The New Sudoku Players' Forum

[ghfick]

I came across this thread a while back. It is noted in this thread that 9 empty groups might be the maximum and 4 empty blocks might be the maximum. Has this matter been explored more recently on this forum? Any definitive arguments in any case?
Gordon

[JPF]

A "group" may be a box, a row or a column.
It's impossible to have more than 3 empty rows.
same thing for columns.
Then, to get 10 empty groups, one need at least 4 empty boxes.
More than 4 empty boxes is not possible for valid* pattern; see here
Finally,10 empty groups would require 4 empty boxes, 3 empty rows, 3 empty columns.
From here the only pattern that could be valid is this one or an equivalent:

Code: Select all

+---+---+---+
|...|...|...|
|...|...|.xx|
|...|...|.xx|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+


I leave it to Serg to remind us why that pattern is not valid!

*valid pattern : pattern containing at least a valid puzzle

JPF

[Serg]

Hi, JPF!

Code: Select all

+---+---+---+
|...|...|...|
|...|...|.xx|
|...|...|.xx|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+
|...|...|...|
|.xx|.xx|...|
|.xx|.xx|...|
+---+---+---+


I leave it to Serg to remind us why that pattern is not valid!

This pattern is not valid because it is subset of maximal invalid pattern P141

Code: Select all

        P141
+-----+-----+-----+
|. . .|. . x|. . .|
|. x x|. . x|. . .|
|. x x|. . x|. . .|
+-----+-----+-----+
|. . .|x x x|x x x|
|. . .|x x x|x x x|
|. . .|x x x|x x x|
+-----+-----+-----+
|. . .|x x x|x x x|
|. . .|x x x|x x x|
|. . .|x x x|x x x|
+-----+-----+-----+

from 40 maximal patterns list.

Serg

[JPF]

To close the topic:

9 empty groups

* with 4 empty boxes, here are all the possible ed patterns:


Code: Select all

      A               B               C               D
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|   |...|...|.xx|   |...|...|...|
|...|...|xxx|   |...|...|.xx|   |...|...|.xx|   |...|...|.xx|
|...|...|xxx|   |...|...|.xx|   |...|...|.xx|   |...|...|.xx|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|   |...|...|...|   |...|...|...|
|.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
|.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|   |...|...|...|   |.xx|.xx|...|
|.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
|.xx|.xx|...|   |.xx|xxx|...|   |.xx|.xx|...|   |.xx|.xx|...|
+---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+



I'm sure Serg will find good reasons to prove that these patterns are invalid.

* with 3 empty boxes, there are only two ed patterns:


Code: Select all

      E               F
+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|...|.12|.34|   |...|...|.xx|
|...|.56|.78|   |...|...|.xx|
+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|.74|...|.21|   |.xx|.xx|...|
|.83|...|.47|   |.xx|.xx|...|
+---+---+---+   +---+---+---+
|...|...|...|   |...|...|...|
|.68|.21|...|   |.xx|.xx|.xx|
|.52|.64|...|   |.xx|.xx|.xx|
+---+---+---+   +---+---+---+


The first one E, valid, has been found by tso.
I doubt that F is valid...

JPF

[ghfick]

Hi JPF,
This is very interesting. Can you elaborate on why there can be no more than three empty rows or three empty columns?
Gordon

[mith]

More than three empty rows implies two empty rows in the same band, and any solution would have a second where those two rows are swapped. (Likewise for columns.)

[JPF]

About his puzzle,

The puzzle has a unique solution and is not difficult.

Here is the highest rating I could get for the same equivalent pattern:

Code: Select all

+---+---+---+
|...|...|...|
|...|.12|.34|
|...|.53|.16|
+---+---+---+
|...|...|...|
|.45|...|.67|
|.87|...|.23|
+---+---+---+
|...|...|...|
|.32|.65|...|
|.58|.47|...|
+---+---+---+    SER=8.9


JPF

[ghfick]

A beautiful puzzle and very tough too!
Fascinating properties.