------------------------------------------------------*
 | 167   4     5     | 2     16    9     | 3     68    678   |
 | 68    9     13    | 13    68    7     | 2    *45   *45    |
 | 678   2     37    | 35    4     568   | 67    1     9     |
 |-------------------+-------------------+-------------------|
 | 17    8     127   | 4     3     26    | 5     9     267   |
 | 49    5     27    | 8     679   1     | 467   3     267   |
 | 49    3     6     | 59    579   25    | 147   28    1278  |
 |-------------------+-------------------+-------------------|
 | 5     6     8     | 19    2     4     | 19    7     3     |
 | 3     1     4     | 7     589   58    | 69    26    256   |
 | 2     7     9     | 6     15    3     | 8    *45   *45+1  |
 *-----------------------------------------------------------*
At this stage you can either find some xy-wings, or take advantage of, say the uniqueness rectangle in r29c89 to crack the puzzle.
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Myth Jellies
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Posts: 42
Joined: Tue Apr 04, 2006 7:07 am

Post by Myth Jellies »

Code: Select all

*------------------------------------* 
 | 8   8   . | .   8   . | .   8   8 | 
 | 8   .  -8 | .   8   . | .   8   8 | 
 |#8   .  *8 | .   .  *8 | .   .   . | 
 |-----------+-----------+-----------| 
 | 8   8   8 | .   .   . | .   .   8 | 
 | .   .   . | 8   .   . | .   .   . | 
 | 8   .   . | .   .   . | .   8   8 | 
 |-----------+-----------+-----------| 
 | .   .  *8 | .   .  *8 | .   .   . | 
 | 8   8   . | .   8   8 | .   .   . | 
 | .   .   . | .   .   . | 8   .   . | 
 *------------------------------------*
Of course it is also a finned (fin = r3c1) r37c36 x-wing as well. A colored box-box interaction is just another way of finding these.
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Ruud
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Posts: 601
Joined: Fri Dec 30, 2005 10:21 pm

Post by Ruud »

Hi MJ,

The finned thingy was intentional. 8-)

Wasn't the method you describe called 'grouped colouring'?

cheers, Ruud
“If the human brain were so simple that we could understand it, we would be so simple that we couldn't.” - Emerson M Pugh
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Myth Jellies
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Posts: 42
Joined: Tue Apr 04, 2006 7:07 am

Post by Myth Jellies »

Hmm, well after re-examining the problem it seems that "normal" grouped coloring would find the solution as well. This would be an example of normal group coloring...

Code: Select all

 *------------------------------------* 
 | 8   8   . | .   8   . | .   8   8 | 
 | 8   .  -8 | .   8   . | .   8   8 | 
 |A8   .  A8 | .   .  a8 | .   .   . | 
 |-----------+-----------+-----------| 
 | 8   8   8 | .   .   . | .   .   8 | 
 | .   .   . | 8   .   . | .   .   . | 
 | 8   .   . | .   .   . | .   8   8 | 
 |-----------+-----------+-----------| 
 | .   .  B8 | .   .  b8 | .   .   . | 
 | 8   8   . | .   8   8 | .   .   . | 
 | .   .   . | .   .   . | 8   .   . | 
 *------------------------------------*
...A's or B's must be true, thus the 8 which sees all A's and B's can be removed. So perhaps coloring the box-box interaction is unnecessary after all. C'est la vie.
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David Bryant
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Posts: 86
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Location: Denver, Colorado
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Contact David Bryant

I think it's Nishio

Post by David Bryant »

Hi, MJ! Hi, Ruud!

I felt pretty good about finding this pattern, but I saw it as a classic "Nishio".

Code: Select all

 *-----------------------------------*
 | 8   8   . | .   8*  . | .   8   8 |
 | 8   .   8*| .   8   . | .   8   8 |
 | 8   .   8 | .   .   8 | .   .   . |
 |-----------+-----------+-----------|
 | 8   8   8 | .   .   . | .   .   8 |
 | .   .   . | 8   .   . | .   .   . |
 | 8   .   . | .   .   . | .   8   8 |
 |-----------+-----------+-----------|
 | .   .   8 | .   .   8*| .   .   . |
 | 8   8   . | .   8   8 | .   .   . |
 | .   .   . | .   .   . | 8   .   . |
 *-----------------------------------*
r2c3 = 8 ==> "8" in row 8 is in box 7 ==> r7c6 = 8 ==> r1c5 = 8

Now there's no way to place an "8" in box 3, so r2c3 <> 8. dcb
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